Arts, Faculty of
Music, School of
DSpace
UBCV
Sawatzky, Grant Michael
Doctor of Philosophy - PhD
University of British Columbia
In the influential 1989 study Phrase Rhythm in Tonal Music, William Rothstein developed a method for analysis of eighteenth and nineteenth-century European common-practice music that describes how melodic gestalts and their supporting harmonic progressions relate to the metrical alternation of strong and weak beats, especially hypermeter (metrical structure beyond the notated bar). The interaction of these two kinds of temporality constitutes the “phrase rhythm” of the music. Interacting temporal structures exist in much other music, but application of Rothstein’s method to a broader corpus is hindered when the music is without the formulaic harmonic progressions that help define gestalts, or does not feature regular pulsation at certain levels of its meter. To overcome these limitations, I propose a way to generalize Rothstein’s insights. I reconceive phrase rhythm as arising from the “grouping-complex”: a network of interrelationships among different ways of partitioning the same span of musical time. These ways include event affiliation, gestalt segmentation, rhythmic/motivic parallelism, and partitioning by phenomenal accent, among others. Theorists, I show, recognize these more general partitions, but have not investigated how they can interact independently of alignment to well-formed meter or involvement in tonal prolongation. This approach makes it possible to identify and describe distinctive phrase-rhythmic effects (e.g., overlap, elision, and phrase expansion) in music that does not feature the melodic, harmonic and metrical schemata of the common-practice corpus. I demonstrate these effects in passages by Debussy, Mussorgsky, and Bartók. Moreover, I avoid the “conformational” analytical frame of much writing about phrase structure and meter, wherein a passage is described in terms of its conformity to some a priori model that is taken to be “normative.” The “generative” analytical frame I favour allows for the description of emergent phrase-rhythmic structures of considerable nuance and complexity, whether or not “tonal,” and with or without meter. Analyses of music by Stravinsky, Luther, Schubert, Bach, and Bartók demonstrate. This also makes it possible to conceive of the music in terms of grouping-complexes—dubbed “multivalent”—wherein different affordances of the same passage may be appreciated as mutually informing rather than as competing interpretations (argued via analyses of Schumann and Mozart examples).
PHRASE RHYTHM AS GROUPING COMPLEX: HEARING STRUCTURE IN “IRREGULAR” MUSIC by GRANT MICHAEL SAWATZKY B.Mus., University of Saskatchewan, 2005 M.A., University of British Columbia, 2013 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Music - Theory) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2024 © Grant Michael Sawatzky, 2024 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Phrase Rhythm as Grouping Complex: Hearing Structure in “Irregular” Music submitted by Grant Michael Sawatzky in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Music - Theory Examining Committee: Dr. John Roeder, Professor Emeritus, Music Theory, School of Music, UBC Supervisor Dr. Alan Dodson, Professor, Music Theory, Department of Music, Mount Allison University Supervisory Committee Member Dr. Leigh VanHandel, Associate Professor, Chair of Music Theory Division, School of Music, UBC Supervisory Committee Member Dr. James T. Enns, Professor and Distinguished University Scholar, Department of Psychology, UBC University Examiner Dr. T. Patrick Carrabré, Professor, Composition, School of Music, UBC University Examiner iii Abstract In the influential 1989 study Phrase Rhythm in Tonal Music, William Rothstein developed a method for analysis of eighteenth and nineteenth-century European common-practice music that describes how melodic gestalts and their supporting harmonic progressions relate to the metrical alternation of strong and weak beats, especially hypermeter (metrical structure beyond the notated bar). The interaction of these two kinds of temporality constitutes the “phrase rhythm” of the music. Interacting temporal structures exist in much other music, but application of Rothstein’s method to a broader corpus is hindered when the music is without the formulaic harmonic progressions that help define gestalts, or does not feature regular pulsation at certain levels of its meter. To overcome these limitations, I propose a way to generalize Rothstein’s insights. I reconceive phrase rhythm as arising from the “grouping-complex”: a network of interrelationships among different ways of partitioning the same span of musical time. These ways include event affiliation, gestalt segmentation, rhythmic/motivic parallelism, and partitioning by phenomenal accent, among others. Theorists, I show, recognize these more general partitions, but have not investigated how they can interact independently of alignment to well-formed meter or involvement in tonal prolongation. This approach makes it possible to identify and describe distinctive phrase-rhythmic effects (e.g., overlap, elision, and phrase expansion) in music that does not feature the melodic, harmonic and metrical schemata of the common-practice corpus. I demonstrate these effects in passages by Debussy, Mussorgsky, and Bartók. iv Moreover, I avoid the “conformational” analytical frame of much writing about phrase structure and meter, wherein a passage is described in terms of its conformity to some a priori model that is taken to be “normative.” The “generative” analytical frame I favour allows for the description of emergent phrase-rhythmic structures of considerable nuance and complexity, whether or not “tonal,” and with or without meter. Analyses of music by Stravinsky, Luther, Schubert, Bach, and Bartók demonstrate. This also makes it possible to conceive of the music in terms of grouping-complexes—dubbed “multivalent”—wherein different affordances of the same passage may be appreciated as mutually informing rather than as competing interpretations (argued via analyses of Schumann and Mozart examples). v Lay Summary Musicians commonly refer to medium-length musical passages as “phrases.” The term “phrase rhythm” refers to the particular emphases within and proportionality among consecutive phrases. It is a musical characteristic analogous to how distinct poetic forms (sonnet, limerick, haiku) involve different stress patterns, distribution of rhyming words, and/or syllables per line. Existing methods for phrase-rhythm analysis compare phrase lengths and harmonic progressions (akin to line/stanza lengths and end-rhymes) to predetermined accentual patterns as a kind of conceptual “grid” called musical meter. That method works well but only where harmonic progressions and regular accentual patterns are found. In order to describe the phrase rhythm that emerges in more “irregular” music—analogous to “free-verse” in poetry—I develop a more general method for describing relationships between different ways of “grouping” the same passage. Interrelationship of these groups allows for description of phrase rhythm without the expectation of conformity to regular patterning. vi Preface This dissertation is original, unpublished, independent work by the author, Grant Michael Sawatzky. vii Table of Contents Abstract ...................................................................................................................................... iii Lay Summary .............................................................................................................................. v Preface ....................................................................................................................................... vi Table of Contents ..................................................................................................................... vii List of Examples ......................................................................................................................... ix Acknowledgements .................................................................................................................. xiii Chapter 1: Phrase Rhythm and “Irregularity” ........................................................................... 1 1.1 Introduction: Phrase Rhythm after Rothstein, and Lerdahl and Jackendoff ..................... 1 1.2 Phrase-rhythmic Effects in “Regular” Common-practice Examples ................................. 5 1.3 Phrase Structure and “Irregularity” ................................................................................. 39 1.4 Phrase-rhythmic Effects without (hyper)Meter (and without “basic phrase”) .................. 56 1.5 Summary ......................................................................................................................... 75 Chapter 2: Contrasting Conceptions of Meter in the Literature: Generative or Conformational, Inter- or Intra-opus ................................................................................................................... 77 2.1 Theories that Accommodate Irregularity into a Meter-as-grid Outlook .......................... 77 2.2 Precedents for Phrase Rhythm without Meter in the Literature .................................... 110 2.3 Meter as Grouping ......................................................................................................... 135 2.4 Summary ....................................................................................................................... 167 Chapter 3: Phrase-rhythmic Particularity in Bach’s Sarabande Double BWV 1002 (with special emphasis on “harmonic image”) ............................................................................................. 169 3.1 Introduction ................................................................................................................... 169 3.2 Grouping as Determinant of Generative Metric Structure(s) ......................................... 171 3.3 Emergent Metric and Phrase-rhythmic Structure in Bach’s Sarabande Double BWV 1002 ....................................................................................................................................... 184 3.4 Conclusion ..................................................................................................................... 210 Chapter 4: Multivalent Phrase Structure in Two Tonal Examples ......................................... 214 4.1 Introduction: Ambiguity vs. Multivalence ..................................................................... 214 4.2 The Grouping Complex and a case for Multivalent Phrase-rhythmic Structure in Schumann’s Volksliedchen (op. 68 no. 9) ........................................................................... 218 viii 4.3 The Grouping Complex and a case for Multivalent Phrase-rhythmic Structure in Mozart’s Menuetto I (K.282) ............................................................................................ 230 4.4 Conclusion (or, what do harmonic images need to be?) ................................................. 252 Chapter 5: The Grouping Complex and Generative Phrase Rhythm in a passage from Béla Bartók’s Violin Sonata no. 2 ................................................................................................... 254 5.1 Introduction ................................................................................................................... 254 5.2 Phrase 1 Grouping Complex ......................................................................................... 256 5.3 Phrase 2 as Expanded Restatement of Phrase 1’s Piano Introduction ........................... 279 5.4 Phrase 3 as Multivalent Restatement of Phrase 1’s Violin Material (with expansion) .... 286 5.5 Summary ....................................................................................................................... 296 Chapter 6: Metaphorical Reflections, Prospective Directions ................................................. 297 6.1 Reflections on Measuring Musics (and metaphors thereof) ............................................ 297 6.2 ‘And thus do we of wisdom and of reach, / With windlasses and with assays of bias, / By indirections find directions out.’ ..................................................................................... 306 Bibliography ............................................................................................................................ 311 Appendix A: IFUNC Tables for ‘Nun freut euch…’ (Chapter 2) ................................................ 324 Appendix B: Extra Bach Diagrams (Chapter 3) ...................................................................... 331 Appendix C: Extra Mozart Diagrams (Chapter 4) .................................................................. 338 Appendix D: Grouping Complex Analysis – Emergent Phrase Rhythm in Bartók’s Violin Sonata no. 2 mvt. II, Fourth Phrase (Chapter 5) ..................................................................... 345 ix List of Examples Example 1-1: GTTM’s Grouping Overlap as Counter to Grouping Well-Formedness ............... 7 Example 1-2: Grouping Overlap in Clementi op. 33 no. 1, mm. 11–15 ................................... 11 Example 1-3: Elision in Haydn Symphony 104 mvt. 1, mm. 13–19 (adapted from GTTM ex. 3.27) ................................................................................................................... 14 Example 1-4: Elision in Beethoven op. 2 no. 2 (blue hyperbeat counts from PRiTM p. 53) ..... 18 Example 1-5: Composed-out Deceleration in Brahms Scherzo op. 4 mm. 101–118 .................. 24 Example 1-6: Mozart’s K.331 Menuetto mm. 1–10 as Eight-measure Basic Phrase with Internal Expansion ........................................................................................................... 32 Example 1-7: Elision in Beethoven op. 2 no. 2, again (now including mm. 1–20) .................... 45 Example 1-8: When is Rothstein’s Hypermeter Extrinsic to the Basic Phrase? (an elision thought experiment) ........................................................................................... 50 Example 1-9: Tenney and Polansky’s Temporal Gestalt Segmentation Analysis of Debussy’s Syrinx mm. 1–5 ................................................................................................... 58 Example 1-10: Temporal Gestalt Segmentation with Grouping by Parallelisms and Event-affiliation in Debussy’s Syrinx, mm. 1–5 .............................................................. 61 Example 1-11: Elision in the ‘Promenade’ from Mussorgsky’s Pictures at an Exhibition ............... 66 Example 1-12: Phrase Expansion in Bartók’s Etude op. 18 no. 3 ............................................. 71 Example 2-1: Luther’s Nun freut euch, lieben Christen g’mein [Melodie C, WitK 1533, Bl.24v–25r] ........................................................................................................................... 84 Example 2-2: ‘Nun freut euch…’ Three-level Dot Diagram with and without Metric Reinterpretation ................................................................................................. 87 Example 2-3: Horlacher’s Metric Reinterpretation Symbols .................................................... 87 Example 2-4: ‘Nun freut euch…’ Four-level Dot Diagram assuming “4/1” Metric Structure ..... 97 Example 2-5: ‘Nun freut euch…’ Four-level Dot Diagram assuming “6/1” Metric Structure ..... 97 Example 2-6: ‘Nun freut euch…’ Phrase Rhythm as Interaction between Melodic Grouping and “5/1” Structure of P-series ............................................................................... 107 Example 2-7: Schachter, GTTM, and Dahlhaus on Schubert’s Valse Sentimentale op. 50 no. 13 ......................................................................................................................... 123 Example 2-8: Grouping Complex in Stravinsky’s Three Pieces for Clarinet Solo mvt. I ............... 144 x Example 2-9: Conventional Metric Patterns as illustrated by Dot Diagram and by Hasty’s Symbols ............................................................................................................ 153 Example 2-10: Hasty’s C-Major Scale Analysis (with added “harmonic images”) .................. 155 Example 2-11: Alternate C-Major Scale Analysis (with added “harmonic images”) ............... 159 Example 2-12: Grouping and Meter in London’s Invented Example ..................................... 162 Example 2-13: London’s Invented Melody with Harmonic Images, and Hasty’s Symbols illustrating London’s Intended Metrical Interpretation .................................... 165 Example 2-14: London’s Invented Melody with Harmonic Images, and Hasty’s Symbols illustrating Lester’s Metric Interpretations ....................................................... 166 Example 3-1: Grouping According to Gestalt Segmentation (or accent of image shift) and Durational-accent-as-group-beginning in an Invented Rhythm from Benjamin’s Fig. 7 ................................................................................................................ 173 Example 3-2: Four Possible Metrical Interpretations generated from Grouping Boundaries, Accents, and Parallelisms in an Invented Rhythm from Benjamin’s Fig. 7 ...... 176 Example 3-3: Benjamin’s ‘Figure 1’ (1984, 356–7), Riemann (b) and Westergaard (a) on “Natural” Metric Interpretation for Long-short Pattern .................................. 178 Example 3-4: Three Rhythmic Interpretations of Duration Series <3, 1, 2, 3, 1, 2> assessed according to Longuet-Higgins and Lee’s Syncopation ..................................... 180 Example 3-5: Three Rhythmic Interpretations of Duration Series <3, 1, 2, 3, 1, 2> as inferred from different Aspects of the same Grouping Complex ................................... 182 Example 3-6: Harmonic Image and Contour (stepwise segment) Diagram, Sarabande Double BWV 1002, mm. 1–8 ....................................................................................... 189 Example 3-7: Stepwise String Terminus Events Correlated to Pulse Classes Modulo-n ......... 198 Example 3-8: Sarabande Double BWV 1002 (complete score) annotated with Harmonic Image Groups and Stepwise String Termini as Group-initiating Events .................... 201 Example 3-9: Phrase-by-Phrase Tallies of Stepwise Terminus Event Distribution by Pulse Class ......................................................................................................................... 205 Example 3-10: Harmonic Image and Contour (stepwise segment) Analysis of Sarabande Double BWV 1002, mm. 25–32 ................................................................................... 206 Example 4-1: Robert Schumann’s Album für die Jugend op. 68 no. 9 Volksliedchen, mm. 1–8 ..... 219 xi Example 4-2: Co-present Grouping Structures in Robert Schumann’s Volksliedchen (Album für die Jugend op. 68 no. 9 mm. 1–8) ............................................................................ 222 Example 4-3: Mozart’s K.282 Menuetto I – P-series, R-series, and A-series Groupings ........... 231 Example 4-4: Mozart’s K.282 Menuetto I – Pulse Streams emphasizing an Anti-periodic Aspect of P-series ......................................................................................................... 235 Example 4-5: Mozart’s K.282 Menuetto I – Pulse Stream emphasizing a Periodic Aspect of A-series ................................................................................................................. 235 Example 4-6: Grouping Complex (P-series, R-series, A-series) as Phrase Structure in Mozart’s K.282 Menuetto I, mm.1–12 .............................................................................. 237 Example 4-7: Mozart’s K.282 Menuetto I – P-series, R-Series, and A-series with Harmonic Images (H-series) .............................................................................................. 242 Example 4-8: “3/2” aspect of Mozart’s K.282 Menuetto I as Reference to K.282 Movement I ......................................................................................................................... 244 Example 4-9: Mozart’s K.282 Menuetto I – P/R/A/H-series with S-series Accents and Highlighted Non-chord Tones ......................................................................... 249 Example 4-10: Mozart’s K.282 Menuetto I A-series and S-series Accents creating Syncopation into Cadence .................................................................................................... 251 Example 5-1: Score Excerpt – Bartók’s Violin Sonata no. 2 mvt. II, mm. 1–43 ..................... 258 Example 5-2: Basic Grouping – Bartók’s Violin Sonata no. 2 mvt. II, mm. 1–5 (vocables) .... 259 Example 5-3: Basic Grouping – Bartók’s Violin Sonata no. 2 mvt. II, mm. 5–12 (vocables) .. 261 Example 5-4: Basic Grouping – Bartók’s Violin Sonata no. 2 mvt. II, mm. 1–12 (vocables) .. 264 Example 5-5: Grouping Complex Analysis – Emergent Phrase Rhythm in Bartók’s Violin Sonata no. 2 mvt. II, First Phrase ..................................................................... 270 Example 5-6: Grouping Complex Analysis – Emergent Phrase Rhythm in Bartók’s Violin Sonata no. 2 mvt. II, Second Phrase ................................................................ 280 Example 5-7: Grouping Complex Analysis – Emergent Phrase Rhythm in Bartók’s Violin Sonata no. 2 mvt. II, Third Phrase .................................................................. 289 Example 6-1: Alexander Calder – ‘Vertical Foliage’ (1941) .................................................... 300 Example 6-2: Bressan’s (2006) “Christmas Wall-of-blocks” Illusion ....................................... 305 Example 6-3: Demeter and Metanira ..................................................................................... 310 xii Example A-1: ‘Nun freut euch…’ with numbered time-points for half-note pulse ...................... 325 Example A-2: Detail of IFUNC table for ‘Nun freut euch…’ (timepoints 0–21) ......................... 325 Example A-3: Detail of IFUNC table for ‘Nun freut euch…’ (timepoints 0–41) ......................... 327 Example A-4: Complete IFUNC table for ‘Nun freut euch…’ (timepoints 0–73) ....................... 330 Example B-1: Harmonic Image and Contour (stepwise segment) Diagram, Sarabande Double BWV 1002, mm. 1–8 ....................................................................................... 332 Example B-2: Harmonic Image and Contour (stepwise segment) Diagram, Sarabande Double BWV 1002, mm. 9–16 ..................................................................................... 333 Example B-3: Harmonic Image and Contour (stepwise segment) Diagram, Sarabande Double BWV 1002, mm. 17–24 ................................................................................... 334 Example B-4: Harmonic Image and Contour (stepwise segment) Diagram, Sarabande Double BWV 1002, mm. 25–32 ................................................................................... 335 Example B-5: Stepwise String Terminus Events Correlated to Pulse Class Modulo-n ............ 337 Example C-1: Heinrich Schenker’s Annotated Score of Mozart K.282 Menuetto I ................. 339 Example C-2: Fundamental Bass and Rhythmic/Motivic Grouping Analysis Informed by Schenker’s Annotated Score ............................................................................ 339 Example C-3: Schoenberg’s Analyses of Mozart’s Piano Quartet K.478, mvt. I, mm. 57–61 342 Example C-4: Anti-parallelism in “Dissonance” String Quartet K.465, Menuetto mm. 17–20 344 Example D-1: Grouping Complex Analysis – Emergent Phrase Rhythm in Bartók’s Violin Sonata no. 2 mvt. II, Fourth Phrase ................................................................. 349 xiii Acknowledgements First of all, I acknowledge and whole-heartedly thank Liz, who, years ago, encouraged me to pursue my academic interest in music and music theory, and who, all this time later, remains my closest companion and partner in life. She continually believed in me, even during periods when I was losing my way. I simply would not have completed this project without her. I am also deeply grateful to my family and friends whose continued support made it possible to keep going during the stages where progress felt especially challenging. Working and learning under the direction of John Roeder has been an invaluable experience. This project benefited greatly from his analytical insights, theoretical and methodological open-mindedness, and keen editorial skill. As an advisor he offered patient willingness to hear out even the most scrambled-up versions of my thoughts, and provided gentle guidance all throughout my slow march toward completion of this dissertation project. The clarity, depth, and generous spirit of his music-analytical writing and thought will remain a model toward which I intend to strive. Sincere thanks to my committee members Alan Dodson and Leigh VanHandel; and also, to Richard Kurth who, while a prospective committee member, provided a much-appreciated sounding board for the exploratory big-picture thinking during the early stages of this project. I am especially appreciative of Alan Dodson’s close reading. The deeply considered commentaries that he provided for each drafted chapter certainly rank among the most valuable parts of my graduate study. I am thankful to all of my teachers; especially to Gregory Marion who first introduced me to what music theory can be; to Gyula Csapó and Zsuzsanna Takáts whose influence was pivotal in the re-opening of my musical imagination; and to Michael Tenzer who inspires with xiv a literary and cultural frame of reference that is wonderfully wide-ranging. Thank you to William Benjamin whose music-theoretical and philosophical thought is an extraordinarily deep well of knowledge: I count myself as very fortunate to have been introduced to some of it firsthand. Much of this dissertation was written in the Firehall Branch of the Vancouver Public Library—I am thankful for the quiet company and welcoming space provided by its kind staff and friendly patrons. Finally, I express warmest appreciation to my graduate-student friends and colleagues. Even after our paths have diverged, I continue to benefit from working out music-theoretical thoughts in the uniquely beneficial form of imagined dialogues, debates, and discussions like those we were once privileged to share in the Old Auditorium: heart-felt thanks to Antares Boyle, Hei-Yeung (John) Lai, Chantal Lemire, Nancy Murphy, and James Palmer. xv there was always another and another and another shall we match the like knuckle-players with bones or stones or counters, the fatality of numbers? the first? the last? and the many, many in-between H.D., Helen in Egypt, book 5.1 (1961) 1 Phrase Rhythm and “Irregularity” Appeals to nature or to cosmology have been used by many theorists over the centuries to find a basis for then-current musical conventions. In all cases, changing styles have revealed the inadequacy of the appeal. The normalcy of duple or four-measure groupings also fails to stand up to scrutiny. … [T]here is no reason for assuming the primacy of one type of grouping over another. Indeed the assertion that nonduple divisions arise as adjustments in a duple framework often leads to absurd arguments. Joel Lester, The Rhythms of Tonal Music (1986) 1.1 Introduction: Phrase Rhythm after Rothstein, and Lerdahl and Jackendoff William Rothstein’s Phrase Rhythm in Tonal Music (1989) theorizes what had previously been a somewhat nebulous aspect of musical structure: how consecutive segments of music, with differing lengths and patterns of accentuation, are strung together akin to the way syllables, words, and sentences are strung into paragraphs to create pacing and flow in artful prose. Rothstein calls this aspect of music “phrase rhythm,” which he defines as the interaction between grouping structure—by which he means melodic segmentation and certain aspects of melodic and harmonic closure (i.e., what musicians informally call “phrasing”1)—and a 1 While Rothstein does offer his own take on what, precisely, constitutes a musical phrase (1989, 5–7), the principal focus of his theory has more to do with attending to the rhythms of and between phrases than with specific details of phrase construction. Schmalfeldt (2019) offers a thoughtful and deeply referenced overview of conceptions of the musical phrase. She points to various sources who have described the “time-oriented” aspect of defining the phrase and phrase group, often drawing upon rhetoric and semiotics as an analogue: whether Kofi Agawu’s “beginning–middle–end paradigm” (Agawu 1991, 51ff.); or Aristotle’s definition of the rhetorical periodos as “an utterance that has its own beginning and end, and an easily comprehensible magnitude” (Caleb Mutch’s translation of Aristotle Rhetoric, III.9, 1409a34-b1 (Mutch 2015, 17)). Mutch’s history of the cadence (ibid.), Hasty’s “Phrase Formation in Post-Tonal Music” (1984), and Howland’s “Formal Structures in Post-Tonal Music” (2015) are examples of theories concerned with phrase, but which are not theories of phrase rhythm in Rothstein’s sense, since they attend primarily to the factors that define phrase segments (i.e., “construction of sentences” in my prose metaphor) more than the proportions and relationships among phrase segments (i.e., the pacing and flow of sentences and paragraphs, in the prose metaphor). 2 co-present hypermetric structure—by which he means the more-or-less regular structure of nested pulsations, especially at and beyond the level of the notated barline. By reading any irregular groupings according to a more regular “basic phrase” (and its associated hypermeter) that he takes to be prototypical (Rothstein 1989, 64), this conception of phrase rhythm provides an elegant way to describe such common musical effects as grouping overlap, elision, and expansion, as I will outline below. The scope of Phrase Rhythm in Tonal Music (hereafter PRiTM) is limited to Western common-practice art music, and more specifically to examples where hypermetric structure is more-or-less regular and where a listener trained in the Classical style may infer a phrase prototype to which the music refers. Musical passages featuring highly irregular rhythmic organization or idiosyncratic harmonic language fall outside its purview. A practical reason for this is to ground his theory of phrase rhythm in the same sort of metric and grouping structure as described in Lerdahl and Jackendoff’s influential Generative Theory of Tonal Music ([1983] 1996, hereafter GTTM), which likewise restricts its focus to the tonal common-practice musics that, its authors claim, are perceived with reference to an underlying regularity. In this dissertation, I will argue that a theory of phrase rhythm—one in which phrase overlap, elision, and expansion are conceivable phrase-rhythmic effects—does need not be constrained in this way if one adopts a view of meter that is wider than GTTM's. Such a view has been offered by several theorists who do not think of grouping and meter as necessarily existing in an oppositional relationship, and who do not consider regular pulsation (or “entrainability”) to be an essential 3 property of metric organization.2 Section 1.2 of this chapter includes examples of musical passages whose phrase rhythm can easily be described in terms of Rothstein’s mode of phrase-rhythm analysis. With those section 1.2 examples (from Clementi, Haydn, Beethoven, Brahms, and Mozart) I demonstrate the essential features of Rothstein’s approach while also defining the specific phrase-rhythmic effects I am interested in. My discussion of these preliminary examples also previews some aspects of the “alternate” perspective of meter I favour, before Chapter 2 provides a more detailed theoretical treatment of the “meter-as-grouping” alternative. Ultimately, I wish to describe “phrase rhythm” in music that neither establishes regular “well-formed” (hyper)meter3 nor straightforwardly alludes to regular schematized “basic 2 I sought a wider definition of “meter” in the work of theorists such as Berry ([1976] 1987), Benjamin (1984), and Hasty (1997), to be discussed in detail in Chapter 2. In brief, Berry explicitly takes meter to be a special subcategory of grouping. Benjamin’s theory of meter resembles Berry’s, though he uses the term “grouping” slightly differently to explain the various the ways music can establish equivalences among time points. In Hasty’s Meter as Rhythm, grouping is discussed primarily when positioning his theory relative to other well-known and influential theories. Generally, Hasty does not describe his projective theory of meter in terms of “grouping,” yet neither does he attribute to meter “the sort of independence or autonomy that would place it in opposition to or in conflict with grouping or the articulation of phrases and phrase constituents” (Hasty 1997, 122). I feel justified in affiliating his theory with Berry’s (and Benjamin’s) because of the way his “beginning,” “continuation,” “anacrusis,” and “hiatus” are characterizations of moments that involve nested decisions about event affiliation—a kind of “grouping”—within the complexity of musical flow. Hasty describes these affiliations and orientations in terms of “projective durations.” Berry’s “impulses” and Benjamin’s “groupings” are similar, in that they represent a way of describing temporal qualia and nested interrelations. 3 Scott Murphy uses the “rather awkward construction ‘(hyper)meter’…to embrace all perceptible periodicities slower than the tactus” (2012, 92 fn[8]). I am using the same term here, for similar reasons. My usage of “(hyper)meter” refers to meter-like structures beyond the level suggested by the notated barlines, which GTTM would call part of “metrical structure” and which PRiTM calls hypermeter. The parenthetical is a reminder of differences between the way PRiTM and GTTM employ these related concepts. Lerdahl and Jackendoff make no categorical or conceptual distinction between the parts of metrical structure that occur above and below the notated barlines: it is all “meter.” Rothstein, on the 4 phrases” or any normative phrase prototypes. There are certain examples in Rothstein’s book where his theory is applied to relatively “irregular” examples from within the Western common-practice corpus. Offering commentary on two of them (Beethoven op. 2 no. 2, and Mozart K.515), my section 1.3 highlights some ontological issues that arise from PRiTM’s handling of “irregularity.” That section will form the basis for an investigation of the relationship between theories of phrase and schema (be they metrical templates or prototypical phrase structures), which I develop further in Chapter 2. I then conclude the first chapter by offering brief analyses of three examples (Debussy, Mussorgsky, and Bartók) that are considerably less metrical (in the conventional sense) and also less straightforwardly tonal than Rothstein’s. While meter and grouping are less regular (i.e., less periodic, more idiosyncratic) in those examples, I demonstrate that they nevertheless feature phrase-rhythmic effects very similar to the ones PRiTM theorizes for repertoire that is more regular/prototypical.4 In this way, the first chapter sets the stage for analyzing phrase rhythm in other “irregular” musics by dispensing with the requirement of (hyper)meter (in the strict conventional sense) for description of phrase rhythm and phrase-rhythmic effects. other hand, uses hypermeter to refer to meter-like structures beyond the notated barline, which represent a distinct aspect of metrical structure that is informed partly by its hierarchic periodicity and regular alternation of accent (as with meter proper), but also having to do with the proportion and normativity of a referred-to phrase prototype. Working out the ontological status of Rothstein’s hypermeter is the subject of a latter section of this chapter (1.3.2) and part of the next (2.2.2). 4 As the examples in section 1.2 illustrate, harmonic considerations are quite important to Rothstein’s phrase rhythm. Pitch and harmonic factors remain an important factor in most of the analytical examples in later chapters, even where the harmonies are not functional/syntactic in the usual Western common-practice sense. 5 1.2 Phrase-rhythmic Effects in “Regular” Common-practice Examples In this section I will draw out the essence of three particular phrase-rhythmic effects that Rothstein’s theory5 can describe in music of the Western common-practice: namely grouping overlap, elision, and (internal) expansion. The description of these effects in Western common-practice music is of course nothing new, but I work through these effects in a such a way as to de-emphasize their degree of conformity to presumed “regularity” (whether metrical or phrase-prototypical), and emphasize, instead, the way in which these effects arise because of the particulars of multiple concurrent groupings that emerge from the piece itself. The groupings I am referring to are not only the groupings that arise from melodic segmentation (which is most of what “grouping” means in GTTM and PRiTM) but also the groupings that arise because of other factors such as harmony, texture, and rhythmic/motivic parallelisms. 5 Strictly speaking, PRiTM is perhaps described better as mode of analysis (based on existing theories of rhythm and harmony from Koch, Schenker, Schachter, Lerdahl and Jackendoff, et al.) more than as a theory per se. I, however, refer to PRiTM as a “method of analysis” and as “[Rothstein’s] theory” interchangeably and do not mean to make something of this distinction. In PriTM, Rothstein incorporates sophisticated distillation, re-presentation and synthesis of historical and contemporary theoretical sources into his deeply considered analytical demonstrations. (The “Theoretical Contexts” and “Rhythm and Meter” sections in his recent Musical Language of Italian Opera, 1813-1859 (2023, 25–63; 87–141) are structured similarly—surveying theoretical sources both historical and contemporary, and seamlessly adapting and applying them to highly sensitive analyses.) The significant portion of my chapters 1 and 2 given to a critique of PRiTM should not be read as faulting PRiTM for some lack of high-concept theoretical rigour (note, Rothstein specifies that its intended readership includes music historians, undergraduate music students, and “thoughtful performers and listeners” (PRiTM, iix)). Rather, the critique that I offer is a larger one that uses Rothstein’s PRiTM as a kind of foil. I glimpse existing theories of meter and phrase through the aperture of PRiTM’s practical approach to phrase-rhythm analysis in order to expose certain biases, inconsistencies, limitations and assumptions of existing theories within its frame. If the reader will indulge an extended metaphor, I hope to process these negatives and get them developed in the analytical demonstrations of my chapters 3–5. 6 1.2.1 Grouping Overlap PRiTM follows GTTM in recognizing a special irregular grouping condition that is encountered relatively often in tonal music. A “grouping overlap” occurs when an event (or series of events) is understood as belonging to two different groups at the same level of structure (and which may be likewise overlapped on additional structural levels). For Lerdahl and Jackendoff this constitutes “an important class of counterexamples to the grouping well-formedness rules.” According to their fourth grouping well-formedness rule “if a group G1 contains part of a group G2, it must contain all of G2” (GTTM, 55).6 Yet, listeners will often encounter situations where there is “a discrepancy between the predictions of [grouping well-formedness] … and certain actual musical intuitions according to which musical groups overlap” (ibidem). Lerdahl and Jackendoff account for this paradox by differentiating between well-formed “underlying” grouping structure (where overlaps are non-syntactic) and “surface” grouping where overlaps can and do occur, as illustrated in my Example 1-1, based on their formal definitions (GTTM, 60–61). In general, Rothstein’s PRiTM accepts the premises of GTTM’s conceptions of grouping and metrical structure. However, in PRiTM, Rothstein reserves the use of the term “grouping overlap” for those situations where the moment of overlap does not cause (or otherwise coincide with) a disruption to the underlying regularity of the hypermeter.7 When a grouping 6 Other theorists of grouping (e.g., Tenney and Polansky 1980, 218) insist on a strict definition where groups are discrete, that is, any given event may belong to only one group at a given level. 7 Here we see a hint of how Rothstein’s take on grouping and meter differs slightly from GTTM’s: in PRiTM meter and grouping are often mutually informing (likely a trace of Schachter’s influence). Comparatively, GTTM’s grouping and metrical structure are conceptually separate, and with the former largely understood relative to the latter. “Condition iii” of their “metrical deletion (second 7 overlap does cause (or coincide with) a disruption to the regularity of the hypermeter PRiTM describes the overlap as an instance of “metrical reinterpretation.” Those special overlaps, commonly called “elisions” (or “grouping overlaps with metrical deletions” in GTTM’s terms) will be discussed in the following section of this chapter. Example 1-1: GTTM’s Grouping Overlap as Counter to Grouping Well-Formedness The authors of GTTM and PRiTM treat overlaps as self-evident and tend to provide very little explanation of how they are brought about, trusting that the familiarity of the examples will mean designations of this or that overlap are uncontroversial and intuitive for their intended readership. But because I wish to recognize and formally describe overlap in pieces much less straightforward than the examples in GTTM and PRiTM, I will provide here a detailed narration of precisely which aspects of the music give rise to grouping overlap in the particular example of an excerpt from Clementi’s piano sonata op. 33, no. 1. (Chapter 2 will lay out these criteria more rigorously, grounding them in recent theories of rhythm and meter, and subsequent chapters provide further analytical demonstrations.) In short, my view is that version)” is a notable exception. There, in a reversal of their usual conceptual ordering, a grouping transform relates two conceptually distinct spans to a single surface time-span with implications for metric structure (GTTM, 104). eʹ g2e1 e2 e3 e4 e1 e2e3e4g2g1"well-formed underlying group"g1 ⊂ e1 ∧ e2g2 ⊂ e3 ∧ e4e2 ≠ e3g1 ⊂ e1 ∧ e2g2 ⊂ e3 ∧ e4eʹ = e2 = e3 "surface grouping overlap"g1 - 8 grouping overlaps arise from the relationship between different types of co-present groupings which are simultaneously suggested by different aspects of the same passage of music: for instance, by gestalt segmentation of melody, by accent (of various kinds) as group-initiating events, and according to motivic parallelisms. GTTM and PRiTM refer to such aspects obliquely, and other theorists deal with the conditions that give rise to these kinds of effects more directly.8 My emphasis on “co-presence” of different grouping structures runs counter to the way that grouping is described in GTTM and PRiTM, where final-state grouping structure is one-dimensional. Throughout this dissertation I emphasize how grouping possibilities (informed by different aspects of the music’s events) are co-present and mutually informing as opposed to existing as antithetical alternatives. I will describe the grouping overlap in the Clementi excerpt in terms of the interrelation between melodic grouping (gestalt segmentation), grouping according to durational accents and rhythmic parallelisms in the harmonic rhythm (suggested by the left-hand accompaniment), and strong parallelism within the melody, as illustrated in Example 1-2, below. The first small melodic group, labeled G1.1 in Example 1-2, consists of a <do, re> ascent, lightly ornamented with stepwise neighbours. My use of “open-brackets” for these group beginnings is meant to show the way that accents or gestalt segmentation criteria will propose certain events as the first of a prospective group, without necessarily defining the span and end-point for the group. In the case of G1.1, it is marked as a group beginning as the initial event in the melodic layer 8 Hanninen, for example, has written about these same three factors in the context of her three segmentation criteria: contextual/associative, structural/theoretical, and sonic/psychoacoustic (Hanninen 2012, 19–62). A more recent example of this sort of approach is Jason Yust’s Organized Time: Rhythm, Tonality, and Form, where he makes an argument "for the independence of musical parameters and the importance of structural disjunction, structural conflict between parameters” (Yust 2018, 1). 9 (gestalt segmentation), and also because its dotted-eighth note value receives emphasis as a durational accent relative to the uniform eighth-note pulse of the preceding accompaniment (prospective partitioning of the span according to duration accent). The exclusively stepwise motion during G1.1 is interrupted by a leap onto sol. As the largest melodic leap so far, the contextually large gap proposes a point of gestalt segmentation for the melody, so that a new melodic group beginning G1.2 commences from the high E♮5 (beat 2, m. 12).9 Here, the open bracket G1.2 flags the contextually large melodic interval as a prospective point of gestalt segmentation. Partitioning the span by accent nominate the same point of segmentation on the basis of E♮5 being both a contour peak, and an agogic accent. Stepwise motion is resumed during group G1.2 as the melody descends from the upper register sol toward the tonic pitch A♮4 at the downbeat of measure 13. The pitch A♮4 on the downbeat of m. 13 is the longest duration heard thus far, and so for (temporal) gestalt segmentation reasons that event may be construed as the final event in a group of relatively proximate attacks, G1. While I hinted that “relative long” events (open brackets G1.1 and G1.2) may suggest agogic accent as a proposed group-initiating event (when partitioning by accent), readers will note that I draw the opposite conclusion with respect to the final event of G1 (on the basis of grouping by gestalt segmentation)—more on that shortly. For now, we may appreciate the emergence of G1 as a self-contained thematic group on the basis of its 9 Whether the sixteenth-note E♮4 belongs to group G1.1 (i.e., because the largest melodic interval immediately follows it) or to G1.2 (i.e., since it can be heard as if it is a grace-note that belongs to the subsequent E♮5) is immaterial to the argument I am making about grouping overlap in this example. A more detailed treatment of strict algorithmic methods for melodic segmentation can be found in section 1.4.1 of this chapter, where Tenney and Polansky’s (1980) method of gestalt segmentation is contrasted with other kinds of grouping by event-affiliation. 10 departure-from/return-to A♮4.10 (The closed bracket that ends higher-level group G1 implies also the ending of its constituent subgroup G1.2). The left-hand accompaniment layer, meanwhile, outlines a <I, viio, I> progression in the rhythm h q q to constitute Clementi’s m. 11 (my “group H1”); answered in the next measure by the progression <V, (P), V> in the same harmonic rhythm (my “group H2”). The group beginnings of H1 and H2 are marked by the agogic accent (relative longs) of their initial harmonies. Considering the two layers together, we may then come to characterize G1.1 and G1.2 as “anacrustic” groups when considering them in relation to their concurrent H-series groups. The G-series beginnings are not aligned with an H-series group beginning, but initiate a span that extends across a subsequent H-series grouping boundary. But this “alignment” condition, alone, does not confer the “anacrustic” status to the G-series beginnings; it also requires that the context of their interrelation invites us to measure G-series groups using H-series groups. (The reversed situation would contextualize H-series beginnings as “late” relative to G-series, rather than G-series beginnings as “early” relative to H-series). The thematic group G1 affiliates G1.1 and G1.2 as constituent subgroups on the basis of their loose parallelism (anacrustic groups of similar duration and stepwise makeup), and also because of their balanced departure-return gesture: <do-re, sol-do>. In mm. 13–14 the accompaniment layer exactly repeats its material from mm. 11–12, thereby defining the larger two-measure groups T1.1 and 10 An analytical approach that is informed by gestalt segmentation in combination with melodic expectations and other perceptions of patterned musical “gesture” can be found in Steve Larson’s Musical Forces (2012, especially chapters 2–5). This can be seen especially in the way that Larson layers his “motion” and “force” onto more basic (or “atomist”) parts of musical structure such as those informed by from Eugene Narmour’s implication-realization model (developed in The Analysis and Cognition of Melodic Complexity (1992)). 11 T1.2. At the same time, a very strong melodic parallelism emerges between the melodic material of groups G1 (mm. 11ff.) and G2 (mm. 13ff.), except that in recognizing this melodic parallelism the pitch-event A♮4 (downbeat m. 13) is construed as both the final event of melodic group G1 and also the initiating event of melodic group G2. Example 1-2: Grouping Overlap in Clementi op. 33 no. 1, mm. 11–15 This interpretation of “phrase overlap” involves the interrelationship of two concurrent but distinct varieties of grouping. An awareness of groupings according to melodic closure and gestalt segmentation informs the decision about the end of melodic group G1, while a simultaneous awareness of (co-present) groupings according to durational accent, harmonic rhythm, and (approximate) parallelisms between G1/G2 and T1.1/T1.2 inform the decision to begin melodic group G2 on the same melodic pitch-event that concludes G1. One might also hear overlap of the very same kind on the downbeat of m. 15 (between melodic groups G2 and G3), though, that depends on whether the melodic gesture of group G3 is understood to begin from C♮5 (and thus parallel to the on-the-beat beginning of G2) or from A♮4 (and thus parallel to the anacrustic melodic group from G1.1 and G1.2). That the potential overlap between G2/G3 is less overt than for G1/G2 is instructive here, since it points to the way that the {11 cc& ### G1G1.1 G1.2G2G2.1 G2.2? G3 Ω Ω Ω6 6?###H1T1.1I vii ó IH2V (P) VH3T1.2I vii ó IH4V (P) VH5T2IŒ œ ™ œœœ œ œ ™ œ œ œ œ œ œ œ œ œ ™™ œœœ œ œ ™ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ wœ œ œ œ œ œ œ œ3 12 interrelationship of combined factors of melodic closure, parallelism, and gestalt segmentation are what determine the prospective overlap. The grouping overlap(s) here do not cause or coincide with any disruption to the underlying (hyper)metric pattern, which is based on the regular duple (or perhaps quadruple) periodicity of the H-series groupings, corresponding closely to the measures of Clementi’s notation. 1.2.2 Elision (PRiTM’s “reinterpretation,” GTTM’s “deletion”) The term “elision” is borrowed from literary contexts where it can describe the deletion of a syllable from a word (e.g., the two spoken syllables of “ever” collapsed into monosyllabic “e’er”); or cases where the final syllable of one word serves also as the first syllable of the following word, as in the opening line of Shakespeare Sonnet 129: “Th’expense of spirit in a waste of shame.”11 The term is in relatively wide use in the music theory literature. The conventional definition of [musical] elision I have in mind is quite close to the way it is defined in Benjamin’s “A Theory of Musical Meter” (1984): “the hallmark of [musical] elision is metric distortion, a sense that some span of time has been skipped over or elided—whence ‘elision’—and that where one span appears at the surface, two were present before transformation” (Benjamin 1984, 394). Neither Lerdahl and Jackendoff, nor Rothstein use “elision” in exactly that sense—though they both describe the phrase-rhythmic effect that I am calling “elision” in their own terms.12 The concept in GTTM that most closely resembles “elision” as I am using it, 11 This is the example of poetic “elision” in Oxford Dictionary of Literary Terms (Baldick, 2008). 12 Rothstein does not use the term elision at all in PRiTM, but in broad strokes his “metrical reinterpretation” is very close to what I mean by “elision.” Lerdahl and Jackendoff do use “elision” in a technical sense, but define it quite narrowly mean a particular subset of grouping overlaps that involves marked texture change. “Elisions” in this narrow sense are further differentiated as either “right” or “left” overlapped according to whether “texture 2” replaces “texture 1” precisely at the moment of overlap, or after the moment of overlap, respectively (GTTM, 55–60). 13 is Lerdahl and Jackendoff’s “metrical deletion” (GTTM, 101–104). The equivalent concept in Rothstein is his “metrical reinterpretation” (PRiTM, 52–56). An elision is a special variety of overlap where one timespan comes to represent two differently conceived spans of measured time. Benjamin’s definition conveys this aspect of elision by comparing the elided phrases to an (imaginary) un-elided version, in order to confirm “that where one span appears at the surface, two were present before transformation” (Benjamin 1984, 394).13 We can test this distinction by recalling the grouping overlap (not elided) in the Clementi example from the previous section. There, the note A♮4 (downbeat of m. 13) was shared between melodic groups: the final event of group G1 is simultaneously the first event of group G2, yet the quarter-note timespan shared by the two melodic groups does not represent two differently conceptualized spans of measured time. To “undo” that grouping overlap we can simply imagine a quarter-note A♮4 (as the concluding event of melodic group G1) on the first beat of m. 13 that is followed by a dotted-eighth-note A♮4 on beat 2 of the same measure (as the beginning of melodic group G2). Neither the total duration nor the (hyper)metric count of underlying groups H2 and H3 is affected by the overlapped melody note. An elision is different in this respect because the overlapping of material pertains not only to an event shared between two melodic groups, but also an overlap of the groups/timespans 13 In describing elision via comparison of two versions of the phrase Benjamin’s account resembles Rothstein’s comparison of the sounding phrase (post-transform/overlap) to the “basic phrase” (un-overlapped). Like Benjamin’s elision and PRiTM’s reinterpretation, Lerdahl and Jackendoff’s metrical deletion is also understood as transformed version of a more regular metrical structure. The designation of their “deletion” determined by locating the moment that disruption well-formed metrical structure before and after the overlap (GTTM, 101). Besides Benjamin (1984, 395), other theorists have called this effect “elision,” especially prior to publication of GTTM and PRiTM (e.g., Lester 1986, 189). 14 that determine the (hyper)metric count (or otherwise serve as a “measure”) for the unfolding phrase. An excerpt from Haydn Symphony 104 demonstrates in Example 1-3, below.14 Example 1-3: Elision in Haydn Symphony 104 mvt. 1, mm. 13–19 (adapted from GTTM ex. 3.27) From the outset of Haydn’s Symphony 104 (first movement) the alignment of clear melodic groups, harmonic rhythm, and associated metrical structure establish a norm of consecutive four-measure groups (mm. 1–12 not included in Example 1-3, see Lerdahl and Jackendoff’s ex. 3.26, GTTM, 57). Starting in m. 13 a fourth consecutive four-measure unit is begun. Its first half is defined by the stepwise ascent of the syncopated 5-6 sequence in mm. 13–14, and it is concluded by a two-measure cadential progression whose expected tonic arrival is 14 This same Haydn excerpt appears in GTTM where it is given as an example of their “right-overlapped elision.” The right-overlap occurs when D♮6 (and the full texture of its supporting harmony) replaces the expected D♮5 (and thin texture of its supporting harmony) as conclusion of the previous melody (see the illustration of conceptually distinct measures 16a and 16b). In the same example, they would describe the sense of missing time as a “metrical deletion”: the result of the interruption to an expected quadruple (hyper)metric structure in mm.13–16, where the fourth downbeat (my m. 16a) is “deleted,” leading directly to the first downbeat of a new four-bar metric structure (mm. 16b–19). {{{13 16a16b22bcccccc& ## Ó ∑ ∑ ∑ ∑ ∑ ∑ ∑?## Ó ∑ ∑ ∑ ∑ ∑ ∑ ∑& ## ∑ ∑ & ## ∑ ∑?## ∑ ∑ ?## ∑ ∑& ## ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑?## ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑œ˙˙˙œ œ˙˙˙œœ œ œ œ œœ œ œ œ œ œœ ˙˙œ ˙ œ œ ˙ œ œ œ œ œ ˙˙˙˙˙˙œœœœœœœœ ˙˙˙œœœ œ œœ œ œ ˙˙˙˙˙œœœœœœœœ ˙˙˙œœœ œ œœ œ œœœ˙ œ œœ œœ œœ˙ œ œÓ œ œœ˙ œ œœ œœ œœ˙ œ œÓ œ 15 the downbeat of m. 16 (illustrated in first system of Example 1-3). In the second system of Example 1-3 we find yet another clear four-measure unit (2+2) consisting of the verbatim repetition of a two-measure tonic prolongation figure (mm. 16b–19). In Haydn’s notated version, the conceptually distinct measures “16a” and “16b” occupy one and the same measure. Representing them as visually separate measures helps illustrate the sensation that the overlap in m. 16 is not merely an overlap in the melody (i.e., the shared half note D♮5/D♮6)—but that the host measures too are overlapped. The feeling of “missing-time” arises because m. 16’s tonic harmony represents the initial whole-note duration (initial tonic) for the new group mm. 16–19 while simultaneously representing the fourth-and-final whole-note duration (and final tonic) for the group of measures 12–16. To put it in Rothstein’s terms this is an instance of “metric re-interpretation” because the material that begins in m. 16 bring with it a four-measure hypermetrical structure that interrupts the regularity of the four-measure hypermetric counts from the preceding measure-groups 1–4, 5–8, 9–12, and 13–16 such that m. 16’s anticipated hyperbeat 4 becomes actual hyperbeat 1. Nota bene the different emphasis (but same analytical result) of describing this elision/deletion/reinterpretation as resulting from a dual function attributed to one event (i.e., final tonic becomes initiating tonic); as opposed to the description in terms of the interruption to presumed (hyper)metric regularity. Because elision often correlates to moments in a piece where a regular periodicity is interrupted, the (hyper)metric irregularity is sometimes described in such a way that the (hyper)metrical irregularity itself is the defining trait—or even the cause of the elision. This conflation of cause and effect would not be a problem except that to define elision as contingent on disruption of periodicity means the effect could never be described in music where a sufficiently periodic structure has not been established. My aim is to generalize phrase-rhythmic effects, 16 including elision, so that they can be accounted for even where a periodic structure like Rothstein’s hypermeter is not established. With the next example, from Beethoven op. 2 no. 2, I will provide a clear definition of elision that is in line with Rothstein’s definition; but adjusted so that it can also be generalized to situations that Rothstein’s method in PRiTM does not easily accommodate. As with the preceding Haydn example, the excerpt from Beethoven Piano Sonata op. 2 no. 2 reproduced in my Example 1-4 illustrates an elision by vertically aligning two conceptually distinct measures—32a and 32b—that occupy a single notated measure in the score, and a single timespan in the sounding experience of the phrase. Rothstein uses this example as a demonstration of his “metrical reinterpretation,” and begins the analysis by asserting that “the hypermeter of the excerpt is two-bar” (PRiTM, 53). His duple hypermetric count (started from m. 23), is reproduced in boxed blue text within my Example 1-4. In Rothstein’s telling, the elision (his metric reinterpretation) occurs when—noting the “unmistakable” two-measure units starting from m. 32—the listener has “retrospectively … reinterpreted m. 32—which was initially weak metrically—as the first strong bar of a new hypermeasure” (PRiTM, 53). Above and below the staves in Example 1-4 my labeled brackets flesh out some grouping considerations for the passage, taking Rothstein’s observations about harmonic rhythm and parallelism as a starting point. With these brackets I mean to provide a fuller picture of the kinds of grouping—harmonic, and melodic—that set up the elision. The G-series groups above the staves are a kind of melodic segmentation much like GTTM and PRiTM’s [melodic] grouping structure. A notable difference, though, is that I propose independent 17 segmentations for the right- and left-hand parts, which GTTM or PRiTM would not do.15 The particulars of determining the points of segmentation of the G-series groups for the right- and left-hand parts are not central to the demonstration of elision, so I leave it to the reader to scan the example and note that gestalt segmentation principles govern my decisions for partitioning the melodies into these basic segments. Note that prior to m. 32, all the melodic groups (in both right- and left-hand parts) are anacrustic (i.e., starting after a downbeat, and extending through a subsequent downbeat) in their orientation relative to the notated bar. Then, in m. 32 and following melodic groups are switched to begin on downbeats. The bass voice’s arrival on downbeat of m. 32 is understood as a grouping overlap when its pitch event A♮2 serves as both the final event of melodic group G.lh.5 and the first event of group G.lh.6. Whether that moment of overlap is also an elision will require us to consider aspects of structure other than the left-hand’s G-groups alone. For Rothstein, that other aspect of structure is hypermeter. For me, that other aspect is a kind of grouping determined by harmonic rhythm and motivic parallelism—which in this particular case is so periodic as to correlate exactly to the two-bar hypermetric structure of Rothstein’s “1–2” counts. 15 Lerdahl and Jackendoff are explicit in GTTM that their approach to grouping essentially treats all music as if it is homophonic and concede that their theory of grouping structure is inadequate for contrapuntal textures (GTTM, 37). In PRiTM, Rothstein follows suit, and restricts grouping analysis to a single layer of the composition—most often the melodic segmentation uppermost voice. Both theories can and do treat segmentation of the music according to (functional) bass lines of tonal music as a separate stratum, but do not call that aspect of the composition “grouping.” I return to this topic in Chapter 2 section 2.2.2 where I provide commentary on GTTM’s (homophonic) grouping analysis of a polyphonic example (Schubert’s Valses Sentimentales, op. 50 no. 13 (GTTM, 250–253)); and in section 2.3 where I develop a “meter-as-grouping” outlook modeled on Benjamin’s theory of musical meter (1984). 18 Example 1-4: Elision in Beethoven op. 2 no. 2 (blue hyperbeat counts from PRiTM p. 53) With the following narration, I demonstrate that the G-series melodic groups and H-series harmonic groups interact to create and break parallelisms, thereby affording the sensation of elision at measure 32. With some effort this effect can be described without an appeal to the phrase’s conformity to an expected hypermetric regularity as a prototypical structure. The beginning of the first group H-series group, H1, is marked by a textural switch from rapid arpeggio to stepwise motion in staccato articulation. The eight events of H1 are affiliated with one other because of the consistent articulation and steady stepwise ascent; and because, in this overtly tonal context, the ascending scale implies a single underlying tonic A-major harmony that intimates one whole-note duration in the harmonic rhythm. The leap down to E♮3 in the left hand is marked for attention because of the way it breaks the scalar {{{2329 32a32b 332424&###[1]∑[2]∑[1]pG.rh.1[2]G.rh.2fp[1] [2]?###G.lh.1H1 . . . .sfpH2 <V, I>. . ..G.lh.2H3 <V, I>G.lh.3 G.lh.43&###[1]G.rh.3[2] [1]. G.rh.4.[2 becomes 1]f.?###sfpH4 <V, [I, IV]>G.lh.5H5 <V, I>...&###&###[2 becomes 1]∑[2]∑[1]G.rh.5[2] [1] etc.. . . . .333 3?### ? ###fG.lh.6H6 fH7 <V>. . . . . . . . . . . .33 333œ œ œ œ‰ ‰œœœœœœœœœœ œœ ™ œjœœ œ œ œ œŒœ œœœ œ œœ œ œœ œ œ ™ œJœ œ œ œ œœ œ œœœœ œ‰œ œ œœ œ œœJœœ œ œœ œœœœœœœœœ ŒœJœ ™ œJœœ œ œ œœœŒœ œœ œ œœ œœ œ œœ œ œ œ œ œ œœ œœ œ œœ œœ œ œœ œ œ œ œ œ œ œ œ œ œr 19 ascent and switches the melodic contour. Additionally, the E♮3 receives a durational accent (dotted-quarter note after a string of eighth notes) and a dynamic accent (sfp). Considered alone, the left hand’s E♮3 does not itself mark a definitive harmonic change from the preceding tonic A-major harmony of H1; however, as soon as the right hand joins the texture it becomes clear that the left hand’s E♮3 is not the chordal 5th of an A-major harmony, but rather the root of E-major (V in A). Thus, a harmonic change is also attributed to the event E♮3 in m. 25, which I have indicated as the beginning of group H2, accordingly. The left hand’s prominent <sol, fa, mi> gesture in mm. 25–26 implies a <V, I> harmonization, in half-note harmonic rhythm. The subsequent measures repeat the same <sol, fa, mi> gesture (now in the right hand) with contrapuntal accompaniment in “alto” and “bass” voices implying the same <V, I> harmonic support (again in half-note rhythm) for group H3 as for group H2. Next, the material in mm. 29–30 begins as if a third iteration of the same <V, I> idea from H2 and H3 is underway, except that a novel pre-dominant IV harmony is squeezed into the H4’s harmonic rhythm on the last eighth note of measure 30. Then, in mm. 31–32, group H5, repeats the <V, I> progression in half-note rhythm once more—this time adding a cadential six-four suspension to the dominant harmony, and for the first time presenting the harmony in block chords as opposed to suggesting the harmony by three-voice counterpoint. The idiomatic cadential six-four suspension in H5, and the brief appearance of pre-dominant functional harmony in H4 lend special importance to the tonic arrival mid group H5. Upon the tonic arrival in m. 32, though, the left hand presents a completely new two-measure motivic idea commencing from the moment of harmonic and melodic arrival. Dynamic accents (forte markings) now appear regularly on each of the even-numbered measures’ downbeats. As before, the H-series groups (H6, H7) propose two-bar units. In 20 Rothstein’s terms, the overlap in m. 32 causes a disruption to the otherwise regular metrical structure that would correlate to a “2/1” time signature for the passage, starting from m. 21, interrupted and re-started from measure 32. In my terms, the conditions that define H-series groups provide a measure relative to which the action of the melodic grouping can be evaluated. The melodic groups (G.lh.5 and G.lh6) are overlapped at m. 32; but the phrase-rhythmic effect of elision arises because of the concurrent overlap in H-series groups (H5, H6). While true that the H-series groups are two-measure in duration for most of the excerpt such that their periodicity could become metric (in the conventional sense), the elision results from the sense of time-span overlap that occurs when the expected second-half of group H5 (as final tonic) becomes the first half of group H6 (as initial tonic). The interrupted periodicity helps to highlight the special overlap of H-series groups but is not its cause. 1.2.3 Internal Expansion: Composed-out Deceleration (or fermata), Repetition, and Parenthetical Insertion “Phrase expansion” is another category of phrase-rhythmic effect theorized in PRiTM, where it is defined as “a class of rhythmic operations, any of which may transform a given phrase into a larger one” (PRiTM, 64). Expansion, for Rothstein, is based on comparison between two phrases: one is the music as sounded/notated, the other existing only as a conceptual model of the same phrase, imagined in its “original” unexpanded form. Rothstein explains: The original, unexpanded phrase is called the basic phrase. The specific portion of the basic phrase that is subject to transformation—since expansion usually affects only a part of the original phrase—is called the prototype. The transformed version of the prototype is called the expansion, and the new version of the phrase as a whole is termed an expanded phrase. … [I]f a transformation is to be perceived, the original and transformed versions of the 21 phrase must be heard as different representations of the same thing. That “thing,” in Schenkerian terms, is the structural skeleton common to both phrases (including a rhythmic pacing of events that is closely similar in some corresponding parts of the two phrases). (PRiTM, 64) The ontological status of Rothstein’s basic phrase is rather knotty: not least because to “hear” (i.e., conceive of) the “original” version of the phrase, one first must encounter the phrase in its already expanded (i.e., transformed) version. (His somewhat unwieldy past tense verb “un-expanded” betrays this unusual reversal of the usual conceptual order, originalàtransform.) Although the “basic phrase” is worked out from the expanded phrase, the basic phrase is also informed by considerations separate from (and conceptually prior to) the encounter with the specific passage under consideration. First, it is premised on some notion about the normative amount of harmonic and motivic content for a phrase; and second, it is premised on an understanding that periodic constructions, most often duple, represent typical rhythmic pacing and proportion for the component parts of a phrase (e.g., an eight-measure phrase presumed 4+4; whose four-measure units are presumed 2+2). The former premise is informed primarily by the theories of Heinrich Schenker, and the latter is informed by theories of Heinrich Christoph Koch and Hugo Riemann among other European theorists of the late 18th and early 19th century (see PRiTM, 64–65). Rothstein differentiates between “internal” and “external” expansion, but I will focus mostly on internal expansions, which are those that increase the total duration of the phrase by altering material after the phrase has begun but before the phrase proper is brought to a close.16 I am primarily interested in types of expansion that 16 Rothstein’s external expansion involves situations where “prefix” or “suffix” materials extend the phrase by being affiliated with the phrase proper, while ultimately not understood to be constituent of the basic phrase itself. Examples include short introductory materials prior to the presentation of the 22 Rothstein understands as directly altering the assumed regularity of the basic phrase (which external expansion does not do). Rothstein recognizes three varieties of internal phrase expansion: expansion by composed-out deceleration or fermata, expansion by (redundant) repetition within the phrase, and expansion by parenthetical insertion (PRiTM, 64ff.). For my purposes, PRiTM’s expansion by repetition and by parenthetical insertion can be understood as two manifestations of the same general effect: both determine certain material within the phrase to be extraneous to the “structural skeleton” and as an interruption of the presumed (hyper)metric regularity of the basic phrase.17 His expansion by “repetition” is where the extra material resembles preceding material, while expansion by “parenthetical insertion” is where it does not. As with previous examples in this chapter, my commentary on the following Brahms and Mozart passages will introduce the phrase-rhythmic effects as defined by Rothstein while also previewing the kind of main motivic idea (prefix), or echoes of a concluding cadential idea which serve as a codetta (suffix). (PRiTM, 68–73). Supposing a “basic phrase” was eight measures in duration, and was expanded only via external expansion (prefix, and suffix): the “basic phrase” is still present as eight consecutive measures in the sounding/notated music, only it is preceded, and followed by affiliated, non-essential materials. In this sense, the “regularity” of the phrase is not altered internal to the phrase proper, hence “external expansion.” 17 Rothstein follows Koch, who defines “Parenthese” as “the insertion of unessential melodic ideas between the segments of a phrase” (Koch [1786] 1983, 53). To Koch’s model, Rothstein adds the Schenkerian concept of middleground prolongational structure as part of determining whether the inserted materials are “essential” (PRiTM 88–90). There are some extended parenthetical insertions that contain quite substantial harmonic material, but that are nevertheless considered expansion by parenthetical insertion (and not elaboration of the essential middleground) on the basis of their harmonic remoteness from the surrounding material. In commentary on Tovey’s “purple patches” and the correlation of harmonically remote asides to phrase expansion by parenthetical insertion in common-practice music, Rothstein suggests that “the more remote harmonically a “patch” is, the more likely it is to be rhythmically unessential” (PRiTM, 91). 23 “analysis by emerging, co-present grouping structures” that I will use later to describe phrase-rhythmic effects in music that is less metrical (in the conventional senses of meter-as-grid, or as entrainable pulsation on two or more levels). In PRiTM’s terms, composed-out deceleration occurs when a listener who infers the proportions and expected duration of a “basic phrase” (i.e., a prototype to which the notated/sounding phrase refers) finds the actual phrase to occupy a longer span than the referred-to basic phrase would have. A clear example of composed-out deceleration is illustrated in the following example from early Brahms (Scherzo op. 4, Example 1-5). In it, a quarter-note anacrusis (segment A0) precedes two consecutive accented chords whose dotted-half-note duration defines the first notated measure (or, segment A1). The descending motion initiated in mm. 102–103 (A1) gains momentum as it continues a downward trajectory in a faster triple subdivision of the measure with new staccato articulation (A2). The descending motion concludes when the uppermost voice reaches the tonic E♭5 on the downbeat of m. 105 (end segment A2). Together, segments A0–A2 constitute a single larger melodic group “A” in Example 1-5. Note, in this analysis I use the terms “segment” when referring to A0, A1, and A2, and designate A0–A2, together, as group A. This is noteworthy because, despite the visual similarity, my brackets above the staff here are not indicating “grouping structure” in GTTM’s sense. The beginning of segment A1 is defined by a contour peak, a phenomenal accent, agogic accent, and articulation change (relative to the trivial one-event segment A0); the beginning of segment A2 is marked by the articulation and rhythmic change relative to A1 Next, melodic group B includes a closely parallel pattern of begin with anacrusis (B0) followed by stepwise melodic descent (B1 and faster B2), though its material sounds one octave lower than it had in group A (save for the surprising and dramatic upward leap to its concluding pitch). 24 Rhythmically groups A and B are exact repetitions. A third iteration of this same motive, group C, differs slightly from A and B in terms of pitch material (descending from F♮5, rather than from C♭6/C♭5), and in terms of rhythm. However, these differences are not enough to break the obvious parallelisms between groups A, B, and C. Example 1-5: Composed-out Deceleration in Brahms Scherzo op. 4 mm. 101–118 Groups A, B, and C share the same general rhythmic contour <short, long, long, short, short, short, long> and differ only when the relative durations within the rhythmic motive are described in more precise terms.18 The rapid descending gestures within segments A2 and B2 18 Using 0 to represent the shortest duration within a span, 1 to represent the next shortest, and so on, the rhythmic contour of A = B = <0, 2, 2, 0, 0, 0, 1>; while C = <0, 2, 2, 1, 1, 1, 3>. Applying formal contour theory to the descriptions of durational series in this way first appears in Marvin (1995). {{{{102110119126&bbbbbbAA0.A1> >A2. . ..BB0.B1> >B2. . . >?bbbbbb.1>2>3. . .4. . .1>2>3. . .4>&bbbbbbC0C.C1>>C2> > >>U. .?bbbbbb1>2>3> > - >4U> - >. .&bbbbbb(4) (?) (?) (1) (2)>(3) (4)?bbbbbb &?>&bbbbbb(1). . .(2)...(3).. .(4). . .(1).. .(2)...(3).. .(4).. .?bbbbbb.. . . . . . . . .. . . . .... .. . . . .œœœœ˙˙˙™™™ ˙˙˙ ™™™ œœœœœœœœœ œœœŒœœœœ˙˙˙˙™™™™˙˙˙˙™™™™œœœœœœœœœ ˙˙˙˙nnœœœnn˙˙˙ ™™™ ˙˙˙˙ ™™™™ œœœœœœœœœ œœœœœœ˙˙™™˙˙™™œœœœœœ˙˙nnœœ˙˙˙˙ ™™™™˙˙˙˙™™™™˙˙˙˙œœœnn œœœ˙˙˙˙˙˙˙ ™™™™ œœœœ ‰œ# œn œbœj ‰ Œ Œ Œ ‰œn œ œnœj‰ Œ ŒŒ˙˙˙™™™˙˙nn ™™˙˙˙œœœnœœœ˙˙˙˙˙nn ™™™œœœ ‰ œ œ œnœJ‰ Œ Œ Œ ‰ œ# œnœ œJ ‰ Œ ŒŒ ‰ œ œœœn œ# œ œbœ œn œ œnœ œ œœœ œ œ œ œn œ œ œ œ œ œ œn œ œ œ œ œ œ œn œ œ œ œ œŒ ‰ œn œœœn œ œb œœœ# œnœ œ œn œ‰ œ œ œ œb œn œœ œ œ œ œ œn œ œ œ œ œ œ œn œ œ œ œ œœœœœ œœœœœœœœœœœœ œœœœœ œœœœ œ œœœœ œœœœœœ œ œœœœ œ œœœœ œœœœœ œœœœœœœ œœœœ œ œ œœœœ œœœœ œœœœœœ œ œ œœœœ œ3 25 consists of three relatively short events that are equal to each other and equal to the short durational value of the anacrusis in A0 and B0. This duration is one third of the durational value of the relative long events within A1 and B1. In group C the descent that initiates segment C2 still articulates a faster pulse (relative to the dotted-half pulse established in C1), but its descent in half-note rhythm is double the durational value of the parallel descending passages in A1 and B1. This rhythmic augmentation does not interfere with hearing the events internal to C as closely paralleling the events internal to A and B, even though the resulting music occupies a different number of notated measures and exhibits different internal proportion among its segments. A quantitative approach to hypermeter, such as Rothstein’s, accommodates this sort of irregularity by “pausing” the hypermetric count after m. 112’s “hyperbeat 3” and then resuming the quadruple count as of measure 114. In this practice Rothstein resembles GTTM where (hyper)meter is understood as a kind of structure to which the music can be compared, rather than as a special type of grouping that arises from events and timespans that the music brings into affiliation. One of the broad aims of this project is to shift emphasis away from hypermeter as kind of patterned structure that exists conceptually prior to the music (which the music might distort); and toward a kind of structure that emerges from the music’s events. Hypermeter, in this view, becomes a special kind of grouping where temporal quale (such as beginning, anticipation, continuation, relative strong, relative weak, etc.) are assigned to certain events or points of segmentation on the basis of emerging group interrelations and affiliations (Berry [1976] 1987, 303–6, 317–24). Metric grouping of this kind may correlate to periodic structures (such as well-formed (hyper)meter) but does not require them. 26 In this re-conception of (hyper)meter, “hyperbeat 1” in Example 1-5 is not simply “when/where each cyclic count begins,” but is instead a quality attributed to those timepoints because of the way that the first event in segment A1, B1, and C1 is marked for attention by durational, contour, and phenomenal accent. Hyperbeat 3s are not “strong” because the expected alternation of weak and strong beats in a quadruple structure makes them so, but rather because they are group-beginnings marked by changes in attack-density and articulation that occur when the dotted-whole values are replaced by relatively rapid descending events that begin groups A2, B2, and C2 (exact duration variable). Likewise, hyperbeat 2s are not “weak” because beat two is known to be a weak “place” within a quadruple metric structure; but rather because they represent the continuation of the still-unfolding groups (A1, B1, etc.) begun one measure prior. In the same way, strong hyperbeat 3s, here, can be described as weak relative to hyperbeat 1s—and yet strong relative to hyperbeat 2s—because their accent-defined beginnings initiate groups that are ultimately subsumed into the continuation of the larger-scale groups A, B, and C, which are defined by thematic parallelism. Rothstein’s PRiTM would not employ this kind of blow-by-blow account of emerging grouping structures as the basis for arguing in support of an observed instance of composed out deceleration, though certainly these kinds of considerations are the basis for why he chooses to begin, pause, and resume his hypermetric counts when and where he does. A PRiTM-style approach to this Brahms passage would instead describe it in terms of a four-bar hypermetric norm (presumed for most common-practice music and confirmed here by the overt four-bar units in mm. 102–5 and 106–9) from which the music of phrase C departs. But recognizing the existence of “composed out deceleration” in the more general terms I propose will enable us to adapt Rothstein’s approach to accommodate analysis of phrase-rhythmic effects where the 27 determination of “a basic phrase” is more opaque. For example, in the Brahms excerpt, the evaluation of phrase C relative to the proportion and duration of A and B is enough to account for the composed-out deceleration; so, recognizing the expansion does not require the abstract extrapolation of a four-measure basic phrase prototype: the comparison can be made directly within the passage. Other times a passage of music presents us with a series of phrase segments in “expanded” forms such that no sounding music corresponds exactly to the proportion and duration of its basic phrase units. Mozart’s Menuetto from the K.331 piano sonata provides one such example: its first ten measures might be described in PRiTM’s terms as an eight-measure basic phrase (4+4) subject to two one-measure expansions. If we are to generalize Rothstein’s “expansion,” some material needs to be identifiable as extraneous to the basic phrase, but so identified without appealing to the phrase’s degree of conformity to some a priori hypermetric structure as normative. Rothstein’s basic phrase is determined by identifying “essential harmonic and rhythmic content,” not by metrical/rhythmic proportional criteria alone. It should be possible, then, to account for the measures extraneous to a passage’s “basic phrase” by appealing to a combination of rhythmic, harmonic, and motivic criteria, rather than simply identifying the measure(s) that exhibit a degree of non-conformity to whichever a hypermetrical pattern of “strong and weak” alternation one might assume is most relevant for the passage.19 19 Here, I am working to ground this in Rothstein’s terms, but already my method contrasts parts of PRiTM’s mode of analysis which, for “familiar” common-practice repertoire, would fast-track the analysis by evaluating events and groups in terms of their degree of conformity to the basic phrase prototype and its associated (hyper)metric count. Indeed, this is how phrase analysis is commonly taught, informed by the norms of theme construction in the Classical style, e.g., according to Caplin (1998), or Hepokoski and Darcy (2006). These practical methods require some notion of the 28 Example 1-6A reproduces Mozart’s notation for the K.331 Menuetto (mm. 1–10) with annotations to illustrate a PRiTM-style account of hypermeter and phrase expansion. In Example 1-6A hypermetric counts appear as boxed Arabic numerals below the staff. The expansion during the first half (or “presentation”) is highlighted in orange, where I characterize mm. 1–5 as a four-measure basic phrase that is expanded via composed-out fermata to occupy a total of five measures. In the second half of the phrase (or “continuation”) two slightly different accounts of expansion by (redundant) repetition or by insertion are illustrated in in green and purple, respectively. (A third alternative account for the continuation instead posits a five-measure basic phrase without expansion, shown in grey.) Above the staff the brackets labeled “a” through “f” segment the phrase into a series of one-bar motivic units. These are grouped further into “thematic phrase groups” of various kinds, illustrated with brackets X, Y and Z. Example 1-6B reproduces most of the same information from Example 1-6A, but adds harmonic information (in roman numerals) for quick reference. (Note, the shading in Example 1-6B is meant only to highlight (dis)continuities of various kinds, it does not continue the same colour-coding used in Example 1-6A’s brackets and text labels.) In my commentary, I argue that various different kinds of groupings are involved in creating the sensation of beat qualia that accrues to certain group beginnings as the music unfolds, ultimately resulting in the complex grouping information that informs decisions about the when and where of the prototypical phrase as existing conceptually prior to the evaluation of the music which presents a logical problem since the basic phrase is also something that arises from the music considered. In order to generalize phrase expansion techniques, it is necessary to dispense with the idea that sounding music is measured against anything other than structures that arises from/within the music itself. I return to this philosophical question, framing it in terms of “generative/conformational” and “intrinsic/extrinsic” dichotomies throughout the literature survey and analytical demonstrations of Chapter 2, especially in section 2.2. 29 hypermetric counts offered. I refer to the diagrams Example 1-6 throughout the analytical narration that follows. Measures 1–4 cohere as a group for several reasons. The phrase begins with unit a, a lightly ornamented descending gesture, <do, sol> in <long, short> rhythm. Measure 2 (unit b) continues with a lightly ornamented <mi, do>, in <short, long> rhythm. Together, the two small gestures combine into a single larger one, group X1, which is defined by the <short, long, long, short> palindrome rhythm20 and also by a chord-factor “double neighbour” gesture about A♮4’s do (much like scale degree neighbours ti, and re often do). Next, in m. 3, we encounter an immediate repetition of motivic unit b, now transposed up a by diatonic third (its “long” duration filled in by octave C♯s in quarter-note rhythm). Then, unit c offers a novel stepwise melodic gesture <fi, sol> in the same basic <short, long> rhythm as motive b. The <b2, c> material of mm. 3–4 coheres into group X2 on account of its parallels to the preceding X1—once again outlining a chord-factor double neighbour gesture (about E♮5 in mm. 3–4). Together, groups X1 and X2 combine into a four-measure group for a number of reasons: one is the phrase-structural synecdoche that results when the palindromic aspect internal to X1 is writ large between X1 and X2 because of the immediate repetition of unit b on either side of the 20 This impression is based on inferring a simple rhythm from gestures b1 and b2 as if played without any surface ornamentation: <C♯5, A♮4> and <E♮5, C♯5> in rhythm. Unit b1’s A♮4 is not a literal half-note in duration—the half-note rhythmic value is attributed to the A♮4 in m. 2 when no subsequent attack is sounded until the downbeat of the following measure. Neither is unit b2’s “long” a literal half-note duration; yet a half-note value is suggested by the pitch-class continuity of m. 3’s two consecutive quarter-note C♯s, one octave apart. 30 point of segmentation that separates them.21 Another (more familiar) basis for the affiliation of X1 and X2 is the continuity created by the middleground arpeggiation of the tonic triad from A♮4 (m. 1) to C♯5 (m. 2) and finally to E♮5 (mm. 3–4). Example 1-6A’s annotated score uses blue circles to highlights these principal notes of the melody (as is also done for the rest of the phrase). Various factors—including motivic parallelism, texture, and contour—propose a contrasting second group that begins in measure 6. (I will return to the question of m. 5’s grouping affiliation shortly). The unit e gesture is stated three times in direct succession across measures 6–8. The direct parallelism strongly associates mm. 6–8 to form thematic group Y. Therein, the series e1–e3 introduces a novel compound melody in the right-hand part where an “alto” voice begins a stepwise ascent through an ornamented <G♯4, A♮4, B♮4> gesture (circled noteheads in Example 1-6A, mm. 6–8), while a “soprano” voice holds steady in the upper register to prolonging E♮5 as a pedal.22 The action that unfolds across units e1–e3 further differentiates itself from the X groups in that each and every sounded quarter-note beat within group Y is an accented non-chord-tone; whereas every sounded quarter-note beat in groups X1 21 Admittedly, this aspect of X1 and X2’s affiliation is quite abstract. I include it because I have come to hear it and find it rather beautiful. Further, with regard to the general theme of this project, I want to argue that more abstract (and less tonal) varieties of event affiliation—such as this one—are among the grouping conditions that, across many compositions, give rise to now-familiar idioms such “the presentation phrase.” In other styles, such affiliations may give rise to idiosyncratic forms, and/or to idiomatic constructions yet to receive formal treatment. I like to strike this attitude in order to combat (in myself) any lingering notion that a (Platonic) form such as “basic idea,” “presentation phrase,” or “sentence,” might be said to “exist” in any sense other than as the conceptual residue of the many musical utterances that, together, realize them. 22 The pedal is sustained by the chord tone E♮5s in the upper voice’s melody in mm. 6-7; and is extended into m. 8 where it is prolonged (more abstractly) by way of F♯5 as incomplete upper neighbour. 31 and X2 is a chord tone. (Beyond group Y, every sounded quarter-note beat in the remainder of the continuation is likewise an accented non-chord tone, save for m. 9’s downbeat F♯4.) Measure 9’s unit f sharply contrasts the preceding group when it replaces Y’s compound melody with a new Alberti bass texture and discontinues Y’s series of consecutive on-the-beat non-chord tones. At the same time, unit F extends the same pre-dominant harmony from m. 8 into m. 9, adding a chordal seventh to the continuing b-minor amid its stepwise descent (just as m. 5 does for m. 4’s E-major triad). The phrase concludes in the next measure where a <V, I> progression (m. 10’s motivic unit c2) re-uses the same gesture that “ended” the presentation in measure 4. Units f and c2 (mm. 9–10) are affiliated—as Group Z—by their shared chordal accompaniment texture, and by the idiomatic <ii6/5, V7, I> cadential idea. Groups Y and Z are affiliated—as the “continuation phrase”—for a number of reasons. These include: the harmonic continuity from e3 (prospective end of Y) into unit f; the resumption of on-the-beat non-chord tones (from Y’s e1–e3) in m. 9 (beat 2–3) and m. 10 (beat 1); the registral transfer of the “soprano” voice’s F♯5 (e3) to F♯4 (unit f) across the prospective Y/Z boundary; and the sense of unit c2’s principal melody note A♮4 as the completion of a (middleground) stepwise neighbour gesture initiated by group Y’s principal notes <G♯4, A♮4, B♮4>. 32 Example 1-6: Mozart’s K.331 Menuetto mm. 1–10 as Eight-measure Basic Phrase with Internal Expansion A) Annotated Score B) Schematic Representation {1 2 3 4 5 6 7 8 9 103434&###"Presentation"X1 a b1 N Nb2 X2 N Nc U[ ]PdP Pe1 Y"Continuation"AP (N)e2 A Ae3 AAfZ(A) Pc2 AP?###hypermetric count(s)non-chord tones indicated by small upper-case italics:N = neighbour (including incomplete, double)P = passingA = appoggiaturaAP = accented passing1 2 3&expansion of presentation's 4-measure basic phrase (by composed-out fermata)4 —∑ ?1expansion of continuation's 4-measure basic phrase (by insertion/repetition) option 12expansion of continuation's 4-measure basic phrase (by insertion/repetition) option 2continuation as 5-measure basic phrase (no expansion)"extra"333"extra"445œ œ˙ œ œ œ œ œ Œ œ œ œ œœ œ# œ ™ œ#J œ œ œn œn œ œ œ œ œ œ Œ œ œ œ œ Œ œ œ œn œ Œ œ œ œ œ œ œ œ œ Œ˙˙œœœœ œœ Œ Œ Œ œœ œœ œœ Œœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ Œ Œ œœ œœ Œlink/lead-in ← ? →Measure Numbers 1 2 5 6 7 8 9Motivic Units a b1 d e1 e2 e3 fHarmony I I I vi V7/V V(9) V7 I ii ii V7(–) 3(–)4(I) vi V7 (V7) I ii6 ii6/53Hypermeter Count(Basic Phrase)Vii6/5I V7 Ib23I321IV7(–)4 2145X2Vc1 c2Y4 10HarmonicReduction"Presentation" "Continuation"Thematic Phrase Groups X1 ZI 33 The account of grouping and motivic affiliation just offered bypasses any decision on whether to affiliate m. 5 with the emergent grouping structure of X1–X2 (as presentation) or with Y–Z (as continuation). On its surface, m. 5’s motivic unit d does not reference, echo, or otherwise continue the pattern of strong pairwise motivic connections that characterized measures 1–4. As the first and only measure entirely without left-hand accompanimental support, unit d is an outlier in terms of texture. Harmonically, it does extend the dominant arrival of the previous measure’s unit c1; but it also anticipates the (initiating) dominant harmony that begins group Y. Melodically, while its stepwise motion “leads in” toward e1, any sense of it as a (potential) larger-scale group beginning is quickly overshadowed by the accentual weight that is attributed to e1 (as beginning of group Y). This has to do with the total duration of and strong parallelism internal to group Y: its repeated unit e gestures, together, define a group that is longer in duration than the presentation phrase’s initial thematic group X1. Further, the introduction of a new compound melody along with a “thickened” Alberti bass accompaniment texture confers a degree of accent-by-textural-contrast to e1 that outweighs the relatively slight textural-contrast accent attributable to unit d’s having “thinned-out” the texture relative the preceding material. As a result of these grouping conditions my sensation is that while m. 6’s unit e1 receives the descending motion begun by unit d as “lead in,” unit d does not itself begin any larger group so much as it extends X2. Some voice-leading features of mm. 4–5 further bolster the argument for unit d as extension of X. For example, the wonderful expressive chromatic passing-tone E♯5’s expected F♯5 arrival is delayed by an intervening G♯5 that appears on the downbeat of measure 5. On the surface, E♯5’s tonicization of F♯5 fleetingly characterizes the downbeat G♯5 as an appoggiatura, though G♯5 is immediately (re)conceptualized as the chordal third of 34 E-major (as V7) extended from m. 4 into measure 5. With that harmonic prolongation in mind, I can focus my hearing toward the middleground, conveying m. 5’s unit d as a downward dominant arpeggiation <G♯5, E♮5, (D♮5,) B♮4> that is balanced against the ascending tonic arpeggio that is unfolded across mm. 1–4. With these motivic, harmonic, and textural details laid out, we can provide a full account of Mozart’s mm. 1–5 as being understood to representing a four-measure basic phrase, and no pre-existing notion of four-measure normativity is required in order to characterize its m. 5 as an expansion by composed-out fermata/deceleration.23 Rather, it is emergent grouping information that defines the “basic phrase” unit in a local context—which happens to be of the conventional four-measure variety. The four-measure unit is defined according to coherence of the principal melody and the harmonic progression occurring within mm. 1–4, among other grouping considerations. Of course, we could have gotten to this point much more quickly had we appealed to stylistic and music-theoretical convention and accepted the relevance of a normative 4+4 structure for the passage from the start. But the level of detail offered here is necessary precisely because I wish to generalize PRiTM’s approach to phrase-rhythmic expansion to passages that do not obviously refer to a priori basic phrase prototypes of certain pre-determined “regular” lengths.24 23 Rothstein provides an example of a lengthy cadenza notated within the time of a fermata in the Rondo of Beethoven piano concerto no. 4 show how this kind of elaboration “pauses” (hyper)metric counts, and then argues music that likewise pauses measured time sometimes occupies regular notated bars in the notation (PRiTM, 42). William Benjamin treats issue of composed-out fermata within his “A Theory of Musical Meter” pointing to historical accounts of the same at least as far back as Riemann’s System der musikalischen Rhythmik und Metrik (1903, 259–270; as cited in Benjamin 1984, 397–402). 24 There is a question even within the European common-practice repertoire as to whether non-duple hypermeasures/phrase-lengths could be normative for a piece where irregular lengths are frequently 35 The potential benefit of generalizing phrase rhythm in this way can already be observed when considering the continuation phrase of the same K.311 Menuetto example. A PRiTM-style account would almost certainly convey this five-measure continuation as an expansion of a four-measure basic phrase unit. As already discussed, mm. 6–8 are strongly affiliated as group Y on account of the triple statement of motivic unit e; first as V7, then as I, and finally as ii6. For sentence-forms of the Classical style, the prototypical “continuation” begins with a twice-stated one-measure gesture (Caplin's "continuation idea" or "c.i." (1998, 47)). In this regard, unit e3 might be said to represent “one too many”—and thus, construed as “extraneous” to the basic phrase that brings about expansion by redundant repetition (as illustrated by the green annotations below mm. 6-10 in Example 1-6). To me, this account of the phrase as expansion by (redundant) repetition is plausible, but not entirely satisfactory. First, more generally, this is because it deems the third iteration of unit e material as extraneous based on an appeal to inter-opus stylistic norms over and above emergent (intra-opus) structure. While perhaps germane within the narrowly defined Classical style, this type of appeal will not serve the goal of this project’s generalization of PRiTM’s phrase rhythm. Second, more specifically, this is because when I audition a prospective “basic phrase” that excises e3 and imagines proceeding directly from e2 into the <f, c2> cadential idea of mm. 9–10, the continuation phrase loses two encountered. Rothstein’s answer (citing Schachter and Schenker as precedent) is that yes—non-duple basic phrases are possible, if relatively rare (PRiTM, 34). The only direct examples Rothstein provides in PRiTM are a five-measure basic phrase in Dvořák’s Serenade for Strings op. 22 (PRiTM 34–35), and a reference to Schachter’s analysis of Schubert’s Wandrers Nachtlied D.768 (as a twice-stated five-part structure that is expanded to fill 14 notated measures (Schachter 1999, 89–93). I return to this question a few times in this and the following chapter: in section 1.3 of the present chapter I discuss a possible five-measure basic phrase in relation to Mozart K.515; and in sections 2.1.1 and 2.1.2 of the following I survey and demonstrate theories that accommodate “irregularity” into a meter-as-grid framework. 36 of its most important characteristic features: the middle voice’s middle-ground melodic ascent <G♯4, A♮4, B♮4> across e1–e3 (whose principal tones I hear as a stepwise parallel to the ascent in chordal skips <A♮4, C♯5, E♮5> in mm. 1–3); and on a more surface level, the upper voice <G♮5, F♯5> which beautifully colours the phrase by modal mixture. These considerations give me considerable pause in deeming e3 as extraneous to the continuation’s basic phrase.25 An alternative account of expansion within the continuation phrase might instead nominate m. 9’s unit f as extraneous, as illustrated by the purple annotations below mm. 6–10 in Example 1-6A. This reading hinges on the harmonic similarity between measures 8 and 9. It characterizes m. 9 as extraneous to the basic phrase on account of being a (redundant) harmonic extension from the previous measure (and thus, representing a similar kind of expansion as in mm. 4–5). As a result of excising m. 9’s unit f, this reading would affiliate unit e3 with c2 (as part of group Z, harmonically) as much or more so than affiliating e3 with the other unit e motives of group Y. While not conventional, differently associating the (strongly parallel) units e2 and e3 with groups Y and Z, respectively, is not so implausible because of the way that the immediate repetition of b1 and b2 within the presentation phrase had contextually defined that pair being on opposite sides of the X1/ X2 grouping boundary in the middle of the presentation. Excising unit f, though, introduces a rather glaring melodic discontinuity that 25 Prolongational reduction further supports an interpretation where m. 8’s unit e3 is essential to the phrase. Schenker’s deep middleground graph of K.331 (2nd mvt.) shows material from mm. 2, 8, and 10 as the constituting essential elements during initial tonic prolongation through to the background dominant arrival in m. 11 (Der freie Satz, Fig.35/1). My account of “basic melody” pitches (an informal quasi-Schenkerian reduction) are too close to the foreground for the details of Schenker’s graph to provide any useful comparison; except that it can be noted that my prolongational analysis had informally understood E♮5 as prospective Kopfton, where Schenker nominates a C♯4/C♯5 complex. His scale-degree three Kopfton makes more sense on the scale of the entire minuet, where my informal reduction pertained only to mm. 1–10, where E♮5 is the goal of a more local initial arpeggiated ascent. 37 results when imagining a basic phrase that proceeds from m. 8 directly into measure 10. Neither account of this continuation as an expanded four-measure (basic) phrase unit is entirely convincing, yet both at times occupy my imagination when I consider to the proportions and duration of the continuation relative to the presentation (as expanded). The third alternative is to entertain the possibility that no measure within the continuation phrase is deemed unessential to its basic phrase unit, and that the continuation phrase is a five-measure basic phrase without expansion. The ten-measure phrase as four-measure presentation (expanded to five), with five-measure continuation (unexpanded) is an unlikely prospect within any approach to phrase construction that is informed by GTTM and PRiTM, both of whom take equal two-part phrase constructions to be the default. But having accounted for expansion in the presentation solely by a close consideration of grouping interactions (and without any appeal to normative phrase-length), what kind of problem remains when the same grouping considerations propose a five-measure basic phrase in the continuation? I do not outright reject the interpretations that would convey both presentation and continuation as expansions of four-measure basic phrase structures to five; only that—once phrase rhythm analysis is emancipated from hypermeter’s orthodoxy—it ain’t necessarily so.26 26 I include, here, the tuneful words of Sportin’ Life (in Gershwin’s Porgy and Bess (1935)) to represent a pithy and somewhat irreverent challenge to orthodoxy. Quite fortuitously, it led me on to Hilary Putnam’s very entertaining article “It Ain’t Necessarily So” (The Journal of Philosophy 1962). In it, his memorable “automaton cats radio-controlled from Mars” thought experiment offers an argument for replacing the common top-down notion that the paradigmatic case (this is a cat…) is derived from a true-and-essential meaning (…because it correctly expresses the essential characteristics of the category: cat) with a bottom-up model for the construction of meaning where paradigmatic cases (this is a cat, that is a cat…) constitute the category and its essential meaning (…so what their essential characteristics are found to be will construct the meaning of the category to which they belong). More than an entertaining aside, this previews the distinction between “generative” and “conformational” analytical methods that I introduce in Chapter 2 sections 2.2.2 and 2.2.3. 38 1.2.4 Phrase-rhythmic Effects Summary Within the prevailing view of phrase rhythm, informed by PRiTM and GTTM, each of the phrase-rhythmic effects discussed in the preceding examples have been defined in terms of a certain relationship to (hyper)metric structure that is presumed to be regular. Rothstein’s position is that an awareness of the relationship between the sounding music (with expansion, say) and what could have been a regular “basic phrase” (without expansion) is essential to appreciating phrase-rhythmic irregularity. Note that when a sounding phrase differs from the proportions of the basic phrase to which it refers, Rothstein most often understands “the hypermeter” of the sounding phrase to be a distorted/aperiodic version of the regular hypermetric structure that is associated with the basic phrase. For example, a four-measure basic phrase which is expanded to five measures might still be described as “having quadruple hypermeter,” except where one of its hyperbeats occupies two notated measures rather than the usual one. In that specific sense, the hypermeter is a quality intrinsic to the structure of his basic phrase, and not an extrinsic grid-like measure of the periodic aspect of the actual sounding/notated measures (as it is understood GTTM). Grouping overlaps are those notes/groups which seem to have dual group memberships, but which do not bring with them a sense of distorting the otherwise regular metric underpinning. Elisions are those notes/groups with dual group membership that do bring with them a distortion of the otherwise regular metric underpinning. In conventional examples, phrase expansion is easy to spot when it results in a perturbation to (hyper)meter (or to the expected matching proportion between a phrase and its basic phrase prototype). Examining these examples as I have done—with careful description of the conditions which give rise to grouping structures of various kinds and their associated phrase-rhythmic effects— 39 invites the question: just how essential are grid-like meter and “presumed regularity” for the description of these phrase-rhythmic effects? The following section puts a finer point on this question by arguing that PRiTM’s general presumption of grid-like regularity leads to an untenable ontological position in relation to analysis of irregular phrase lengths. This limits the applicability of Rothstein’s theory in a way that would not be necessary if its emphasis were shifted away from (hypermetric) regularity and toward the grouping conditions that afford the phrase-rhythmic effects. Generalizing these phrase-rhythmic effects becomes difficult when the difference between elision and grouping overlap hinges on locating moments of disturbances to otherwise regular periodicities (i.e., metrical quantities). The same problem pertains to determination of whether an irregular phrase is expanded. But elision, overlap, and expansion are open to generalization if we understand that phrase-rhythmic effects involve expectations of certain metrical qualities of time spans, rather than their quantities. These qualities are informed by various kinds of groupings which are independent of (if often correlated to) hierarchical periodic structures. 1.3 Phrase Structure and “Irregularity” 1.3.1 PRiTM’s Hypermeter, and its Limits Across many theoretical discussions of meter and phrase structure, theorists tend to draw on two distinct senses of the term “irregular.” On the one hand, it is taken to mean something like aperiodic (e.g., featuring unequal phrase-group lengths, irregular accentual or durational patterns); and on the other, it connotes “atypical” or “non-normative” (e.g., accounts where phrase structural irregularity is conveyed as “departures from” or “deformations of” a prototype). These two meanings of musical irregularity pertain to two distinct kinds of measuredness: one measuring musical activity according to whether it sustains 40 equidistant pulsation, and the other measuring the sounding music against some prototypical structure defined by inter-opus stylistic norms (and their “typical” lengths). For example, David Temperley invokes the former sense of irregularity and measuredness in his description of Bartók’s Mikrokosmos, vol. 5 no. 133 “Syncopation,” writing that “while the eighth-note pulse … is clear enough, any higher levels are, at best, highly irregular” (Temperley 2001, 300). That is, the regular eighth-note pulses are grouped in way that does not likewise suggest a periodic series of (slower) pulses. Charles Rosen invokes the latter sense of irregularity and measuredness when he describes the “eccentric five-bar rhythmic structure” in the introductory phrases of Mozart’s String Quintet K.515, where phrase structure reverts “to the more orthodox four-bar groups” only later on in the movement (Rosen [2003] 2016, 5). Here, the “orthodoxy” of four-bar groups is a norm that is not established by Mozart’s K.515 theme, itself. Rather, Rosen is drawing on a normativity that appeals either to what is typically done (among Mozart’s contemporaries, or some other corpus), or to a blanket assertion that any nested duple organization is de facto more “natural” than a quintuple organization (the starfish would like a word!). The dual-aspect deployment of “irregularity” within prevailing metric and phrase-rhythmic frameworks poses a considerable challenge for describing music that is regularly irregular: that is, any music that is typically aperiodic (e.g., much of Bartók), or music that is periodic in an atypical way (e.g., the Mozart K.311 Menuetto’s <5+5>, as described in the previous section). First, this is because most work on phrase rhythm has been conducted on music where it is assumed that aperiodic phrases are atypical. As a result, it is difficult to know how or whether these theories could be extended to other musics where this is not the case. Second, this is because failure to distinguish “atypical” from “aperiodic” irregularity can result 41 in an almost imperceptible drift of meaning from one sense to the other. Nuanced arguments about phrase rhythm and phrase-rhythmic effects will require precise language about expectations, periodicities, and to establish the basis by which some form or gesture is deemed typical or atypical. The terminological drift common to uses of “irregular,” then, is a problem precisely because “periodic” and “typical” are overlapping but non-equivalent concepts in common-practice repertoires; and because—relative to a wider corpus of musical styles where uniform periodic structures are not universally expected or presumed—the concept of “typical” phrase length/organization is undefined. Rothstein’s PRiTM aims to occupy a middle position in its attitude toward so-called “irregularity” of phrase rhythm. The following extended quote provides a helpful summary of his outlook on this issue. Most theorists, whatever their conceptual framework and vocabulary, have recognized that phrases of two, four, eight, and sixteen measures enjoy a privileged status in tonal music. There is no general agreement as to why this should be so. Some might think it merely a statistical phenomenon; these phrase lengths simply occur more often than others. At the other extreme, authors such as Riemann and Prout have held that all phrases of non-duple length—those of other than two, four, eight, or sixteen measures—can best by understood as variations of “normal” phrases of duple length. … There is also a middle position on this question, and—like many middle positions—it is the most sensible one. It was first advanced by the 18th-century theorists Riepel, Kirnberger, and Koch. All three believed that duple phrase lengths are the “best” or “most natural.” All believed, nonetheless, that phrases of other lengths may be used to good effect. Some of these non-duple phrases may be produced by modifying regular (that is, duple) phrases in various ways; others however, cannot be so produced and must be considered as irregular phrases independent of duple models. Analytically, too—especially for Koch—many but not all non-duple phrases may be traced to duple origins. (PRiTM, 33) 42 There are two ways that Rothstein’s “middle position” represents a flexible approach that can accommodate certain non-duple phrase structures. Rothstein does see regularity (i.e., periodic duple organization) as, generally, the inter-opus norm, but considers also how the sounding phrase refers to—and may depart from—the local “norm” provided by the prototypical structure that is his “basic phrase” (PRiTM, 64). To the extent a “basic phrase” is defined on an intra-opus basis, the theory can accommodate aperiodic structures even as the bias toward duple organization pulls in the other direction. Allowing for the possibility of non-duple basic phrases goes against the grain of Riemann and of Lerdahl and Jackendoff, whose theories make regular duple construction a more basic requirement. This is a notable, if somewhat underdeveloped feature of PRiTM. I say it is underdeveloped only because in most of PRiTM’s examples, Rothstein’s hypermetric structures resemble the well-formed metrical structures from GTTM.27 Conceding that basic phrases may be non-duple introduces complicating repercussions for the theory, especially since—within the context of presumptive periodic structure—identifying instances of phrase-rhythmic effects (i.e., elision, written-out deceleration, and internal phrase expansion) is often done simply by scanning for the (hyper)metric irregularity with which they coincide. The limit of this theory comes into view in two circumstances: [1] where music is stylistically idiosyncratic enough that it becomes untenable to refer each of its segments to a well-defined basic phrase; and [2] where music presents consecutive segments whose basic phrases are differing lengths. Only a modest degree of irregularity (aperiodicity) is needed for 27 More recent studies of irregular phrase lengths and hypermeter have likewise allowed the possibility of normative non-duple hypermeters: for example, Ng (2018) “On the Oddness of Brahms’s Five-Measure Phrases,” and Murphy (2012) “Septimal Time in an Early Finale of Haydn.” 43 these complications to come into view, as already previewed by question of whether there are one or two instances of expansion in the excerpt from Mozart K.331’s, Example 1-6. Now, I provide two further examples: first briefly revisiting and expanding on Rothstein’s example from Beethoven op. 2 no. 2 (PRiTM 53–54), and second, summarizing Rothstein’s account of quintuple hypermetric structure in Mozart K.515 (PRiTM 55–56). Consider for a second time Rothstein’s example from Beethoven op. 2 no. 2—discussed earlier (in section 1.2.2) as a straightforward elision, and now reconsidered (in Example 1-7) as a complex example of phrase-rhythmic irregularity. Rothstein commences his analysis starting in m. 21, which is treated as a hypermetric downbeat in a two-measure hypermeter. The elision in m. 32 is noted when its anticipated hyperbeat 2 becomes actual hyperbeat 1 for the subsequent material. Another way of putting it is that Rothstein’s two-measure duple hypermeter imagines the passage as if Beethoven had notated it in 4/4 or 2/2 rather than in 2/4 barring. In that regard, the elision, or—metric reinterpretation, in Rothstein’s terms—is the moment where the 4/4 (or 2/2 time) signature is resumed after one irregular measure (of either 5/4 or 1/4). I do not disagree that the feeling of reversal is a key part of this particular example of elision: however, describing it solely in terms of the interruption to preceding hypermetric regularity is not entirely satisfactory. First, as I have already argued, it overstates the periodic requirement and understates the relevance of the basic phrase and its groupings toward realizing the phrase-rhythmic effect. Second, the two-measure hypermetric structure he posits (while convenient for describing the interruption to the half-note pulse stream that co-occurs with the overlap) lacks explanatory power for other interesting aspects of phrase rhythm in the surrounding music. 44 To illustrate, my Example 1-7 starts by extending Rothstein’s duple hypermeter (backward) to begin from the first measure of the movement. The “1–2 count” is shown as boxed numerals in black text immediately below the staff, where they are grouped by two-measure brackets. Below that, I have added boxed numerals in blue text as a hypermetric count that is grouped into longer and more irregular spans based on textural similarities and motivic parallelisms. For instance, when I think about of mm. 1–8, I cannot help but conceive of its introduction phrase as two four-measure spans defined by the explosive two-measure gesture (twice), and the downward arpeggio, respectively. One may or may not wish to call that 4+4 structure “hypermeter,” but either way the whole-note period these groups propose acquires relevance for the grouping and metric evaluation of the music that immediately follows. My blue numerals and brackets below the second system illustrate the result of my holding on to the four-measure structure from mm. 1–8 as a “measure” for what unfolds in mm. 9–16 and following. At first, the music in m. 9 and following does nothing to contradict the four-measure organization. From mm. 13ff. a contrapuntal “echo” of m. 11 is exchanged between right- and left-hand parts, as the invertible counterpoint every two measures, slightly attenuating my sense of four-measure groups after the downbeat of measure 13. The quasi-hypermetric numbering and bracketing in Example 1-7 illustrate this by extending the “four-measure” group begun in m. 9 beyond m. 13, where mm. 13–14 and 15–16 reiterate the material from hyperbeats 3 and 4. When a new texture presents itself along with the initiation of the cadential progression in m. 17, the latent sensation of the double-whole-note pulsation of four-measure groups returns to the foreground; suggesting a division of the theme into two segments—two and one double-whole in length, respectively. Next, m. 21 reuses the figure from m. 1 as a sort of punctuation 45 mark between the phrases of mm. 9–20 and 23ff., surprisingly heard alone rather than as part of pair as in mm. 1–4. The sense of interruption caused by mm. 21–22’s intervening fragment pertains to the double-whole-note pulsation established in mm. 1–20 and cannot be described as an interruption of a two-measure hypermeter (or whole-note pulse). Importantly, the interruption to the double-whole pulsation at m. 21 is not an elision. (Whether it might be described as a “metric reinterpretation” is a slightly different question, but beyond my present purposes.) Example 1-7: Elision in Beethoven op. 2 no. 2, again (now including mm. 1–20) The music of mm. 1–30 nominates the double-whole-note as a relevant value by which to measure certain aspects of phrase-rhythmic action. Hearing according to these four-measure { 9 11 13 15 17 19 21& ### . . . . . . . . # # #3?###Phrase 1 (12 measures)11~~~22∑13241"3"2"4"1"3"2"4"11221324Intervening Fragment (2 measures)1122œ œ œ œ œœ œ ‰œ œœ œ œ ™ œjœ œ œ œJ ‰œj ‰ ŒŒ ‰ œœ œœ œœ œœ œ œœ œ ™ œjœ œ œ œ œ ™œ œ œ œœJ œ œœœ œ œ œ œ œœœ œ œ œ œœœ œœœ œœœ ‰œœJ œœ ‰ œ œ œ œ œ ‰‰ œœœœ Œ ‰ œ œ œ œJ œ œ œ œ ™ œJ œ œ œ œ œ œ œ œ ˙œ œ œ œ ˙œ œ œ œ œ œ œ œ œœ œ œ ‰ œJ œ ‰ œ œ œ œ œ ‰{ 23 25 27 29 32 34& ### ∑ ∑ ∑ . . . ∑ . . . . .3 3 3 3œ œ œ œ .œ .œ .œ .œ?###//Phrase 2 (10 measures)11. .22. . . .13. .241"3"2"4"1"3"2"4"1"5"..Phrase 3 2 = 1"6" = 12213. .24. . . . . .33 3333œ œ‰œœjœœ œœ œœ œ œœ œ ™œ œ œj œœ œ œ ŒŒ ‰‰ œjœJ œ œœ œ œJ œœ œ œ œ œœ œœœœ œœœ Œ œ œ œ œ œ œ œœ œ œ œ œ œ œ œ œ œœJœ œ œ œ œœ œ œ œ œ œ œ ™ œJ œ œ œ œ œ œ œ œ œ œ œ œœ ™ œJ œ œ œ œ œ œ œ œ œ œ œ œœ œ œ œ œ œ œ œ œ{ 1 3 5 72424& ### p . . . . . . . .?###Introduction (8 measures)1122132411. . .22. .13. .24.œ œœœJ œœ ‰ œ œ œ œ œ ‰œœJœœ ‰ œ œ œ œ œ ‰œœJœœ œœ œœ œœ œœ œ œ ‰œJ œ ‰ œ œ œ œ œ ‰ œJ œ ‰ œ œ œ œ œ ‰œJ œ œ œ œ œ œ œ ‰2 46 groups affords more complex sensations of phrase rhythm than an evaluation according to two-measure spans can illuminate. Whether the elision at m. 32 can be described in terms of the prospective four-measure structure is worthy of consideration: it would be ideal if the elision could (still) be described in terms of the prospective hypermetric groups that have the best explanatory power for the rest of the passage. The music in mm. 23–30 closely parallels phrase 1’s material from mm. 9ff., including the sense of an “extended” four-measure group that results from the whole-note period created by the invertible counterpoint echo of mm. 25–30 as in mm. 11–16. Phrase 2’s m. 31 breaks this parallelism when it begins a swift cadential progression that reaches its tonic arrival in the very next measure (whereas the first phrase stretches its cadential progression across four measures to sustain the double-whole-note pulse). My boxed count in blue text captures this sensation by extending a single group from mm. 23–32, conveyed as six measures of material expanded to ten measures, by repetition. Conceptualizing the phrase-rhythm of this passage in terms of the four-measure, ten-measure, and 12-measure groups that my Example 1-7 proposes would, in one respect, characterize the elision/metric interpretation at m. 32 as taking place on a “lower” level of structure than that to which the basic phrases refer (i.e., the four-measure group, subject to expansion of various kinds). In that respect, the elision at m. 32 is cast as being “weaker” than it might have been, had it occurred in a context that was uniformly periodic at a “higher” hypermetric level. My intuition, though, is that expansions and irregularities of this kind, occurring prior to the moment of elision, can in certain circumstances heighten the effect rather than diminish it. This is because the delay in reaching the phrase’s concluding event raises the bar as to what kind of “arrival” will satisfactorily conclude the phrase. I will save further critique of what I see as GTTM and PRiTM’s overdependence on regularity as a criterion for 47 elision for later in this chapter (where I will also demonstrate some elisions in “irregular” musics that would be invisible to GTTM’s “metrical deletion” and PRiTM’s “metrical reinterpretations”), and in a broader discussion of regularity and schema in Chapter 2. 1.3.2 Hyperbeat as Quality of the Basic Phrase vs. Hyperbeat as Measure for the Sounding/Notated Phrase Problematizing the elision in Beethoven op. 2 no. 2 (as in Example 1-7 and surrounding discussion) highlights the outsized role metric regularity serves for PRiTM’s evaluation of phrase-rhythmic effects, when its “basic phrase” could allow more flexibility in that regard. Rothstein shows off the application of his theory to non-duple phrase lengths when he argues for an instance of metric reinterpretation (my elision) in the five-measure phrases in the first movement of Mozart's string quintet K.515 (PRiTM, 55–6). (These are the same phrases Rosen deems “eccentric” according to his description that characterizes them relative to prototypical four-measuredness which he takes to be a global norm for the Classical style (Rosen [2003] 2016, 5)). Rothstein’s analysis understands the recurring quintuple periodicity of phrase lengths in the K.515 example not merely as an aspect of grouping, but as establishing a period regular enough to establish quintuple hypermetric structure. This is notable because the five-measure (hyper)metric structure he describes is not one that GTTM would recognize as metric. The metrical reinterpretation he observes occurs when m. 15—the expected fifth hyperbeat of Rothstein’s third hypermeasure—becomes the first hyperbeat of the fourth (see his ex 2.30, PRiTM 55–56).28 28 I have not reproduced the example here—in part because I do not share Rothstein’s view that m. 15 of the movement is in fact an elision. However, even without the example, the general description of 48 The implication is that five-measure hypermeter, which is an irregular 2+3 (or 3+2) at one level, does not prevent the phrase-rhythmic effect of metric reinterpretation/elision because the moment of overlap causes a disturbance to the otherwise regular five-measure periodicity. This, again, raises interesting questions as to the relationship between Rothstein’s “basic phrase” and his “hypermeter”—with significant bearing on how his approach to phrase rhythm might be generalized. Presumably he takes the five-measure phrases in the Mozart K.515 example to be an instance of a (relatively rare) non-duple basic phrase—otherwise he would not describe it as a five-beat hypermeter, but instead, an essentially quadruple hypermeter expanded to occupy five measures. The problems associated with the terminological drift between “irregular as atypical” and “irregular as aperiodic” are at play here. Trying to untangle them reveals certain inconsistencies in Rothstein’s theory of phrase rhythm; specifically, as to whether hypermetric structure is intrinsic to the basic phrase whose uniform hyperbeats may be as “stretched” or “compressed” to occupying a different number of measures in the actual music; or whether hypermeter is an extrinsic structure: that is, a measure that provided by some conceptually prior “allowable” pattern of hierarchic pulsation (or strong/weak alternation) against which the music’s durational patterns are tested for “fit.”29 The series of (abstract) phrase-structure diagrams in Example 1-8 present a thought experiment designed to illustrate the nature of this problem. Each of them represents a passage this effect is illuminating toward a better understanding Rothstein’s conception of what a basic phrase is, its relationship to hypermetric structure, and what kinds of phrases may be called “irregular.” 29 A recent paper by Jarvis and Peterson (2019) offers an approach to phrase-expansion that de-emphasizes the concept of “extraneous” measures, which Rothstein’s expansions rely on. Treating examples by Mendelssohn, they account for expansion using the metaphor of “alternative paths” which “offers a flexible and sensitive approach that allows the analyst to describe the passage’s unique characteristics without conceptually removing any of its content” (idem, 212). 49 of music consisting of two complete segments and the beginning of a third, labelled X, Y, and Z, respectively. These bracketed phrase segments enclose boxed numbers (italic) which represent the measures occupied by each phrase (10 or 11 measures each). Beneath the measure numbers and within the lower bracket enclosing each phrase segment, I show a cyclic count of hyperbeats intrinsic to each segment’s basic phrase. Because some of these phrases represent music elongated via expansion techniques, the number of actual measures for a segment may be greater than the number of its basic phrase’s hyperbeats. Finally, below each bracketed phrase segment, numbers boxed in dashed lines offer a (different) hypermetric count extrinsic to the basic phrase: that is, a count that naïvely measures from segment boundary to segment boundary according to cycles of as uniform a period as possible (Essentially this is like GTTM’s metric structure and the grouping preference rule that parallel groups should receive the same metrical interpretation. The exception here, though, is that my phrases E and F involve hypermetric counts that do not correspond to GTTM’s well-formed meter). The diagrammed phrase in Example 1-8A includes neither elision nor expansion, showing there is no difference between the intrinsic and extrinsic conception of hypermeter for a phrase of its kind. In Example 1-8’s phrase B there is likewise no hypermetric irregularity internal to the basic phrase units for segments X, Y, or Z. However, a hypermetric irregularity is found to occur according to the extrinsic hypermetric count in phrase B when an elision conveys the fourth hyperbeat of segment Y as also being the first hyperbeat of segment Z. GTTM would say Y’s (hyper)beat 4 is “deleted.” PRiTM would say Y’s hyperbeat 4 is, retrospectively, reinterpreted (during Z) as hyperbeat 1 for Z. My Example 1-8 marks (extrinsic) hypermetric irregularities like these with a double exclamation point. 50 Example 1-8: When is Rothstein’s Hypermeter Extrinsic to the Basic Phrase? (an elision thought experiment) In phrase C (loosely based on the Beethoven op. 2, no 2 example discussed earlier), it appears as if the PRiTM-style approach sets aside the close correspondence between the hypermetric structure of the basic phrase and the extrinsic hypermetric count as was observed in phrases A and B. Phrase C’s segments X and Y both refer to four-measure basic phrase units, though segment Y is expanded to six notated measures. When the end of Y is elided with segmentation according to basic phrasenotated measures 1 2 3 4 5 6 7 8 9 10 …hypermetric count intrinsic to basic phrase 1 2 3 4 1 2 3 4 1 2 …hypermetric count extrinsic to basic phrase 1 2 3 4 1 2 3 4 1 2 …segmentation according to basic phrasenotated measures 1 2 3 4 5 6 7 8 9 …hypermetric count intrinsic to basic phrase 1 2 3 4 1 2 3 41 2 …hypermetric count extrinsic to basic phrase 1 2 3 4 1 2 3 1 2 …segmentation according to basic phrasenotated measures 1 2 3 4 5 6 7 8 9 10 …hypermetric count intrinsic to basic phrase 1 2 3 4 1 2 3 – 4 –1 2 …hypermetric count extrinsic to basic phrase 1 2 1 2 1 2 1 2 1 1 2 …1 2 3 4 1 2 3 4 1 1 2 …segmentation according to basic phrasenotated measures 1 2 3 4 5 6 7 8 9 10 …hypermetric count intrinsic to basic phrase 1 2 3 4 1 2 3 – 41 2 …hypermetric count extrinsic to basic phrase 1 2 3 4 1 2 3 4 1 2 …segmentation according to basic phrasenotated measures 1 2 3 4 5 6 7 8 9 10 11 …hypermetric count intrinsic to basic phrase 1 2 3 4 5 1 2 3 4 51 2 …hypermetric count extrinsic to basic phrase 1 2 3 4 5 1 2 3 4 1 2 …segmentation according to basic phrasenotated measures 1 2 3 4 5 6 7 8 9 10 11 …hypermetric count intrinsic to basic phrase 1 2 3 4 – 1 2 3 – 41 2 …hypermetric count extrinsic to basic phrase 1 2 3 4 5 1 2 3 4 1 2 …A)B)C)D)X Y (expanded +2)X Y ZX YZ!!basic phrase lengths: <4, 4, ...>expansion: nometrical reinterpretation: noextrinsic hypermeter: quadruple (regular)(invented example)basic phrase lengths: <4, 4, ...>expansion: nometrical reinterpretation: yesextrinsic hypermeter: quadruple (irregular)e.g., Haydn Symphony 104(ex. 1-3)basic phrase lengths: <4, 4, ...>expansion: yes (Y +2)metrical reinterpretation: yesextrinsic hypermeter: duple (irregular)e.g., Beethoven op. 2 no. 2 (ex. 1-7; PRiTM p. 53)!!ZF)basic phrase lengths: <4, 4, ...>expansion: yes (X +1; Y +1)metrical reinterpretation yes?extrinsic hypermeter: quintuple? (irregular)X&Y resembles Mozart K.331 (ex. 1-6); but addition of Y/Z elision is an invented examplebasic phrase lengths: <4, 4, ...>expansion: yes (Y +1)metrical reinterpretation no?extrinsic hypermeter: quadruple (regular)e.g., Beethoven op. 2 no. 3 mm. 9ff. (PRiTM pp. 44–45)!!ZX (expanded +1) Y (expanded +1)!!ZX YX Y (expanded +1)ZE)basic phrase lengths: <5, 5, ...>expansion: nometrical reinterpretation yesextrinsic hypermeter: quintuple (irregular)e.g., Mozart Quintet K.515 mm. 6–15 (PRiTM p. 55) 51 the beginning of Z in the tenth measure of phrase C, it becomes difficult if convey the Y/Z overlap as a metrical reinterpretation in the sense of an interruption to otherwise regular (quadruple) hypermeter, since the quadruple structure has already been made irregular as of measure 9 (via the elongation of the basic phrase’s hyperbeat 3). To describe m. 10 as interrupting the regular period of a quadruple hypermeter would be to invoke a structure that is conceptually distinct from the quadruple structure that is subject to expansion in segment Y. Example 1-8C supplies a duple (extrinsic) hypermeter beneath the basic phrase diagram to match the way Rothstein accounts for metric reinterpretation in an analogous phrase (from Beethoven op. 2 no. 2). The implications of this intrinsic/extrinsic problem may not yet seem all that consequential, since a reasonable reading of passages like Example 1-8C might conclude that the metric reinterpretation (or elision) in such passages simply involve a “strong/weak reversal” at the two-bar level, even if the four-measure hypermeter is otherwise the relevant unit of measure for phrase structure. But the consequences of this problem come into sharper view with a hypothetical phrase structure as illustrated in Example 1-8’s phrase D.30 Like phrase C, phrase D expands its segment Y and overlaps the last measure of basic phrase Y with the first of basic phrase Z. Unlike phrase C, phrase D’s expansion of Y lengthens it by only one measure (as opposed to two). By recognizing phrase D’s segment Y as being subject to expansion, a PRiTM-style analysis describes hyperbeats intrinsic to the basic phrase as being 30 While I am discussion this scenario as a hypothetical, I am doing so to avoid making this chapter unnecessarily long, not because examples of this kind cannot be found easily found in common practice repertoire. One example analogous to the structure my Example 1-8D is found in of Beethoven piano sonata op. 2 no. 3 mm. 6–13; cited in PRiTM as an instance of grouping overlap (PRiTM, 44–45). 52 expanded to occupy two notated measures, e.g., Y as <1, 2, 3 –, 4>. My diagramed phrase D shows an elision (or metrical reinterpretation) in the phrase’s ninth measure when Y’s hyperbeat 4 serves also as Z’s hyperbeat 1. But when Rothstein analyses an example with this structure, he describes it as being uniformly quadruple (see his ex. 2.20, PRiTM, 44–45). In declining to recognize, as elsewhere, that a “4=1” metric reinterpretation has occurred, Rothstein pivots to an (GTTM-style) account of hypermetrical structure extrinsic to the hypermetrical structure proposed by the basic phrase. The PRiTM-informed account of phrases A, B, and C involves a balanced awareness of both intrinsic and extrinsic hypermetric structures when assessing whether expansion and/or metric reinterpretation has occurred. But by not recognizing elision as occurring in instances like Example 1-8D, Rothstein allows extrinsic and uniformly periodic hypermetric structure to supersede the intrinsic hypermetrical structure common to his account of expansion techniques. The absence of commentary on this problem undercuts the strength of his theory as it is presented in PRiTM, and leaves questions as to whether or not certain phrase-rhythmic effects can occur when certain kinds of irregularities are present.31 Namely: [1] can phrases that are made irregular by expansion still afford phrase-rhythmic effects such as elision? Rothstein’s examples would suggest that when duple basic phrases are expanded by an even number of measures, the answer is yes (Example 1-8C), but when a duple basic phrase is expanded by an 31 I do not wish to unfairly criticize PRiTM for this, since Rothstein expressly intended it as a non-technical introduction to phrase rhythm for a non-specialist audience. Yet his treatment does raise these interesting and difficult problems. What I mean is that the fact these complications have, in my view, not been adequately worked out in the wider music theory literature is part of what imposes an unnecessary limit on the wider application PRiTM’s general concept. 53 odd number of measures, the answer is no (Example 1-8D).32 And [2] can non-duple basic phrases be subject to expansion techniques? If yes, according to what basis could an interpreter reach a decision about whether the basic phrase to which it refers is non-duple? Example 1-8E (analogous to the previously mentioned example of Mozart Quintet K.515 cited by Rosen and by Rothstein) provides an illustration of a quintuple hypermeter conveyed as being regular enough to afford a metrical reinterpretation in the tenth measure, while Example 1-8F imagines a passage of exactly the same proportions, but whose component five-measure phrases are understood as being an expanded version of four-measure prototypes. It is unclear—based on Example 1-8A through 1-8E—whether an approach informed by PRiTM would or could invoke an extrinsic quintuple periodicity to recognize the tenth measure of phrase F as a metrical reinterpretation (my elision), since it would rely on responding to irregularity that disturbs a quintuple periodicity, but for music whose basic phrases (and hypermeter) are understood to be quadruple, not quintuple. When elision is understood as the special kind of overlap that occurs only where there is a disturbance to the periodicity defined by measure-groups of exactly uniform (notated) length (i.e., PRiTM’s reinterpretation, GTTM’s deletion), the definition all but eliminates the possibility of that phrase-rhythmic effect’s occurrence in music where segments are regularly subject to expansion, or in music where phrase segments are otherwise aperiodic. In my 32 Joel Lester’s Rhythms of Tonal Music includes a similar illustration, where he wonders whether non-duple phrase units may be overlapped/elided like duple segments are. He provides an example from Haydn where eight-measure phrase segments are overlapped to cause an irregular sequence of seven- and eight-measure spans. Then, with an example from Mozart, Lester shows that a series of overlapped five-measure structures results in a sequence of timespans that is commonly described as conforming to quadruple organization, despite the fact that—if understood by the same logic as the Haydn example—it would be better understood as a chain of (elided) five-measure units (Lester 1986, 188–89). 54 estimation, though, Rothstein’s occasional pivots from intrinsic to extrinsic metrical structure undercuts what is potentially the most powerful aspect of his theory of phrase rhythm: that the structural meter-like qualia at the phrase level (his [intrinsic] hypermeter) are informed primarily by factors other than their equidistance in time or their occupying an identical number of notated measures. A major complication that arises from understanding irregular phrase rhythm vis-à-vis extrinsic hypermeter and/or prototypes, is that—since an irregular passage may not always display allegiance to some ideal against which it is to be judged—it becomes nearly impossible to determine how and when we would know whether the comparison of some passage to some prototype is inappropriate. The extent of this problem increases exponentially as a piece becomes more and more “irregular” or idiosyncratic. For instance, although Rothstein accepts the possibility of non-duple phrase lengths, at what point would one know for certain that the basic phrase is indeed quintuple, and not the result of an expanded quadruple basic phrase (e.g., Example 1-8’s phrases E and F)? Additionally, can atypical, non-duple basic phrases be subject to expansion techniques? On what bases could an interpreter ever be certain that six notated measures of music are a one-bar expansion of a five-bar basic phrase, and not a two-bar expansion of a four-bar basic phrase? 1.3.3 Whither Phrase Rhythm in “Irregular” Musics? Considering the many virtues of the concept of “phrase rhythm” in the analysis of temporally regular tonal music, it seems worth considering whether it can be applied to irregular music such as is frequently found in the post-tonal repertoire of the early 20th century (and sometimes found within the common practice, where such irregularity tends to be described as rather extreme deformations of “regular” basic phrases to which “most” music of 55 the period more obviously refers). This runs up against a definitional problem. To the extent that we accept certain constraints for what it means to be metric (such as overt reference to metrically regular prototypes, well-formedness rules, standard patterns of “strong / weak” alternation, or tempo thresholds for entraining pulse) we must—at a certain point—regard irregular music as lacking metric structure and therefore, according to Rothstein’s definition, lacking “phrase rhythm” (since phrase rhythm is the interaction between grouping and hypermetric structure). This would not be a problem if there were never a sense that idiosyncratic and irregular (i.e., aperiodic) phrase structures are “expanded,” “overlapped,” or subject to other phrase-rhythmic effects. I am convinced however that many such phrases do exist, and so a theory that can describe phrase-rhythmic effects without reference to a (hyper)metrically regular structure is required. With the examples that follow, I will show that both elision and expansion can be heard in musics whose “irregularity” prevents well-formed meter from being established, and which do not appear to refer to normative phrase lengths. To do this, I argue for a definition of phrase rhythm that reconceives Rothstein’s formulation—that phrase rhythm is the co-presence of, and interaction between, grouping structure and hypermetric structure—like this: Generalized phrase rhythm is the co-presence of, and the interaction between, two or more different kinds of grouping that are simultaneously proposed by different aspects of the same musical signal. In what follows I will refer to this concept as “phrase rhythm as grouping complex.” 56 1.4 Phrase-rhythmic Effects without (hyper)Meter (and without “basic phrase”) 1.4.1 A Less Metrical Account of Grouping Overlap The conventionally tonal and metrical examples earlier in this chapter showed cases where a pitch event is understood to participate as the terminus of one melodic idea while also understood to initiate a new melodic idea, and leaving the metric underpinning undisturbed. It might be concluded that—in the absence of harmonic factors which helped determine the grouping overlaps in those tonal examples, and the absence of schematized phrase types (with their associated hypermetrical structure)—grouping overlaps could not arise. (At least not according to the same criteria.) Tenney and Polansky’s temporal gestalt segmentation (1980) offers a method for segmentation analysis that works well for defining melodic groups in all sorts of music: their algorithmic method of segmentation/grouping considers two kinds of proximity for musical events: pitch height proximity measured in semitones, and timepoint proximity measured in some durational unit. It then synthesizes the two into a weighted value that helps quantify segment boundaries based on these two kinds of proximity/distance. Likewise, GTTM appeals to gestalt principles—and even invokes a gestalt psychology metaphor to demonstrate why grouping overlaps might arise, albeit on an ad hoc basis (GTTM, 59). But a distinctive aspect of how grouping structure is discussed in GTTM and PRiTM (as opposed to Tenney and Polansky’s segmentation) is that GTTM and PRiTM consider grouping to be in a dynamic inter-relationship with other aspects of structure, whereas Tenney and Polansky are describing a (purposefully) naïve approach to segmentation, alone. It is largely the dynamic interaction among grouping and other structures (determined by varieties of event-affiliation such as meter, and motivic parallelism) that allows PRiTM and GTTM to recognize overlaps 57 where Tenney and Polansky—who focus only on boundaries/points of segmentation—do not.33 When considering whether a segment boundary exists between consecutive events t1 and t2, Tenney and Polansky consider the pitch-interval size and inter-onset interval (duration) between the events (weighting the pitch and duration intervals independently, using a context-specific method that is not detailed in their paper). In general, this naïve approach posits that in most cases big leaps and/or long pauses correlate to segment boundaries, and that hierarchical segmentations might be determined by this same method. As an example, their temporal gestalt analysis of the opening passage of Debussy’s Syrinx is reproduced in Example 1-9 below. It divides the first five measures of Syrinx into twelve “clangs” (comparable in scope to the “motivic units” I proposed in the Mozart K.331 analysis earlier in this chapter); and then a higher level of grouping structure collects those clangs into four discrete clang-groups called “sequences” (comparable to phrase segments proposed earlier in this chapter). The algorithmically determined phrase lengths (measured in eighth notes and ignoring times added by the fermata) for sequences 1 through 4 are decidedly aperiodic: 24, 15, 7, and 10, respectively. The beginning of a new clang is, in most cases, determined when an event is 33 Antares Boyle (2018) surveys music theory literature on segmentation (especially for post-tonal musics) and highlights the fact that Tenney and Polansky “note explicitly that in order to make their theoretical framework ‘operational’ in practice, they undertook ‘a shift in emphasis from the unifying effects of proximity and similarity to the segregative effects of temporal separation and dissimilarity’” (1980, 208). Boyle notes that other important theories of segmentation in post-tonal repertoires (e.g., Hasty 1981; 1984; Hanninen 2012) likewise end up adopting a position that foregrounds boundaries themselves the over and above the segments of music: a position she argues is problematic for musics which are known to avoid “crisp” boundaries (Boyle 2018, 38–39). 58 preceded by a pitch interval that is a local maximum.34 For the Debussy excerpt in Example 1-9 this formula determines all clang beginnings except for clang 5, which—despite being preceded by pitch interval 0—they designate as a new clang because of the particularly large inter-onset interval between the two consecutive B♭5s. Example 1-9: Tenney and Polansky’s Temporal Gestalt Segmentation Analysis of Debussy’s Syrinx mm. 1–5 I find many of the proposed segment boundaries from Tenney and Polansky’s segmentation analysis of Debussy’s Syrinx (reproduced in Example 1-9) easy to perceive in a real or imagined performance of the passage. Certainly, I can try to attend exclusively to pitch and durational proximity to guide my listening and perceive a structure exactly like the one in their diagram. However, my understanding of the piece is guided also by grouping structures which 34 Local maximum defined as such: given consecutive events <tn, tn+1, tn+2, …>, ti marks the beginning of a new clang iff |ti – ti-1|>|ti-1 – ti-2|⋀|ti – ti-1|> |ti+1 – ti|. [mf ]Très modérép3p534&bbbbbsequences:clangs:11 2- 3- 4- U ,&bbbbb526- 7- 83- -,9 10411333&bbbbb12 , œ ™ œn œn œb ™ œn œn œb œ œn œ œb ™ œ œb ˙bœ ™ œn œn œb ™ œn œn œb œ œn œ œ ™ œ# œn œnJ œ œ# œ œ œ œ fiœ# jœ œn œœ ™ œ# œn œn œb 59 are proposed by factors other than pitch and temporal proximity. Chief among these are: consideration of motivic parallelism, and the groupings suggested by my sense of “middle-ground” melodic structure (i.e., where some groups of pitches are understood to represent an ornamented version of a single pitch at a more fundamental level). It is these other aspects of grouping (by affiliation, rather than by segmentation (see Benjamin 1984, 358–61)) that, when put into dialogue with the segments proposed by Tenney and Polansky’s method, reveal a possible grouping overlap in the passage. This understanding introduces more than one kind of grouping structure into consideration which can be understood to be in dynamic (if abstract) relationship with one another independent of whether periodic metric structure is also present. For any method that nominates groups/segments as a result of identifying the boundaries that separate them, “grouping overlap” is inevitably non-syntactic and undefined. However, grouping overlaps are perfectly logical from the grouping-by-event-affiliation perspective whenever a reasonable argument can be made for the dual affiliation of the shared event. I suggest that one such argument can be made for the B♭5 that occurs on beat 1 of m. 4 in Syrinx. Because the passage does not feature clearly articulated harmonic rhythm or schematized phrase types, its grouping overlap is perhaps not as immediately apparent here as it was in the Clementi excerpt from Example 1-2. Further, the distinction PRiTM makes between overlaps that do not alter the underlying periodicity and those that do (i.e., his “grouping overlap” vs. “metrical reinterpretation”) is moot for an example like one, which does not establish regular periodic structures on either side of the prospective overlap. Despite this, I suggest that a clear argument for the grouping overlap can be made, though it requires a careful explanation of the different aspects of grouping structure that combine to create the effect. 60 My Example 1-10 replaces Tenney and Polansky’s numbered clangs with labeled brackets W, X, Y, and Z; and replaces their numbered sequences with labeled brackets A1, A2, and B. This change in labeling practice assists in reflecting motivic parallelisms that are outside the purview of their simple segmentation analysis: for example, my W and Wʹ for their clangs 1 and 4, and my A1 and A2 for their sequences 1 and 2. For mm. 1–2 (Example 1-10, first system) my groupings correspond exactly to the points of segmentation Tenney and Polansky propose according to their proximity/distance algorithm. To aid comparison, my Example 1-10 uses black brackets for groupings that corresponding exactly to one of their clangs or sequences. Where my groupings differ from their segmentation brackets appear in blue (with the Tenney and Polansky segments faintly reproduced in grey below the staff). My grouping analysis differs because in addition to consideration of proximity and distance, it considers also how the segments—as a series of inter-connected motivic units—begin to accrue syntactic relationships because of parallelism, departure-return gestures, and inference of a melodic substructure (of which the sounding pitches are an elaboration). In the case of phrase A1, motivic units W and X introduce an ornamented descending step from B♭5 to A♭5. The boundary between W and X is nominated because of its being a local maximum intervallic leap. Additionally, motive X emerges as a distinct low-level group because of its rhythmic and melodic parallelism to W. End of X/beginning of Y is determined by the same local-maximum interval criteria as between W and X. Unit Y is less parallel to X than X was to W, yet despite rhythmic and contour difference, a loose sense of parallelism is retained. This has to do with Unit Y’s initial pitch being one whole tone below the initial pitch of X (just as X began one whole tone below the initial pitch of W). As it unfolds, material of motivic unit Y accelerates the stepwise descent initiated by units W and X before leaping up back into the 61 original register (B♭5) for the return of the W motive, now elongated (Wʹ) via an additional half-note B♭5. Operating strictly on the basis of grouping by motivic parallelism, Wʹ could have ended with C♭6, with the half-note B♭5 suggesting a possible new W-motive—but when that prospective W is left incomplete (into the fermata), its half-note B♭5 is subsumed into the preceding (on the basis of gestalt segmentation). The slightly elongated motivic unit Wʹ traces a low-level departure-return gesture within its measure 2. On the broader scale of mm. 1–2, W and Wʹ bookend X and Y to enact a phrase-level departure-return from B♭5, which gesture—together with gestalt segmentation considerations—is the basis for my phrase-group A1. Example 1-10: Temporal Gestalt Segmentation with Grouping by Parallelisms and Event-affiliation in Debussy’s Syrinx, mm. 1–5 The subsequent downbeat B♭5 (m. 3, W) presents itself as a beginning because of its approximate parallelism with motivic unit Wʹ and its exact parallelism to the initial W unit. The entirety of m. 3 is exactly parallel to measure 1. The exactly parallel restatement of <W, X, Y> is the basis for understanding mm. 3ff. as phrase-group A2, expected to have similar [mf ]Très modérép3p534&bbbbbA1W X- Y- W'- U ,&bbbbbA2W X- Y- W''BZ1- -, Z2 Z3333&bbbbbW''' , œ ™ œn œn œb ™ œn œn œb œ œn œ œb ™ œ œb ˙bœ ™ œn œn œb ™ œn œn œb œ œn œ œ ™ œ# œn œnJ œ œ# œ œ œ œ fiœ# jœ œn œœ ™ œ# œn œn œb 62 structure to A1. The exact parallelism ends when B♭4 arrives (in place of A1’s B♭5) on the downbeat of notated measure 4. Some sense of approximate parallelism remains, though, since group A2 can still be understood to start and end on B♭, just as A1 had done. A sense of melodic closure in A2 is provided by the nearly complete scalar stepwise descent through the octave (skipping only a penultimate C♮5/C♭5). From a segmentation perspective, the appearance of B♭ in the lower octave here has implications for determining the end of A2’s segment Y. In A1, the end of Y was determined by a local maximum interval from D♭5 to B♭5. In group A2, the end of Y is less sharply defined, at least according to pitch proximity/distance. (Tenney and Polansky’s segment boundary is between B♭4 and F♯4 at start of m. 4, the local maximum interval size). I can focus my hearing on a novel gesture started from F♯4s, but when focusing my interpretation according to a search for emergent motivic parallelism(s), I instead prioritize the affiliation of <B♭4, F♯4, G♮4, … > as m. 4’s motivic unit Wʹʹ. The resulting extension of theme group A2 into m. 4 is a reflection of the perceived parallelism between A1 and A2, with Wʹʹ approximating Wʹ, down an octave. The end of Wʹʹ might have been extended one event longer, to include the tenuto B♮4 as an approximation of the Wʹ extension that includes the B♭5 at end of A1. Motivic units Z1, Z2 and Z3 are nominated based on hearing the approximate repetition of the <B♭5, F♯4> interval that began Wʹʹ, altered by an oblique wedge process that transposes the first pitch of the motive upward by a semitone upon each repetition. (The process ends after Z3’s F♯ appears an octave too high, where it is transformed into a grace-note ornamentation on E♮5.) Within the context of phrase A2 (mm. 3–4) the B♭4 of motivic unit Wʹʹ is a goal of the local directed motion descending through the octave. Additionally, the whole of unit Wʹʹ is affiliated with the preceding measure when it extends A2’s parallelism with A1 into measure 4: 63 <W, X, Y, Wʹ[ʹ]>. (In that respect the end of A2 might extend as far as the breath mark in m. 4, to the extent Z1’s B♮4 approximates the half-note B♭5 that concludes A1. The faint dashed line extending A2 in Example 1-10 illustrates). Within the emergent context of group A2 motivic unit Wʹʹ can be understood to serve as the model for the “wedge” motion of Z1–Z3. Additionally, unit Wʹʹ is a beginning for phrase B’s middleground departure-return gesture <B♭4, B♮4, C4, D♭4, B♭4>, faintly mirroring group A1’s whole-tone departure-return <B♭4, A♭4, G♭4, B♭5> but in a chromatic-step version with the inverse contour. To summarize, the argument for a grouping overlap on the downbeat of m. 4 is based on a simultaneous awareness of grouping boundaries proposed by moments of relative temporal/pitch-height distance (which recognizes a group ending after B♭4), and other grouping boundaries which are the result having defined groupings-by-event affiliation: in this case events affiliated because of their participation in parallel, and quasi-parallel departure-return gestures, and linear progressions (at the middleground level). Grouping overlap will not be recognized—in fact is undefined—when strictly considering groupings as discrete segmentation. The Syrinx example shows a grouping overlap to arise out of a dynamic negotiation between the kind of groupings determined by melodic segmentation (naïve, algorithmic), and determined by event-affiliation (context specific, emergent, open to interpretation). 1.4.2 A Less Metrical Account of Elision Earlier in this chapter I provided a definition of “elision” (comparable to GTTM’s metrical deletion and PRiTM’s metrical reinterpretation) as those instances of overlap where the timespan of a new phrase beginning is understood to replace the timespan of the ending (or expected ending) for a prior phrase. Rothstein recognizes that an elision has occurred by 64 identifying the irregularity it causes in the hypermetric structure. If we agree that “missing” or “overlapped” time is the crux of elision as phrase-rhythmic effect we might consider whether the sensation of “missing” or “overlapped” time necessarily correlates to and interruption or irregularity amid otherwise periodic metric structure (cf., Benjamin 1984, 395). The following example, excerpted from the Promenade theme of Mussorgsky’s Pictures at an Exhibition, demonstrates an elision in a less periodic context. As notated, with constantly varying time signatures, Mussorgsky’s tune does not obviously participate in any well-formed (hyper)metric structure that GTTM or PRiTM would recognize. Nevertheless, the notated downbeats take on a special status (independent of emphasis in performance) because of aspects of the music’s grouping structure(s). The first measure can be divided into three small units, labeled in Example 1-11 as motivic units a, b, and c. Measure 2 restates units a, b, and c but in reverse order (and interpolating an extra C♮4 between motivic units b and a). The approximate palindrome (thematic group X1) that results is eleven quarter-note beats long. Thematic group X1’s component parts, groups A and B, are not groups because each is a notated measure (as an extrinsic structure/pattern). Rather, they become audible groups because they are the component parts that participate in palindromic group X1. The immediate repetition of the same tune as X2 in mm. 3–4 (now with harmonization) proposes <A, B> alternation as an emergent local norm. Measures 5–6 (group X3) are an approximate echo of measures 1–2. The A/B alternation (now Aʹ and Bʹ) remains (A and B’s components being defined by their approximate parallelism to X1, rather than by a second instance palindromic structure per se). In mm. 7–8, X4 begins as if a literal restatement of X3, except the Bʹ portion of the group appears a minor third higher in m. 8 than it had in measure 65 6. This transposition loosens the parallelism between groups X3 and X4. From there on, each subsequent Xi grows increasingly distant from the model proposed by X1. In m. 9 group X5 begins with an Aʹʹ motive that approximates a T3 transposition of the Aʹ motive from X3 and X4 (transposition T3 related the B motive from X3 to X4). The Aʹʹ in X5 is further transformed as it—for the first time—is elongated to occupy six notated beats rather than five.35 Likewise, the Bʹʹ motive within X5 only loosely resembles prior B phrases. (Calling it “C” instead would not substantially change the details of my analysis here—the important thing about the latter half of X5 is that it is decidedly “not-A” material.) The sense of “missing” or “overlapped” time associated with elision occurs in mm. 11–12, but without the kind of disruption to a periodic metrical structure that PRiTM or GTTM typically requires. At first, m. 11 is exactly parallel to the Aʹʹ motive from mm. 9, as if to initiate a sixth iteration of an A-B group pair. But the bass line in m. 11 leaps upward one quarter note too early (relative to what it had done in m. 9), and at the same time the expected melodic pitch A♭4 (as would parallel the Aʹʹ from X5) is omitted. Next, on the downbeat of m. 12, the expected continuation of the group-pair—some variant of B—does not appear. Instead, m. 12 begins with a transposed restatement of the Aʹʹ material. This is the first time that two A-material groups have appeared in immediate succession. There are at least two ways in which grouping could suggest the material beginning in m. 12 as replacing the time-span of the end of the prior phrase. By m. 12, the bass-voice gesture that consists of one relatively large upward leap followed by a relatively small descent 35 The extra length in X5’s Aʹʹ could probably be described as an instance of phrase-rhythmic expansion, but I will restrict the discussion of the Mussorgsky example strictly to elision and save the less metrical account of expansion for the Bartók example in the subsequent section of this chapter. 66 (the orange boxes in Example 1-11) has become contextually associated with anacrustic function. One overt example that contributes to this accrual is felt in mm. 9–10 when the bass voice resolves like a functional 4/2 chord.36 The parallelism between m. 9 and m. 11 strongly suggests that the octave G♭s (end m. 11) could likewise descend by step onto F♮ as was done in measures 9–10. Example 1-11: Elision in the ‘Promenade’ from Mussorgsky’s Pictures at an Exhibition 36 In retrospect it is possible also to hear this anacrusis function beginning to accrue in the <F♮, B♭, G♮> motives in m. 4 and m. 6, where the G♮s fall on the strong part of the half-note pulse that emerges during those passages. Accrues Anacrustic FunctionAccrues Initiating Function{{{{1 2 3 45 6 7 89 1011 12 1354 64 54 6454 64 54 6454 64 54 6454 64 54 6454 6454 64&bbX1 (11 beat notated duration)Aa- -b-c- - Bc- -b- -a- -X2 (11 beat notated duration)A Bn?bb ∑ ∑&bbX3 (11 beat notated duration)A'- - - - -B'- -X4 (11 beat notated duration)A'- - - - -T3 of B'-?bb ∑ ∑&bbX5 (12 beat notated duration)A'' (approx. T3 of A', elongated)bbB''bbbbb?bb&bbX6? (incomplete)curtailed A'' (no beat 6)bb////approx. T4 of A'', (beat 6 restored) X6 (approx. T4 of X5)nT4 of B''n n?bbœ œ œ œ œ œ œ œ œ œ œ œ œ œœœ œœœ œœœ œœœ œ œœœ œœœœ œœœ œœœ œœœ œœœ œœœœœ œœ œœ œœ œœ œœ œœ œœ œœ œœnn œœœ œ œ œ œ œ œ œ œ œœ œœ œœ œ œ œ œ œ œ œ œ œ œ œb œbœb œœ œœ œb œŒ Œ œœœœ œœ œœ Œ Œœœbbœœbb œœ œœœœ˙ ™ œœ œœ ˙ ™œœ œœ œœ œœ œœœ œœœœ œœœœ œb œ œœœœ œ œœœ œœœ œ œ œ œ œœœœ œ œœœ œ œœœ˙˙bb œœ œœ œœ œœb œœ œœ œœbb œœ œœ œœbb œœ œœœœ˙ ™ œœ œœ œ ™ ‰œœ œœ œœ œœ œœ˙ ™ œœ œœ ˙ ™œœ œœ œœ œœ œœœ œœœœ œœœœ œ œ œœœœ œ œœœ œœœ œ œ œ œ œœœœ œ œœœ œ œœœ˙˙bb œœ œœ œœb j ‰˙˙bb œœ œœbb œœbœœbb œœ œœ œœ œœœœœœ œœ œœ 67 By this stage, and pertaining now to grouping in the right-hand part, the three-quarter-note gesture that initiates each of the A groups has appeared in various loosely parallel forms, and has accrued an affiliation with initiating function (highlighted by purple shaded boxes in Example 1-11). The surprise parallelism between m. 12 and m. 11 replaces the expected continuation to octave A♭4/A♭3 in the lead melody, while also replacing the bass voice’s expected arrival onto F♮3/F♮2. As a result, an elision could be described at two different structural levels. On a low level, the quarter-note C♮5 overlaps and replaces the time-span where a quarter-note A♭4 (end Aʹʹ) was expected. This could perhaps be felt as interrupting a regular half-note pulse stream beginning from the downbeat of measure 11.37 The elision I have in mind, though, occurs on a broader scale and cannot be described in conventional terms as a disruption to surrounding metric regularity. The repetition of an Aʹʹ gesture (m. 12) initiates a complete X6 group-pair that approximately transposes X5: the time-overlap occurs when the beginning of this full X6 replaces the timespan that would have been occupied by the latter half of the immediately preceding incomplete X6 (m. 11). To summarize, Rothstein and others characterize elision as a phrase-rhythmic effect dependent on metrical criteria. In the discussion surrounding Example 1-8 I problematized this view in order to set up for an approach to phrase-rhythm that is less tethered to the idea of “regularity” as a default. The GTTM/PRiTM-informed conception is that phrase rhythm is the 37 There are also good reasons, though, to hear the first three quarter-note beats of gesture A as a single dotted-half-note duration (and not initiating a half-note pulse stream). Alternatively, a listener might attend to the approximate half-note pulsation beginning from the second quarter-note beat of each A-gesture, according to which m. 12 is not a disruption at all. For these reasons describing the sense of elision at m. 12 solely in terms of its disruption to half-note (or dotted-whole note) pulsation is not entirely satisfactory: it is part of enhancing the effect, but other factors are involved in bringing it about. 68 interaction between grouping and (hyper)metric structure. I replace this with my notion of phrase rhythm as interaction among co-present grouping structures whether or not one of those structures is periodic enough to establish meter. By doing so I aim to recognize elision-like varieties of overlap which occur in musical passages whose aperiodicity does not afford the kind of meter that GTTM and PRiTM’s elisions (i.e., “metrical deletions” and “metrical reinterpretations, respectively) would require. The “elision” I describe (downbeat m. 12) is surrounded by music that does not feature a regular hierarchic (hyper)metric structure, so it is not possible to account for the elision by referring to the moment where an overlap disrupts the metric regularity.38 Instead, I suggest that the elision results from co-present grouping structures defined by motivic parallelisms in the lead melody, along with syntactic qualities accrued over the course of the passage as the result of perceived interaction and interrelation between right-hand, and left-hand groupings. A narration of co-present groupings (i.e., melodic groupings/segmentation, together with grouping by event-affiliation, accent, and motivic/rhythmic parallelism) such as I have offered here would also describe many of the elisions in the common-practice repertoire (where they do, oftentimes, correlate to interruptions to an otherwise periodic hypermetric structure). However, the correlation is not causation, and my argument here is that the sensation of missing time characteristic of elision can be established in the absence of regular (hyper)meter. 38 This claim is in regard to the theme as a whole, wherein notated five- and six-beat measures are presented in loose alternation. The elision (mm.11–12) is uniquely preceded by three consecutive measures of notated 6/4 whose slow dotted-whole-note pulse is interrupted by the elision at the downbeat of measure 12. My point is that there are more immediate grouping factors that establish a half-note pulse (and perhaps also a dotted whole-note pulse) during that same passage, independent of whether the notation conveniently points to the overlap in via its metric notation, as it does here. 69 1.4.3 A Less Metrical Account of Phrase Expansion Earlier in this chapter I summarized the way that Rothstein’s PRiTM accounts for expansion as a phrase-rhythmic effect by comparing a phrase (sounding, or notated) to a regular hypermeter and basic phrase prototype. In that account, expansion is found to occur when a regular hypermetrical pulse is made irregular (whether by insertion of redundant or extraneous material, or by a composed-out deceleration). One complication for the understanding of expansion as a kind of metrical phenomenon is that hypermeter often involves pulses slower than can be reliably entrained. That is, the equivalent (and presumably “equidistant”) hyperbeats are often separated by timespans at or beyond the ~5–6 second threshold that cognitive-psychology informed music theorists have identified as the (soft) limit for the human ability to automatically attend to equidistant pulse (London [2004] 2012, 27–30). So, to the extent that we nevertheless perceive phrase-rhythmic effects like expansion and elision as a quasi-metric phenomenon, it must be that they are informed by factors besides an awareness of the sustained/interrupted isochrony of the phrase’s prospective hyperbeats. Here, I claim that irregular music can nominate moments of phrase expansion in the absence of hypermetric regularity (i.e., isochrony among notated durations of component parts of the phrase), and in music that does not obviously refer to a (hyper)metrically regular basic phrase. The opening eighteen measures of Béla Bartók’s piano Etude op. 18 no. 3 (1918) will demonstrate (Example 1-12). The etude begins with an “out-of-time” introductory arpeggio that Bartók notates in uneven bars and without time-signature(s). In m. 5 an approximate restatement of the lower half of the initial arpeggio is transformed into a left-hand accompaniment figure in straight sixteenth-note rhythm. The four consecutive iterations of this motive (identified by bracket “a” 70 below the staff in Example 1-12) establish a regular dotted-quarter-note periodicity during measures 5–6. Around the same time, a repeated chord in the right-hand layer (partially) realizes a quarter-note pulse stream from end of m. 4 through measure 6. The pulses that the right-hand stream “leaves out” are the moments where its stream of pulsation would create a simultaneous attack with the left-hand’s dotted-quarter-note pulse stream. A second-level grouping based on the period of the least common multiple of these two streams results, as illustrated in Example 1-12 by the labeled brackets V1 and V2. The equal duration and parallelism of V1 and V2 in turn proposes a dotted-half-note pulse that corresponds to the notated measures 5 and 6. Any sense that V1 and V2 might introduce strict meter is short lived. In the very next measure Bartók instructs the performer to reach a steady (and blisteringly fast) tempo while also introducing new variety into both the left- and right-hand strata as the theme proper gets underway (mm. 7ff.). Measures 7–10 set the stage for an internal expansion via insertion/redundant repetition in measures 12–18. In the left-hand accompaniment of m. 7, accompaniment gesture b consists of an upward sweep (still loosely approximating the initial arpeggio from m. 1) followed by a legato descent in the second half of the notated 6/8 measure. Next, in m. 8, a less regular broken chord gesture—accompaniment gesture c—embeds a slower inner-voice melody into the sixteenth-note accompanimental flurry. Motive c’s new accompaniment texture lasts only until motive bʹ appears (m. 9) as a near transposition of gesture b from measure 7. Likewise, m. 10’s motive cʹ approximates a transposition of m. 8’s motive c. This four-element sequence establishes an expected alternation of accompaniment gestures <b, c> as a local norm, with exact durations of the segments variable. 71 Example 1-12: Phrase Expansion in Bartók’s Etude op. 18 no. 3 {p accel. poco a poco il tempo al12 3 4 5 66868&Rubatodolce.- .- . . . .œœ# œ#œ œ‹ œ#? fespr.∑mf&V1a aV2a aœœ#œ#œœ#œ# œ#wœ# Œ Ó ‰œœœœ###‰ Œ ‰œœœœ#‰ Œœœœ###‰œœœ‰ Œœœœ###‰œœœ‰œ œ#œ#œ#œ œ#œ œœœœ œœ œ#œ#œ#œ œ#œ œœœœ œ2{q. =1007 8 9 10 117868101678681016&1AContour Group 1. ..1B. . .Contour Group 22A. . .3AContour Group 3...3B”“. . .&W1b>c# # n b bW2b'> >>Nc'n #># n n?c'' = T-5(c')b b n b b‰œœœ###‰œœœ##‰œœœ#≈ Œœœœ##‰œœœn#n œœœnnn≈ ‰œœœ###≈œœœ#nn≈œœ##‰ ≈œœœœ#nœœœœn#‰œœœœ ≈ ≈œœœœnbœœœœn‰œœœœnnn≈œ œ#œ#œ#œ œ# œ#œnœ œn œ# œn œ œœnœ#œ œœœœ œœbœnœb œ œ œnœ#œn œ#œ# œ# œnœ# œœn œn œ ™ œœ# œœ#œnœ œ œœ#œ‹œ ™ œœb œœnœnœbœœnœ œ3{12 13 14 15916111678916111678&4AContour Group 4...4B...4C 4D. . .4E ?..?Xd>d'> ^d''> ^e ?^e' ?^œœœ# ≈ œœ‰œœœ##n≈œœœ# ≈ œœ‰œœœ##n≈œœœ# ‰œœœ##n≈œœœ ≈ ‰ Œœœœnn#œœœb Œœbœœn œn œœbœnœnœb œb œb œœn œ œœbœnœnœb œb œœn œnœbœnœnœnœnœb œn œœb œ#œbœnœ œnœ#œœn œ#œnœb œn4{16 17 185891658916&Contour Group 55A..5B. ..5C. ”“.. .?fZ>f>f>≈ œ#jœœœ#nj≈œj œœœbJ‰œnj œœœbbbJ≈œ#j œœœ#J≈œj œœœnbJ ‰œnj œœœbbbJ ≈œ#jœj œjœœœn#≈œœœb≈œœbbœb œœb œnœ œb œb œb œn œnœb œnœb œnœn œb œb œb œn œœb œnœbœbœnœn œb œb œ5 72 Meanwhile in the right-hand layer, mm. 7–8 outline a rise-fall gesture, labeled as Contour Group 1 in Example 1-12. Contour Group 1’s component halves (1A = <+, +>, 1B = <-, ->) loosely align their rising and falling with the duration of accompaniment groups gestures b and c, respectively. The perceived co-operation between the two strata, along with recognition of the near-periodic <b, c> alternation results in a second-level grouping of the accompaniment motives, labeled as W1 beneath measures 7–8. When m. 9’s right-hand material presents a three-chord rising contour (gesture 2A)—approximately equal to duration of accompaniment gesture bʹ—it is reasonable to assume that a parallel group W2 is underway. In accordance with this expectation, the left-hand proceeds into accompaniment gesture cʹ during measure 10. However, the right-hand layer replaces the expected “2B” descent <-, -> with a new contour-segment “peak” gesture (<+, -> labelled as 3A) which leaves contour group 2’s prospective rise-fall gesture incomplete. The sense that an expansion of some kind is underway is formed during mm. 11 and following. When m. 11’s accompaniment motive cʹʹ realizes an exact transposition (down five semitones) of the immediately preceding cʹ, it breaks from the regular (and loosely periodic) alternation of accompaniment gestures, <b, c>, established earlier. The back-to-back presentation of c material in mm. 10–11 means that the succession of W-series groups does not continue in close parallel, when no novel group (W3?) is begun at measure 11. Instead, exact pitch-class transposition that relates gestures cʹ and cʹʹ strongly affiliates measure 10 with measure 11 as a new group N. And because cʹ was continuative of W2, so too is cʹʹ and its enclosing group N, as visually suggested by the open-ended dashed bracket for W2 in Example 1-12. During mm. 10–11 (left-hand group N) the right-hand layer offers twofold presentation of its new “peak” contour-segment <+, ->, labeled 3A and 3B. Together 3A and 3B affiliate the 73 right-hand segments of m. 10 and m. 11 with one another (as Contour Group 3). Individually, they recall the slower and bigger arc-gesture from Contour Group 1 (and perhaps also draw attention to the conspicuous incompleteness of the arc-gesture in Contour Group 2). These grouping considerations contribute to the sense that W2 and the end of Contour Group 2 are drawn out beyond their expected proportion. To the extent that the unexpected repetition of gesture c (within group N) affords this sensation it can be recognized an instance of expansion by (redundant) repetition/insertion. In mm. 12–18 the same kind of grouping considerations—i.e., parallelisms among accompaniment gestures, and among contour groups in the right-hand—can be characterized as an example of phrase expansion via composed-out deceleration. In the left-hand accompaniment pattern in mm. 12–14, <d, dʹ, dʹʹ> are strongly parallel, but vary slightly in length. In measures 12–13, the right-hand layer introduces and repeats a new “valley” contour gesture <-, +>, labelled 4A and 4B in Example 1-12.39 The exact inter-onset durations between attacks in the right-hand part are slightly variable, but they approximate a triple subdivision of the (also irregular) notated measures that are defined by parallelisms in the left hand accompaniment.40 The written-out deceleration unfolds when no new gesture d figuration appears where it is expected mid measure 14, making the end of its accompaniment gesture dʹʹ indeterminate. At the same time in the right-hand layer, the three attacks occurring during m. 14 do not contradict the expectation of a “valley” contour <-, +> that would 39 The “valley” contours of 4A and 4B are the inverse of the “peaks” from 3A and 3B. The inversion relationship is not central to analytical concerns here, though I do feel it contributes in a small way to why I hear the contour segment boundaries when and where I do. 40 For me, this approximates the 3-against-2 pattern that was previewed in mm. 5–6, despite the fact that the divisions of the measures are unequal in measures 12 and following. 74 continue in parallel to groups 4A and 4B. Instead, the parallelism defined by repeated pitch material conveys groups 4C and 4D as truncated versions of 4A and 4B, with their middle event excised. Contour gesture 4D is unlike all the previous contour groups, since its beginning is not aligned with the beginning of a d-gesture group in the accompaniment layer. As soon as contour gesture 4D becomes untethered from its alignment with accompaniment layer grouping boundaries, a process of increasing inter-onset durations between successive right-hand attacks (within 4C, 4D, and 4E) strongly suggests composed-out deceleration (i.e., inter-onset durations from beginning of 4C: <4, 3, 5, 6, 7> measured in sixteenth notes). Finally, in mm. 16–18, 5A, 5B, and 5C present a thrice stated “rising” contour gesture <+, +>, each roughly aligned with an accompaniment gesture (f, f, and f ʹ, respectively). Gestures 5A, 5B, and 5C might be understood as discrete three-element groups on account of the pitch-class parallelism (one octave higher each repetition). But strictly in terms of contour patterning they are grouped together to form single and longest continuously unidirectional motion yet heard in the piece. As a single rising gesture, the three notated measures 16–18 can sound as if they are a “filled-in” and “written out” fermata on a single measure, whose extended ascent recalls the “out-of-time” feeling from the introduction’s ascending arpeggio in freer tempo. The single tripartite ascending gesture in the right hand contributes to this sense as well by returning to contour gestures 1A and 2A. In this way, Contour Group 5 together with accompaniment group Z create a phrase-rhythmic bookend that encloses the more substantive thematic material of mm. 7–12 between the introduction’s slow ascent as “wind-up,” and the long and slowing ascent gesture as “wind-down.” The phrase structure that emerges internal to mm. 7–18 is strongly indicative of both varieties of internal phrase-rhythmic expansion that Rothstein theorized in a common-practice 75 context (i.e., expansion by redundant repetition/insertion and by composed-out deceleration). However, because the pulsation is unsteady and notated time signatures almost ever-changing in and around m. 7, it is not possible to account for the expansion by referring to a moment where the normative regularity of the hypermeter is obscured, since it is never established in the first place. Instead, my description has suggested that independently defined, but interrelated grouping structures which arise from [1] motivic parallelisms in the accompaniment and [2] contour gestures in the lead chord-melody in the right hand are the basis for the sensation of phrase expansion in the passage. These types of groups go considerably beyond what GTTM or PRiTM mean by “grouping structure.” And these groups (namely, the accompaniment groups defined mostly by parallelisms) are not regular enough “count” as metrical structure (even though for the most part they correspond to Bartók’s irregular barlines). Making a case for expansion here would not be possible based on regularity of phrase prototypes alone, nor by describing only one kind of grouping, such as melodic segmentation. In this case, it is the co-presence of groups nominated by contour, groupings nominated by parallelisms in the accompaniment texture, and inference of thematic interrelationships between them that make the description of phrase expansion possible. 1.5 Summary The preceding accounts of emerging grouping structures based on motivic parallelism, melodic reduction, contour, and other kinds of grouping by event-affiliation nominate grouping structures independent of, but co-present with groupings by gestalt segmentation or other varieties of event separation. I have argued that we can conceive of these groupings and interacting, and even mutually informing one another through “irregular” examples from Debussy’s Syrinx, Mussorgsky’s Pictures at an Exhibition, and Bartók’s Etude no. 3. These 76 represent instances where consideration of co-present grouping structures affords the phrase-rhythmic effects that Rothstein, Lerdahl and Jackendoff, and others have described in terms of the interaction between (melodic) grouping structure and (isoperiodic) metrical structures. By articulating other types of grouping that can serve in the same role that hypermeter does for Rothstein’s phrase rhythm, I have shown that the great analytical potential of a PRiTM-informed conception of phrase-rhythm need not be limited only to music that is regular enough to suggest isoperiodic (hyper)metric structures. The next chapter surveys notable theories of meter that accommodate this irregularity and aperiodicity to in various ways and to varying degrees. Throughout the survey I define and contrast “meter-as-grid” and “meter-as-grouping” as two schools of thought in contemporary metric theory which bear different relationships to the new perspective of phrase rhythm introduced in the first chapter. 77 Contrasting Conceptions of Meter in the Literature: Generative or Conformational, Inter- or Intra-opus At the moment of the invention of a melody certain tones that have inner value or emphasis distinguish themselves. These determine the separation of measures in the presentation of the melody, provided that other musicians can understand and perform them … It is apparent that simultaneously with the genesis of an idea, those stressed tones which necessitate this or that metrical division are already determined. Heinrich Christoph Koch, Versuch einer Anleitung zur Composition 2:3–4 (1782) [T]he matter [of proportion among larger sections] ought not to be approached in a purely quantitative way, as if our perception of time could remain uninfluenced by the events that fill the time. Carl Schachter, “Rhythm and Linear Analysis: A Preliminary Study” (1976) 2.1 Theories that Accommodate Irregularity into a Meter-as-grid Outlook In the previous chapter I offered commentary and critique on the prevailing view of phrase rhythm as the dynamic interaction between grouping structure and hypermetric structure (from Rothstein’s PRiTM). Conventional approaches to phrase rhythm take meter to be a regular structure that is conceptually separable and conceptually prior to the music that it measures. The general premise of these approaches—which I characterize as the “meter-as-grid” view—is that meter is an unsounding conceptual structure and/or psychokinetic activity against which the actions of sounding music are measured. In this respect the “meter-as-grid” view is necessarily one that takes meter to be extrinsic from the music. For example, such theories normally assume that a listener knows some library of “allowable” or “well-formed” metric structures that can be described without having to have music “in them” (e.g., the dot-diagram as a picture of metric structure is something GTTM assumes is legible to its reader). Chapter 1 showed that Rothstein’s theory at times drifts between a GTTM-informed 78 understanding of (hyper)meter as the extrinsic grid-like structure (to which the music’s durations do or do not conform), and a different understanding of (hyper)meter as determined by characteristic and idiomatic actions of the music and its basic phrase independent of whether the beats it defines are spaced equidistantly —a view largely informed by Schachter (1976, 1980, 1987). With the concluding examples of Chapter 1 examples I showed the less-metrical excerpts to propose grouping structures (namely groups by event affiliations, such as harmony, texture, and rhythmic/motivic parallelisms, etc.) that are understood to interact with other co-present grouping structures (i.e., melodic segmentations) to afford phrase-rhythmic effects even in passages that do not obviously conform to any patterning that could be understood as “metrical” in GTTM or PRiTM’s sense. In this chapter I will lay out some of the principles of my theory of “phrase rhythm as grouping complex” for describing phrase-rhythmic effects without requiring meter or hypermeter in the conventional sense (where meter is grid-like structure defined separately from and prior to the music it would measure). Part of making this argument involves positioning my approach within the context of the existing approaches in the music-analytical literature that have been used to describing “irregular” phrases. Using a Luther chorale tune as an example, I will begin this chapter by addressing some notable approaches that have sought to modify the “meter-as-grid” view in order to accommodate greater degrees of irregularity and aperiodicity, while still assuming the relevance of well-formed metric structures even for music that does not conform to them (sections 2.1.1 and 2.1.2). Then, using the same chorale tune example, section 2.1.3 devotes special attention to the way that grouping conditions—especially grouping parallelisms of various kinds—are a feature that introduces irregularity, invites metric reinterpretation, and more generally, affords the attribution of qualia to 79 timepoints independent of conformity to known metric structural types. Finally, section 2.3 surveys “wider” theories of meter and phrase rhythm—Schachter (1976; 1980; 1987), Berry ([1976] 1987), Benjamin (1984), and Hasty (1997)—whose definitions of (hyper)meter are quite independent of the “meter-as-grid” views of GTTM, London ([2004] 2012), et alia. 2.1.1 Introduction The modern grid-like view of metric structure—formulated in Komar (1971, 52–54) and Yeston (1976, 65–79) and then expanded upon and formalized in GTTM—has become so immensely influential that even scholars whose interest is describing the temporal aspect of music too irregular to conform to GTTM’s well-formed metrical structures have taken metric well-formedness as their point of departure, adapting the meter-as-grid framework in various ways. As a result, such analyses end up accepting the relevance of regular grid-like structures for music that does not necessarily establish nor sustain the regularity of GTTM’s extrinsic grid structure. However, such approaches consider the listener’s ongoing search for a “metric fit” to be an important part of the appreciation of the music’s time-organization even in cases where no “fit” is ever settled upon. For example, Jackendoff argues that listeners engage in an ongoing “meter-finding” activity he calls the “selection function … a device that continuously evaluates the currently active [metric] analysis for relative plausibility and designates one of them as the currently most salient” (1992, 63). (In a way, Rothstein’s theory of phrase rhythm resembles this mode of continuous meter-finding and (re)evaluation, but at the hypermetric level: the relevance of four- and eight-measure basic phrase lengths is more-or-less assumed even for music that regularly presents phrase lengths other than 2n.) Danielsen (2018), Horlacher (1995), Temperley (2001; 2008), Mirka (2009), and Sullivan (2021, 2023) are more recent examples of theorists who adopt a “meter as extrinsic 80 grid” outlook when they take meter to be a regular hierarchic pattern of beats imagined separately from the music, and then describe the listening activity as a sort of real-time “meter-finding” algorithm by which a listener seeks to match the periodicities of the music as best as possible to a known and regular extrinsic metrical pattern as template. Anne Danielsen (with her RITMO collaborators) appeals to a grid-like structure with her concept of “metre-as-dynamic attending” (Danielsen, 2018) which is developed out of the cognitive psychology work of Mari Riess Jones (1976).41 In the North American literature, the “meter-finding” view develops out of GTTM’s metric structure, and more specifically from Jackendoff’s “parallel multiple-analysis processor” metaphor of meter-finding (Jackendoff 1992). In that vein, Temperley understands meter finding to be a natural mental process that is not “turned off” even for music where no (extended) conformity to a known metric template is to be found (Temperley 2001, 301). Mirka’s Metric Manipulations in Haydn and Mozart is similar but puts greater emphasis on tracing the dynamic changes in a listener’s evolving perception of meter as a piece unfolds. Specifically, her method characterizes the way that irregularities can bring about “retrospective analysis,” and “bewilderment or losing one’s bearings” which she understands as a key part of playful metrical irregularities found in the Viennese Classical style (Mirka 2009, 23). James Sullivan builds on Mirka and Jackendoff’s foundation and adapts the “parallel multiple-analysis processor” approach to irregular rhythmic situations in post-tonal music by Webern, Britten, Barber, and Adès (Sullivan 2021, 2023). Gretchen Horlacher’s approach is similar, though it is less overtly connected to the “parallel multiple-analysis processor” model. Instead, she adapts Rothstein’s “metrical 41 For a succinct definition of “metre-as-dynamic attending,” see (Danielsen 2018, 181–82). 81 reinterpretation” in order to accommodate a greater degree of irregularities than are typically encountered in common-practice examples, while still assuming the relevance of the metrical grid for music that does not feature well-formed metric structures GTTM or PRiTM would recognize. What Horlacher’s method of metric-reinterpretation offers is an efficient notational system that can describe in fine detail particular varieties of sensation based on the specific way that a piece of music mismatches with the expectation of a presumed metrical grid. The analysis of the Luther Chorale tune Nun freut euch, lieben Christen g’mein in the next section of this chapter illustrates the benefits of Horlacher’s method and will also highlight certain limitations. Specifically, the demonstration will point to the problem of choosing exactly which extrinsic “norm” an “irregular” piece of music might be said to depart from when its irregular construction precludes any overt reference to “known” metrical schemas. (Nota Bene, my framing exaggerates the extent to which Horlacher’s interpretations of irregular passages in Stravinsky and Bartók (1995; 2001) are constrained by this limit. Her analyses use the metric reinterpretations symbols—inherited and adapted from a meter-as-grid outlook—as a springboard toward a nuanced conception of emergent and contextually defined metric qualities in music of aperiodic construction (1995, 302–303). The literature survey and analytical demonstrations of this chapter overall retrace the arc of Horlacher’s deft conceptual/analytical move). Although it is stylistically outside the purview of Rothstein’s or Lerdahl and Jackendoff’s approaches (which restrict their focus to the common practice or a subset thereof)—the chorale tune presents “irregularities” at the phrase and sub-phrase level of the sort I am interested in theorizing. Another reason for choosing this example is that very singable idiom is quite easily 82 audiated, which assists toward making the technical details of the analytical method(s) I wish to highlight more immediately digestible. 2.1.2 Analyzing Luther’s ‘Nun freut euch …’ by Adjusting the Metric Grid In this first pass at the chorale tune, I will use the well-known analytical approach which looks for entrainable regular pulse, and then builds up nested pulses in a hierarchic “grid-like” manner, symbolized by a “dot-diagram” visual representations of metric structure similar to those used by Lerdahl and Jackendoff (GTTM [1983] 1996), Horlacher (1995), and London ([2004] 2012), invoke “metric reinterpretation” where needed to accommodate irregularities and avoid anti-parallel metric alignments of parallel melodic material. My application of Horlacher’s symbols for a Luther Chorale tune is decidedly anachronistic. This is not only because her method was developed for analysis of early 20th-century music; but also because Luther (and contemporaries) did not possess an equivalent concept to “meter” in the technical sense(s) it acquires after 1600 (cf. DeFord 2015, Houle 2000). Yet, I find the use of Horlacher’s reinterpretation symbols an efficient analytical tool wherever one wishes to differentiate between the varieties of temporal sensation brought about by interruptions to regular pulsation. Certain Luther tunes (or other central European folk-melodies of the period) resemble Stravinsky or Bartók’s folk-inflected modernist idiom when they feature a basic pulse (tactus level or faster) that is not grouped uniformly into twos or threes at “higher” (slower) levels of nested pulsation.42 Chronology aside, I am not convinced that adapting Horlacher’s symbols to 42 The phenomenon of “subjective rhythmization”—the tendency for human participants in psychological studies to spontaneously group undifferentiated pulse into twos or threes—was observed in the late 19th century by Bolton (1894) and Meumann (1894) (as cited in London ([2004] 2012,14)), and represents one of the sources for the general assumption that takes pulses grouped into twos and 83 describe irregular pulse in a 16th-century folk idiom is any stranger a proposal than adapting GTTM’s metrical structure and PRiTM’s reinterpretation toward the analysis of Bartók and Stravinsky. In its first printed version, the Luther tune is given a mensuration sign of imperfectum minor indicating a grouping of the minims into pairs, and likewise grouping the paired minims into pairs of semibreves, roughly equivalent to a modern time signature of 2/2. The score unambiguously posits the minum (modern half-note in my reproduced figures) as a basic pulse but does not include barlines to correspond to the duple subdivided tempus imperfectum that is indicated by the mensuration sign.43 At line endings in the text the otherwise continuous flow of minims is interrupted by semibreves, with some semibreves followed by a minim rest. Curiously— threes to be the basic building blocks of musical time-organization. Those turn-of-the-century experiments conjectured that tempo thresholds have some bearing on the “how” of subjective groupings; and as Rasmus Bååth (2015) has summarized it, possible explanations as to “why” of the phenomenon fall into two categories. The “preferred tempo” explanation posits that each person has in mind their own basic “comfortable tempo” and will seek out uniform groupings in an undifferentiation pulse as close as possible to their “internal” tempo. Centuries earlier, music theorists offered explanations of “Taktus” that are not unlike the ”preferred tempo” explanation for subjective rhythmization (London [2004] 2012,14; 31). The “resonance theory” explanation offers that “neural oscillatory circuits” with their own periodicity resonate with the auditory signal received in the brain (Bååth 2015, 245). Subjective rhythmization and the surrounding cognitive-psychological literature are quite central to the theory of musical meter in Justin London’s Hearing In Time, which leans toward the “resonance theory” explanation when it offers that “attentional peaks” are at the root of the perception of “strong and weak” beats (London [2004] 2012,14). Because London’s theory of meter prefers metric structures with greater number of nested pulse levels, duple structures automatically receive privileged status because nested duple pulses will always offer a greater number of metrical levels that fall within the upper and lower limits of thresholds for entertainability. By contrast, I follow Benjamin, Berry, and Hasty, among others, when I adopt an understanding of meter wherein “has fewer levels of uniform pulse does not directly correlate to “is less metric.” 43 None of the tunes in the 1533 edition of this Lutheran hymnal have barlines, though it is commonplace for “modernized” notation in hymnals and songbooks to add time signatures and barlines corresponding to the “2/2” structure suggested by Luther’s duple designation. For a critical edition of Luther’s chorale tunes, see Luthers geistliche Lieder und Kirchengesänge (ed. Jenny, 1985). 84 despite the lack of barlines—the earliest printing of the tune is shown to begin with a rest. This is likely a reflection of the iambic organization of the text.44 I will show that it also carries implications for our interpretation of prospective meter and phrase rhythm for the tune. Example 2-1: Luther’s Nun freut euch, lieben Christen g’mein [Melodie C, WitK 1533, Bl.24v–25r] Geistli=||che lieder auffs || new gebessert zu || Wittemberg.|| D. ... , 1533 The Digitized Collections of the Staatsbibliothek zu Berlin https://resolver.staatsbibliothek-berlin.de/SBB0003232300000000 A dot-diagram can be built up from the level of the basic pulse toward “deeper” (i.e., slower) levels of pulsation, as shown in Example 2-2. The half note is the basic pulse, and the question of how to align the next slower level of pulsation—the whole note—is unambiguous: 44 In certain contexts of congregational singing the initial rest may have hosted an opening organ chord. Then again, in the early 16th century relatively few Lutheran churches had organs, and even in those that did its role was relatively limited (Herl, 2004, 46). 85 in order to choose a whole-note pulse stream in which each of the notated whole notes participates, the stream must “begin” on the initial rest. Thus, there is no question as to “fit” of the music to this basic two-level meter-like structure shown in the upper two pulse-streams of the dot diagram in Example 2-2. The issue of metric fit becomes more complicated, though, when and if deeper levels of pulsation are sought. A number of theorists of meter have agreed that there are good reasons for seeking out deeper levels of periodic structure beyond the most basic two-level structure such as the one we have so far identified in the chorale tune’s nested half- and whole-note pulse streams.45 45 Andrew Imbrie, for example, describes a natural “desire for order” which “leads us to seek as many higher [i.e., slower, or “deeper”] levels of metrical organization as our powers of attention and synthesis will allow” (Imbrie 1974, 54). Lerdahl and Jackendoff observe that “typically there are at least five or six metrical levels in a piece. [And] the notated meter is usually a metrical level intermediate between the smallest and largest levels applicable to the piece” (GTTM, 20-21), though they do not provide any particular reason as to why they believe this to be the case. Justin London suggests that “whereas two levels of periodicity are necessary [for meter], three or more are preferable, as this provides an attending framework that allows the listener to track fast, moderate, and relatively slow event onsets” (London [2004] 2012, 16). Another reason, bearing Rothstein’s definition of phrase rhythm in mind, is that in order to potentially describe phrase rhythm of a tune, there should be a level of (hyper)metric structure that is similar, durationally, to the phrase lengths and melodic groups considered. This approximate “matching” of lengths between phrase- or sub-phrase-level grouping and (hyper)metric structure is alluded to within GTTM’s figures and analyses but not defined explicitly. In the context of their analysis of Schubert’s Valse Sentimentale Lerdahl and Jackendoff write that “Usually, the periodicity of metrical structure—whether it is in or out of phase with grouping structure—proceeds in a regular correspondence with grouping structure” (GTTM, 250). In PRiTM Rothstein indicates that, as a default, phrase groups and hypermeter are approximately aligned. For instance, in the context of his discussion of whether hypermetric structure can be deduced from regular phrase grouping lengths Rothstein writes: when phrases and hypermeasures roughly coincide, there is generally no difficulty: even if the two types of unit [grouping and hypermeter] are out of phase with each other, there is a consistent and predictable relationship between them. But when phrase structure and hypermeter diverge significantly, as they sometimes do, a series of equal, non-duple phrases might not imply an equivalent hypermeter. (PRiTM, 38) 86 There are two differing ways one could add a slower “breve” (four half-note) pulse stream to the hierarchic dot-diagram to group its whole-note pulse into a nested duple structure (just as the whole-note pulse did for the half-note pulse). A problem of anti-parallel metric alignment of parallel phrase segments will arise whether one chooses to align the breve pulse starting on the initial rest (as I have done in Example 2-2), or whether the initial breve pulse is instead aligned with the second F♮4, on the word “freut.” The discussion that follows ignores the second possibility, though choosing this is not material to the argument I am making about fittedness of the tune to a metric grid since the same problems arise in both alternatives, only in slightly different locations. I do prefer starting the breve pulse stream on the initial rest instead of from the first note. This is because the grouped pulses that result help describe a prosodic stress parallelism that I hear in the iambic stresses of each line-beginning from sub-phrase A to sub-phrase B. That is, “nun” is a prosodically unstressed syllable sung on a half-note F♮4, is anacrustic in relation to the stressed syllable “freut” which immediately follows it. In a parallel way, the initial sung pitch of sub-phrase B is a prosodically unstressed syllable sung on a F♮4, though a whole-note duration this time. But despite its receiving durational accent, I understand this F♮4 as anacrustic for the following stressed syllable “laßt” sung on A♮4. Relative to the tri-level metric structure, the repetitions of the Stollen phrase (first system of Example 2-2) do not bring about any parallelism problems for melodic groups A and B. An anti-parallel alignment problem does arise, though, between the motivic parallelisms and melodic groups of the Stollen and of the Abgesang phrases. Example 2-2 highlights the anti-parallelism between sub-phrases B and Z with blue boxes. The Abgesang’s sub-phrase Z differs from the Stollen’s sub-phrase B only in that the duration of its final note is doubled. But where 87 the initial whole-note F♮4 of sub-phrase B is aligned in a “weak” position relative to the breve pulse, the initial whole-note F♮4 of sub-phrase Z is aligned in a “strong” position relative to that same pulse stream.46 GTTM’s Metric Preference Rule 1 states that “where two or more groups or parts of groups can be construed as parallel, they preferably receive parallel metrical structure” (GTTM, 75). So, in this sense the tri-level metric structure shown in Example 2-2 is not preferred, on account of the parallelism problems between Stollen and Abgesang segments. Example 2-2: ‘Nun freut euch…’ Three-level Dot Diagram with and without Metric Reinterpretation Example 2-3: Horlacher’s Metric Reinterpretation Symbols 46 A similar problem of opposite metrical alignment between sub-phrases A and Y could be argued, to the extent one hears Y as a loose parallel to A. However, for the sake of visual simplicity this second potential anti-parallelism is not highlighted within Example 2-2. ....Nun..freut.euch,...lieb.en-..Christ.en-...g'mein,. ..und. ...laßt.uns..fröh.lich-...spring. ..en,-. ....dass..wir.ge...trost-.und..all.in...ein,. ..mit. ...Lust.und..Lieb.e-...sing. ..en,-.....was..Gott.an...uns.ge..wend-.et-...hat,. ...und...sein.e-..süss.e-...Wund.er-..tat;-. .....gar.......teu'r..hat.....er's..er.....worb-.......en.-........&WwhAStollen 1B AStollen 2 anti-parallel alignment (B & Z)B&WwhXAbgesang YW anti-parallel metric alignment (B & Z)ZWÓ˙ ˙ ˙ ˙ ˙ ˙ ˙ w w ˙˙ ˙ ˙ w wÓ˙ ˙ ˙ ˙ ˙ ˙ ˙ w w ˙˙ ˙ ˙ w wÓ ˙ ˙ ˙ ˙ ˙ ˙ ˙ wÓ˙˙ ˙ ˙ ˙ ˙ ˙w w ˙˙ ˙ ˙ w WAnticipated strong becomes weak Anticipated weak becomes strong Duration between beats (on given level) longer than anticipated Duration between beats (on given level) shorter than anticipated 88 The melodic grouping/metrical anti-parallelism introduced in “irregular” situations like the one in this Luther tune is a problem that an analyst who is committed to a “meter-as-grid” point of view might accommodate in one of three ways. Option 1: the analyst conceives of the music as relating to a well-formed metric structure of sufficient hierarchic depth, even if it means adopting a non-preferred reading of the tune with anti-parallel groups, as in the regular, well-formed tri-level metric structure shown in Example 2-2. This option is not suitable for describing phrase rhythm in irregular music (especially if not straightforwardly tonal and rhythmically idiomatic) because relating the irregular phrases to a regular underlying metric pattern with which the music has no obvious connection is too arbitrary. Option 2: the analyst adopts an approach such as Justin London’s entrainment model—meter as mode of attending to periodic and hierarchic nested pulses. When the melodic grouping parallelisms are found not to adhere to the patterning of an “allowable” tri-level metric structure the music is deemed “ametric” or only very weakly metric. With respect to the Luther example, this would entail abandoning the search for a breve pulse stream, to leave a “weakly metric” two-level structure). This second option is the conclusion that is often reached for early 20th-century music, e.g., Bartók or Stravinsky, where there exists regular pulsation at a very basic level, but where the grouping of those basic pulses into an additional levels of regular pulsation is not possible, or can only be done at very “shallow” levels of metric structure (see, Temperley 2001, 300–301; London [2004] 2012, 106-7,116-117; London 2013, 285).47 This approach, while useful for 47 Evidence of this includes the fact that London rarely includes analyses of situations that do not afford the kind of entrainment behavior on which his theory of meter is founded. His commentary on Stravinsky’s Les Noces (2013, 280-85) is a notable exception. In the passage from Les Noces London considers, “a nearly continuous level of eighth-note pulses is established and maintained” and yet, the 89 describing certain kinds of metric effects and entrainment behaviours, is not suitable for phrase rhythm analysis since it does not nominate the kind of (hyper)metric structures whose period approximates the durations of the phrase segments under consideration (PRiTM, 38; and see footnote 45). And finally, Option 3: the analyst makes necessary metric adjustments so that as much of the tune as possible is heard in terms of its participation within an “allowable” extrinsic metric structure that is understood to be relevant enough for the music, despite the fact that the music does not perfectly adhere to it. In this mode, adjustments and reinterpretations are made so in order to preserve parallel alignment (instead of hearing the “same” material in a new metric orientation), while still invoking “deep” metric levels even despite the lack of isochronous pulsation at those level. At a glance, this option is the most promising of the three, yet it too introduces problems that make describing phrase rhythm (as the interaction between (hyper)metric and grouping structures) difficult, as the following analyses and commentary will demonstrate. In GTTM, Lerdahl and Jackendoff generally do not concern themselves with the kind of music that would lead to option 2, but do allude to the outcomes I have described above as options 1 and 3, especially as a response to the deeper levels of metric structure in tonal music irregular melodic materials are described as “affect[ing] the beat and downbeat location” to bring about a metric situation he calls “unmoored” and “unstable” (London 2013, 285). London’s feeling that meter is severely attenuated when regularity is not evident at more than one or two basic levels can also be observed in his well-known analysis of Beethoven’s fifth symphony. When, in the development section the subdivisions of the 2/4 measures are no longer articulated and while grouping beyond the measure becomes unclear, London describes the situation as a “metric crisis” that leaves the listener “in a metric fog” (London [2004] 2012, 116). He continues the description in decidedly negative terms, saying: Beethoven’s “bare pulse” is “weak and thready.” and that in the absence of nested pulse “in some sense…there is no meter; [when] there are neither subdivisions of the beat nor patternings of beats into larger measures” (idem, 117). 90 which they admit are “commonly subject to a certain degree of irregularity” (GTTM, 99).48 Their theory can accommodate a degree of metric irregularity in two ways. One is to temporarily suspend the requirement for regularity at the deepest metrical levels in “transitional zones.” In these cases “[irregular] grouping gradually takes over responsibility from metrical structure” and introduces spans that are longer than expected between beats at a relatively deep level of structure (GTTM, 99).49 Or, they observe a “metrical deletion” at a moment where a grouping overlap and/or elision foreshortens the expected duration between strong beats at a relatively deep level of structure (GTTM, 100–104). Horlacher (1995; 2001) proposes modifications to the dot-diagram notation to convey a few different varieties of “metric reinterpretation” (which notations are defined and illustrated in Example 2-3, above). This aspect of her approach stems from GTTM, but her adjustments—and her application of these principles to music outside of the common practice repertoire—allow for reinterpretations to be invoked more liberally toward description of emergent metric structure in irregular music by Stravinsky and Bartók.50 The combination of Horlacher’s “anticipated strong becomes weak” (indicated by empty circle) with a shortened duration between pulses on some level (indicated by square bracket below pulses) is more-or-less equivalent to GTTM’s metrical deletion, except that hers are invoked more liberally and do not 48 Their description of Schubert’s A-minor Valse Sentimentale (GTTM, 250–52) is a rare example where they opt to hear a well-formed metric structure as pertaining to the piece at a deep level of structure despite anti-parallel grouping alignments that result (i.e., my “option 2”). 49 Temperley’s “Hypermetrical Transitions” (2008) represents further development of this idea. 50 Peter van den Toorn’s contributions to analytical scholarship on Stravinsky’s music are noteworthy in connection to Holacher’s method (van den Toorn 1983, 1988). His method for describing Stravinsky’s rhythmic irregularity is a considerably more conservative approach, but his account of “on-beat/off-beat reversal” as a signature feature of Stravinsky’s compositional language represents an important precursor to Horlacher’s more radical approach. 91 require the same special conditions of grouping overlap that result in GTTM’s foreshortened metric span. Horlacher’s encircled dot (anticipated weak becomes strong) and the resulting lengthening of durations between pulses on some level of structure does not have an exact equivalent in GTTM, though it could fall under their (seldom invoked) category of metrical deletions that elongate.51 Settling for options 1 or 2 for the Luther tune—deeming the piece only very weakly metric, or accepting an interpretation that includes anti-parallel melodic groupings—are attractive in their simplicity, but are ultimately not very satisfying so far as they characterize the tune as “metrically shallow,” and call into question why a certain well-formed metric structure should be relevant for the tune whose parallelisms are independent of the periodicities of that meter. These outcomes do not capture the depth of phrase-rhythmic richness I experience when singing or thinking about this tune. The quality I have in mind involves attending relationships between segments whose duration is well beyond the whole-note value (making the “metrically shallow” determination less than satisfactory). And this richness also involves the feeling of natural and effortless flow the tune invokes. That feeling of “ease” runs counter to what is asked of the listener if and when the material of parallel phrase segments B 51 Lerdahl and Jackendoff describe a kind of metrical deletion where an overlap causes the deletion of only a “strong beat” from a well-formed measure, leaving behind the rest of the weak beats of the associated measure from which the strong downbeat was deleted and thereby adding time on one or more level of pulsation. This lengthening by “deletion” resembles the effect of “duration between beats is longer than anticipated” that Horlacher conveys with her arrow; but Horlacher’s concept is considerably tidier than this secondary kind of deletion in GTTM. Lerdahl and Jackendoff provide only one example of this elongating kind of deletion: Schumann’s Liederkreis op. 39 no. 9 (GTTM, 103). It has not, as far as I am aware caught on in the music theory literature borne out of GTTM, whereby comparison many theorists have described this kind of elongation effect using either Rothstein’s phrase expansion, or Horlacher’s metric reinterpretation symbols. 92 and Z are construed as “antiparallel” with respect to a rigid underlying metric structure. The third option—invoking frequent metric reinterpretations—improves the situation. Describing irregular phrase structures using Horlacher’s metric reinterpretations offers an elegant way to diagram certain kinds of metric experiences of a passage, though we will see that the particulars of the metrical experiences it helps describe remain tangentially affiliated with arbitrary appeals to metric structures from without (to the degree that the dot-diagrams structures it subjects to reinterpretation are presumed be composed exclusively of nested duple and triple pulses resembling “well-formed” metrical structures). Returning to Example 2-2, a reading using Horlacher’s reinterpretation symbols is shown within the dashed blue boxes below the second system. The metrical reinterpretations are invoked to “solve” the parallelism problem that arises because of motivic parallelisms separated by intervening irregular (non-duple) phrase lengths. The hollow dot—anticipated strong becomes weak—under the first note of sub-phrase Z illustrates a metric re-calibration that a listener might make having understood sub-phrases X and Y in terms of their alignment with a uniformly periodic breve pulse carried through from the Stollen phrases. That pulse is disrupted only when the listener understands the first sung syllable of Z (“gar”) to be an exact grouping parallel of sub-phrase B’s anacrustic “und”/“mit.” The rightward arrow conveys the sense that the inter-pulse duration at the breve level has, here, been made irregular: the duration between sub-phrase Y’s “Wund-” and sub-phrase Z’s “teu’r” is longer than expected. Depending on whether the tune is repeated, the same kind of metric reinterpretation might also be invoked at the end of sub-phrase Z, in order for the return to the Stollen phrase not to appear in an opposite metrical orientation as a result of the elongated final syllable of the Z sub-phrase. 93 The reinterpretation symbols used in Example 2-2 apply the frequent reinterpretations mode of analysis offers a way to accommodate a greater degree of irregularity while accepting a conventional outlook on meter as regular nested pulse. In this view meter is a grid-like structure that is defined independently of the events of a given tune, and is presumed to be regular and either uniformly duple or triple at each level. In this application the reininterpretation symbols illustrate the particular temporal effects that arise when a listener accepts the relevance of an extrinsic periodic metric structure and makes special note of perturbations to that otherwise regular structure which are necessitated by irregularities in the tune itself. But as it does this, it is also susceptible to a problem faced by many of the prevailing approaches to phrase-rhythmic and metric irregularity: describing an alluring aspect of the music (diversity of its durations and phrase-segment proportions) in negative terms. It takes as its object of analysis the particular temporal sensation brought about in a listener when they bring some “extrinsic metric structure m” to their evaluation of the piece, and find it to be a less-than-perfect match.52 Horlacher addresses this issue directly in her “Metric Irregularity in Les Noces” (1995) in two ways. First, she notes that the initial presentations of certain melodic motives invite quite conventional metric interpretation, locally. These “same” melodic motives, subject to rhythmic transformation, conspicuously “break” their “metrical associations” upon subsequent 52 My object of analysis is different: something like a model of potential interrelationships among distinct grouping affordance(s) for “the piece” as abstracted from a notated score, and independent of one particular hearing or performance. This will become clearer through the analytical demonstrations of subsequent chapters—especially Chapter 4, and then summarized in the concluding metaphorical reflections of Chapter 6. My description of a listener/theorist’s “models” (plural) in continuous and multifaceted relationship to “the piece” (singular) at different levels of abstraction is informed by Benjamin’s “Models of Underlying Tonal Structure: How Can They Be Abstract and How Should They Be Abstract?” (1982). 94 repetitions and restatements, thereby introducing a rhythmic vitality characteristic of Stravinsky’s music (1995, 298). Second, Horlacher argues for piece-specific ways that irregular and fragmented motives combine to create characteristic sequences affording a sense of “background periodicity” that arises when “instances of ostensible metric irregularity together produce a relatively stable perceived meter … not simply with respect to an implied external metrical grid, but with respect to the internal motivic organization of the passage itself” (idem, 301–302). In these ways, her work is a model for making possible “metrical” analysis even in the absence of “meter” in the usual sense. As a report of specific temporal sensations that result when a particular yardstick is brought to the evaluation of a passage, liberally invoked metrical reinterpretations will not account for phrase-rhythmic effects (expansion, overlap, elision) unless coupled with an analytical framework that can account for which phrase-level metrical structure might arise from the organization of the passage, and why.53 This is essentially the same limitation already described in Example 1-8 and surrounding discussion (section 1.3.1 of the previous chapter). There, the description of phrase-rhythmic effects in irregular phrases whose (hyper)metric structure is not a given was further complicated by PRiTM’s slippage between extrinsic (hyper)meter (after GTTM), and (hyper)meter as time-point qualia arising from by the intrinsic tonal features of the composition and its basic phrase (after Schachter). 53 Horlacher’s Les Noces analysis achieves this by arguing for emergent “contextual metric associations” that arise as a result of “each pitch assum[ing] a metric identity based upon a wealth of motivic associations” (1995, 305). In “Bartók’s Change of Time: Coming Unfixed” (2001), Horlacher explicitly juxtaposes “processive” and “fixed” approaches to irregularity. Her analysis balances and integrates insights gained from both perspectives, the former capturing the “emerging qualitative characterizations of timespans” and that latter, suited to the description of the “jolt” that occurs when stable pulses are conspicuously interrupted. (2001, 1.4–1.5). 95 This limitation comes into sharper view for the Luther chorale tune example when we contemplate adding a fourth layer to the metric dot diagram. In the following, I compare the outcomes of having added the fourth level in two possible ways. Example 2-4 reproduces the same tri-level dot diagram from Example 2-2, except the “alternate” metric diagrams with reinterpretation from Example 2-2 are promoted to the main dot-diagram in Example 2-4. Then, Example 2-4 posits a fourth metrical level, the “longa” (=8 half-notes) pulse stream. What the example illustrates is the particular moments of “recalibration” that a listener would experience having come to their evaluation of the piece with an extrinsic metric structure of a “4/1” time signature in mind.54 The special metric reinterpretation symbols represent the ways certain moments elicit a sense that the duration between hypermetric downbeats (on a given 54 The (a priori) metric structure in the listener’s mind is what I am calling extrinsic. This is because the “in/out” of my intrinsic/extrinsic dichotomy refers to whether the meter stems from within/without the flow of musical signal under consideration. I recognize that a listener-centered approach might frame the very same dichotomy using opposite terms: i.e., an a priori metric structure brought to the musical experience as intrinsic to the listener’s mind, and with metrical organization suggested by the music extrinsic to that listener. In a personal email correspondence (prompted by a presentation of this material in earlier form) Richard Cohn alerted me to this possible confusion, citing, for example, the music-perception terms “endogenous/exogenous,” where the within/without duality is defined relative to the mind of a perceiving listener (personal correspondence, 2020). These are two sides of the same coin: some differences in emphasis but not in substance. I chose to stick with my terms in order to trace their roots to a source in Gretchen Horlacher’s analytical framing of metrical irregularity (Horlacher, 1995). Her approach accommodates irregularities by reading them “…with respect to the internal motivic organization of the passage itself” (idem, 302, my emphasis). This qualitative position on meter, wherein “contextual metric associations” accrue to motives and their constituent pitch events (idem, 303). This is opposed to a “quantitative” and “periodic” model (idem, 290–91) such as GTTM, that would relate irregularities and aperiodicities “…with respect to an implied external metrical grid.” (idem, 302, my emphasis). I extend the critique begun in Horlacher’s article in the next section of this chapter where I show how the generative framing of metric structure in GTTM and PRiTM is often sidelined in favour of a conformational approach, as if metric structure is defined independently (and prior to) consideration of a passage. 96 level) will feel longer or shorter than expected as a result of listening according to certain expectations about pulsation and its correlation to parallel motivic features. For example, the final syllable of the first sub-phrase A, “g’mein,” is an anticipated “strong” event based on the a priori determination that the phrase is heard in terms of a four-level duple structure initiating from the initial rest. But when the A♮4 of “g’mein” is a relative long duration, it proposes a segment boundary between A♮4 (as last event of segment A) and F♮4 (as first event of segment B). Segment B’s melodic parallelism with A reinforces the same point of segmentation. When invoking (deepest level) metric reinterpretation construe segment B as (hyper)metrically equivalent to segment A, the “expected strong” at the end of segment A “becomes weak” (on the longa level of pulse) as a result. The rightward arrow shows that, relative to presumed the longa pulse stream, the distance between successive longa beats is now “too long” as the next strong beat on this (approximate) level of pulsation is delayed from the end of sub-phrase A, to sometime in sub-phrase B. Example 2-4 illustrates a listening experience that understands the weak syllable “und” to be parallel melodically, prosodically, and metrically to the first anacrustic syllable “Nun” from sub-phrase A. Thus, the downbeat at this slowest level of pulsation is delayed until “laßt,” which is construed as loosely parallel to sub-phrase A’s “freut.” Next, one regular “4/1” hypermeasure underpins the activity of the remainder of sub-phrase B, leading to a strong beat on the rest just before the parallel Stollen 2 phrase begins. Within Stollen 2, the same “anticipated strong becomes weak” (with associated elongation the span between longa beats) occurs mid phrase, just as it had in the exactly parallel Stollen 1. Within the Abgesang phrase a metric reinterpretation occurs at the end of sub-phrase X, where an anticipated strong (on the longa level) is delayed until the beginning of the Y sub- 97 phrase. Despite superficial melodic differences, this reinterpretation is structurally the same as the one that occurs between sub-phrases A and B in the Stollen phrases: metrically, sub-phrase Y’s “und sein-” iamb is equivalent to sub-phrase B’s “und laßt.” Resuming the regular “4/1” metric structure from sub-phrase Y’s “sein-” the quadruple metric underpinning fits nicely with the prosodic emphasis of the text on “sein-e” and “Wund-er-tat.” But as already described with regard to the tri-level metric grid in Example 2-2, a metrical alignment and motivic parallelism problem arises during sub-phrase Z. Example 2-4: ‘Nun freut euch…’ Four-level Dot Diagram assuming “4/1” Metric Structure Example 2-5: ‘Nun freut euch…’ Four-level Dot Diagram assuming “6/1” Metric Structure .....Nun..freut.euch,...lieb.en-..Christ.en-...g'mein,. ..und. ....laßt.uns..fröh.lich-...spring. ..en,-. .....dass..wir.ge...trost-.und..all.in...ein,. ..mit. ....Lust.und..Lieb.e-...sing. ..en,-......was..Gott.an...uns.ge..wend-.et-...hat,. ...und....sein.e-..süss.e-...Wund.er-..tat;-. ..gar. ....teu'r.hat..er's.er...worb-. ..en.-. ...&W›whAStollen 1WB AStollen 2WB&W›whXAbgesang WYWZWÓ˙ ˙ ˙ ˙ ˙ ˙ ˙ w w ˙˙ ˙ ˙ w wÓ˙ ˙ ˙ ˙ ˙ ˙ ˙ w w ˙˙ ˙ ˙ w wÓ ˙ ˙ ˙ ˙ ˙ ˙ ˙ wÓ˙˙ ˙ ˙ ˙ ˙ ˙w w ˙˙ ˙ ˙ w W.....Nun..freut.euch,...lieb.en-..Christ.en-...g'mein,. ..und. ....laßt.uns..fröh.lich-...spring. ..en,-. ....dass..wir.ge...trost-.und..all.in...ein,. ..mit. ....Lust.und..Lieb.e-...sing. ..en,-.....was..Gott.an...uns.ge..wend-.et-...hat,. ...und....sein.e-..süss.e-...Wund.er-..tat;-. ..gar. ...teu'r.hat..er's.er...worb-. ..en.-. ...&W›wh™AStollen 1BªAStollen 2B&ªW›wh™XAbgesang YWZªWÓ˙ ˙ ˙ ˙ ˙ ˙ ˙ w w ˙˙ ˙ ˙ w wÓ˙ ˙ ˙ ˙ ˙ ˙ ˙ w w ˙˙ ˙ ˙ w wÓ ˙ ˙ ˙ ˙ ˙ ˙ ˙ wÓ˙˙ ˙ ˙ ˙ ˙ ˙w w ˙˙ ˙ ˙ w W 98 In Example 2-5, above, I provide an alternate fourth-level dot-diagram that responds differently to the same grouping anti-parallelism issue. The same tri-level dot diagram is reused from Example 2-4 (and Example 2-2), but with regard to the fourth and slowest level of pulsation to the dot diagram, Example 2-5 uses a dotted longa to imply a prospective “6/1” metrical structure where Example 2-4’s longa pulse layer implied “4/1.” Comparing Example 2-4 and Example 2-5, note that they reach the exactly the same conclusion as to the ultimate location of each hypermetric downbeats at every level of pulsation. But importantly, they arrive at this matching outcome by different means; they describe two distinct listening experiences that are framed by differently selected a priori metric structures. Note also, that the four-level hypermetric structures suggested in Example 2-4 and Example 2-5 convey resultant metrical structures that are almost regular in their irregularity. That is, through the Stollen and into the first half of the Abgesang phrase, a large-scale <3+2, 3+2, …> metrical pattern is discontinued only when the (expected) two-breve duration for the latter half of sub-phrase Y is stretched to three when the strong parallelism between the material at Y/ Z and A/B invites an “anticipated-strong becomes weak” reinterpretation in breve pulse level at the beginning of sub-phrase Z. The listener whose extrinsic measure for the tune was “4/1” (Example 2-4) understands the three-breve structure of sub-phrase A as an elongated 2, and the two-breve structure of sub-phrase B as a regular duple, while the listener whose extrinsic measure for the tune was “6/1” (Example 2-5) understands the three-breve structure of sub-phrase A as a regular triple, and the two-breve structure of sub-phrase B as a foreshortened triple. Is one of these a better “fit” than the other? Does it matter if it were? We have reached an impasse in terms of how to proceed with a phrase-rhythm analysis because these are undecidable alternatives. The metric-reinterpretation method does not offer a way of choosing 99 between these alternatives (not that it needs to, I suppose). When Lerdahl and Jackendoff reach a similar impasse in their analysis of metric structure in Mozart’s Symphony no. 40 (first movement), they “refrain from choosing between…competing alternatives” and recognize that such situations will leave it to the performer(s) to “tip the balance one way or the other for the listener” by deploying accent to convey one metric structure or another (GTTM, 25).55 In that respect, GTTM’s object of analysis becomes an amalgam of the notated score and a particular realization of the score out of which the metric structure is formed.56 The Horlacher-style metric reinterpretation analyses I offered in the preceding paragraphs differ considerably from GTTM in this respect, in that the reinterpretations imagine a listener’s evaluation of a passage’s time-organization (both in-time and post facto) where expectations about “strong” and “weak” beats are formulated by the moment-to-moment re-evaluation of the presence/absence of musical features that would sustain regular nested pulse on multiple levels. While Horlacher’s approach involves symbols and terms that remain loosely tethered to the conformational leanings of GTTM and PRiTM, she ultimately uses them as tools toward the description of meter as emerging from contextually determined time-point qualia (Horlacher 1995, 303). In the next section I consider again the same Luther tune but shifting from an extrinsic a priori notion of meter to an intrinsic and emergent view of quasi-periodic structure (whether 55 Lerdahl and Jackendoff provide an example of two recordings of the symphony—Bruno Walter (Columbia, MS 6494) and Leonard Bernstein (Columbia, MS 7029) —leaning toward their metric structures A and B, respectively (GTTM, 335 fn6). Alan Dodson (2002) is an example of further research in this avenue, connecting timing data extracted from specific audio recordings to alternative interpretations of hypermetric ambiguities. 56 Justin London takes up a similar object of analysis in Hearing in Time, where he conceives of meter as a potential structure within the information of the notated score, which is to be “correctly” communicated to a listener via performance (London [2004] 2012, 23)). I return to a discussion of this ontology of meter toward the end of this chapter in section 2.3.3. 100 or not one wishes to call it meter). At the same time, I shift the analytical object from a specific listening experience (based on a set of expectations informed by some a priori metric structure m) to a model of potential phrase-rhythmic affordance(s) for the music and notated score (independent of one particular hearing or performance). The most interesting result of this is that it then becomes possible to describe differently-accounted-for “hypermetric downbeats” shared by the interpretations illustrated in Example 2-4 and Example 2-5 with a more straightforward (if less conventional) description of those same timepoints as group-beginnings based on periodic structure and deep grouping parallelism (independent of melodic grouping). 2.1.3 Describing Emergent Rhythmic (and Metric?) Structures Intrinsic to ‘Nun freut euch…’ Any approach that accepts the “meter as grid” view, or takes multi-level entrainability and isoperiodicity to be the primary criteria of metric structure (London et al.) will necessarily be stricter in its definition of what it is to be metric than other “wider” approaches like Benjamin (1984), Berry ([1976] 1987), or Hasty (1997). The latter are better suited to describing a variety of temporal organizations as potentially metric (whether entrainable or not). The theories of musical meter that understand meter to be either a result of regular and hierarchic pulsation, a psycho-kinetic behavioural response to near-equidistant pulsation, or a characteristic regular alternation of “strong” and “weak” beats will necessarily cast some musics as being less metric than others—even regarding some musics as not metric at all (GTTM, 297; Temperley 2001, 299ff.; London [2004] 2012, 106). Countless analyses are likewise based on the idea that meter is a model of equal (or nearly equal) regular pulsation that serves as a static template against which the dynamic and perhaps “irregular” rhythmic activity of the music is evaluated—observed in commonly encountered descriptions such as: this music 101 is in a meter; this music sustains the meter; this music pushes, pulls, strains against the meter; this music introduces metric ambiguities; this music creates metric dissonance; this music distorts or destroys the sense of meter. All of these are evocative and apt descriptions of certain varieties of a music’s time-organization, and I do not seek to discredit these approaches as potentially valuable for music analysis. But what all of these metaphors have in common is that [1] they take meter to be a conceptually prior and more-or-less extrinsic structure against which rhythmic activity is measured and [2] they exhibit a bias toward describing the rich variety of irregular phrase-structural activity using language with a decidedly negative valance. When meter is conceived as a mental structure that—once established via entrainment or template matching—is presumed to exist more-or-less independently of activity on the musical surface, it is inadequate for describing phrase structures in which irregularities prevent entrainment beyond the level of the basic pulse, and where idiosyncratic harmonic/rhythmic figures regularly allude to, but do not fully realize, the kinds of balanced or symmetrical phrase structures that are well known common-practice idioms.57 I show that this poses a challenge for the description of phrase-rhythmic structure in idiosyncratic pieces or genres, since the conventional approach to phrase-rhythmic analysis 57 Justin London’s theory of meter does admit a certain amount of “fluctuation” (by which he means that hierarchic layers of articulated pulsation may appear and disappear) but does not allow for much metric “flexibility” (disturbances to periodicities). For instance in his analysis of the Beethoven Symphony no. 5, the “metric flux” involves a situation where “the meter never changes, [but] the depth of subdivision and the presence of levels above the tactus (so-called hypermeter) changes radically over the course of the movement” (London [2004] 2012, 110; emphasis mine) . He elsewhere writes that “over the course of a piece, meters can and often do change…but once a meter is established it constrains the formation of subsequent metrical periodicities in metrically continuous contexts” (idem, 41; emphasis mine). 102 (after PRiTM) is predicated on comparing the groupings and rhythms suggested by the melody and/or harmony to a regular metric underpinning, and/or to “normative” phrase structural types. There are pieces of music both in and outside of the common practice which do not establish the kind of periodic metric structures typically required for the description of phrase-rhythmic structure, and for which comparison to uniformly duple phrase-lengths may not be appropriate. This would not be a problem if the feeling were that such “irregular” pieces do not have phrase rhythm. But I suspect that the reader will agree that it is difficult to imagine insisting that any “irregular” piece of music—such as the Luther chorale tune analyzed in the previous section; the Debussy, Mussorgsky, and Bartók examples in the previous chapter; or the excerpt of Bartók’s Violin Sonata no. 2 example in Chapter 5—is found to lack phrase rhythm, simply because they do not establish or allude to (well-formed) metric structure in the strict sense. One way that phrase rhythm and phrase-rhythmic effects could be described in an irregular example like the Luther tune would be if certain grouping structures (perhaps regular enough to be called a hypermetric structure, though not necessarily so) could be defined for the passage independent of a priori metric well-formedness constraints, and independent of the factors of melodic grouping. Because of its ties to GTTM’s grid, the Horlacher-style metric reinterpretation analysis is not quite suitable for this. We saw in the Luther chorale that this method starts with the presumption that a well-formed hypermetric structure (6/1, or 4/1, for example) ought to underlie the tune, then the moments where the tune distorts this structure (and the temporal effects that arise for the listener as a result) are described via the reinterpretation symbols. To conclude this analytical demonstration, another pass at the Luther 103 tune presents an alternative method, that—in a way—refers back to the promising suggestion in PRiTM that hypermeter need not be duple, or regular. A phrase-rhythmic analysis might look like if the search for “meter” (or hyper-meter-like structure, which may not meet prevailing definitions of metric well-formedness) began from the premise that meter is intrinsic and emerging, rather than extrinsic and a priori. To demonstrate, I apply David Lewin’s rhythmic IFUNC as a tool to help identify prevalent durations according to which the phrase might be measured. Lewin’s “The Form of Rhythm, the Rhythm of Form” begins with his arguing that “the idea of form is not adequately conceived as something ‘spatial’” (2003, 157), such as a preconceived grid-like structure. Instead, he describes how a piece’s rhythmic structure is not a thing that is immediately apparent, but rather that “our perception of it needs some time to develop. And the way in which our sense of the [rhythmic structure] develops over time illustrates what I mean by ‘the rhythm of form.’” (idem, 159).58 In other words, the dynamic and developing aspect of the tune is located in the way that it—over time—makes its own rhythmic forms apparent, gradually. Compare this to the locus of the “action” in the “meter as grid” outlook, where a listener’s changing sense of the fittedness of the music to one of multiple possible fixed extrinsic structures whose form is pre-determined. For the purposes of the theory I develop in this dissertation, I do not focus on the experiential, and in-time aspects of analysis (as Berry, Lewin, Lochhead, Hanninen, Hasty, Zuckerkandl, and others have done). However, theories of rhythm and meter that focus on the experiential and emerging aspect of meter and rhythm 58 The particular rhythmic structure Lewin is describing in this quoted passage is an ostinato, but the square brackets are added for readability in the context of this more general discussion. 104 point toward solutions to problems that the “meter as grid” view introduces when it comes to describing phrase rhythm in irregular contexts. Lewin’s “developing rhythmic interval vector” or, rhythmic IFUNC ([1987] 2007, 88–89, 113ff.) points to emerging rhythmic structures by constructing a table that “counts the number N of rhythmic intervals d at time t, as time runs on (that is, as t gets larger) through the passage” (2003, 161). I do not make the IFUNC tool a regular part of the analytical method used through the rest of this dissertation. However, using it for this last pass at the Nun freut euch example nicely illustrates how presumed allegiance to extrinsic norms can sometimes prevent more straightforward analysis that proceeds according to phrase and sub-phrase level periodic structures intrinsic to the material (and seemed hidden only because they differ considerably from the kind of thing we expected them to be). An IFUNC table tallies the total number of occurrences of each notated duration value d (as of timepoint t). Lewin applies the tool primarily for music of the early 20th century, which uses a wide variety of durational values even at low levels. His simple hypothesis is that the duration-interval(s) that will be most convincing as a “measure” of the musical events for a given passage are the durations that one is given many opportunities to hear.59 The IFUNC table, then, presents an evolving picture of which durations are likely to be become metric for the passage as of any time-point t. In the example of Nun freut euch the basic half-note and whole-note pulses are unambiguously involved with metric organization at its lowest levels. But running the IFUNC tally on the entire piece 59 Lewin’s examples in “The Form of Rhythm, the Rhythm of Form” include excerpts from Schoenberg’s Herzgewächse (1911), Wagner’s Parsifal (1878), Berio’s brin (1990), and a “West African Abadja rhythm” (which is simply the <2,2,1,2,2,2,1> time-line/bell pattern sometimes called the “standard pattern,” and also known by many different names in specific cyclic rotations, and within different music-cultural contexts (see Anku, 2000)). 105 reveals which of the longer timespans become most prevalent and which, relatively speaking, occur less and less frequently as the piece unfolds. It turns out that in Nun freut euch, the duration of ten half-note pulses is already more frequently occurring than durations eight or 12 (i.e., the durations that correspond to the prospective (hyper)measures in Example 2-4 and Example 2-5) by midway through the Abgesang phrase. Additionally, the tally of instances of IOI 20 surpasses running tallies for 16 and 24 even within the initial Stollen phrases. See Appendix A for the full Nun freut euch IFUNC tables with accompanying explanation. The brute force method of IFUNC is a rather crude analytical tool. It does not take motivic similarity, pitch, or melodic parallelism into account at all—only the raw duration interval data. But that the method is less sophisticated and less subtle than Horlacher’s is in certain respects, its advantage. It nominates durations by which the piece might be measured without involving any outside considerations of meter, stylistic norms, motive et cetera. The meter-as-grid approach offered in the previous section had not considered “5/1” or “10/2” as prospective meter-like structures for the passage. Why not? The example of the IFUNC table, pointing to durations five and ten, and 20 as (hyper)metric spans, serves as a reminder of the way that even the most flexible meter-as-grid approaches may obfuscate emergent periodic structures as a result of importing only certain encountered metrical structure(s). Informed by the results of the rhythmic IFUNC tables (Appendix A), my Example 2-6 illustrates a phrase-rhythmic effect—a written-out deceleration—that occurs when the listener attends to the ten-half-note span as an emerging piece-specific norm. An approach to phrase-rhythm founded in GTTM would not accept the possibility of ten (or 20) pulse spans as hypermetric units on account of their not being metrically well formed. Likewise, Rothstein’s PRiTM could describe phrase rhythm here only if a clear enough argument could be made to 106 support an “irregular” basic phrase, and its associated quintuple hypermeter. How or whether a PRiTM informed approach could determine the true “basic phrase” for music outside of the common practice repertoire is an open question. As suggested earlier, the theory I am advancing sidesteps these issues by defining phrase-rhythm as a special kind of interaction between groups and (other) groups, rather than between groups and (hyper)meter. Example 2-6, below, keeps the same melodic grouping segments as were used in previous figures, illustrated with labeled brackets above each staff. Underneath the staves, in grey, a dot diagram shows three nested pulse-streams of one, two, and ten half-note durations. This is not well-formed meter. That some would consider these nested pulses as non-metric (whether on the basis of well-formedness rules or thresholds for entrainability) is immaterial to the argument make here: I am interested in them as grouped pulses which present a distinct kind of grouping from the grouping of melodic segments A, B, X, Y, and Z. In Example 2-6 labeled brackets beneath the dot diagram convey the grouping that emerges when the listener attends to rhythmic and melodic parallelisms and to stressed syllables in the text. These groups are labeled P1 through P7. Every P-group begins with the first emphasized syllable for each line of the song text (i.e., the second syllable within each iambic line). Each of groups P1 through P4 begin with the rhythmic motive <half, half, half, half, whole*, whole> (where, within P1 and P3, the first whole[*] value of this parallel rhythmic motive is subdivided by repeated G♮4s on “Christen” and “all in”). They are defined independently from one another, but the P-series groupings (defined by rhythmic parallelism and prosodic stress) co-mingle with the proportionally similar melodic segmentation groupings A, B, X, Y, and Z. The perceived interrelationships of these different groupings draw out new second-level parallelisms between the constituent sub-phrases. For example, the 107 characterization of melodic groups A and B as loosely parallel anacrustic groups involves their position relative to P-series groups P1/P2 and P3/P4 (despite the differing lengths of groups A and B) during the Stollen phrases. The Abgesang phrase begins with a similar misalignment between unequal melodic groups X, Y (durations 10, 9) and P-series groups P5 and P6 (durations 10+10). Example 2-6: ‘Nun freut euch…’ Phrase Rhythm as Interaction between Melodic Grouping and “5/1” Structure of P-series Melodic group Z is the longest of the melodic groups (13 “half notes” duration), but is almost exactly parallel to sub-phrase B, save for the doubled duration of its ultimate pitch. Supposing a performance of the tune proceeds from end of Z into a new Stollen 1 (of the subsequent verse) as would be done in private devotional, or congregational singing—there are ten verses in Martin Luther’s 1524 version of the text—the group P7 would be extended up until the first strong syllable of sub-phrase A. This can easily be understood as a “written out fermata” in the context of the surrounding “regularity” of the ten-pulse P-groups. If we accept the plausibility of “metric reinterpretations” in a non-metric situation like the nested 1, 2 and 10 half-note pulse streams of the dot-diagram in Example 2-8, this elongating effect might be ...Nun ...freut .euch, ..lieb .en- ..Christ .en- ..g'mein, . ..und . ...laßt .uns ..fröh .lich- ..spring . ..en,- . ...dass ...wir .ge ..trost- .und ..all .in ..ein, . ..mit . ...Lust .und ..Lieb .e- ..sing . ..en,- ....was ...Gott .an ..uns .ge..wend-.et- ..hat, . ...und ...sein .e- ..süss .e- ..Wund .er- ..tat;- . ..gar . ...teu'r .hat ..er's .er ..worb- . ..en.- . ...&WwhV_wAStollen 1P1 (10 h)BP2 (10 h)AStollen 2P3 (10 h)BP4 (10 h)&hwV_wXAbgesang P5 (10 h)YP6 (10 h)ZP7 (12 h)Ó˙ ˙ ˙ ˙ ˙ ˙ ˙ w w ˙˙ ˙ ˙ w wÓ˙ ˙ ˙ ˙ ˙ ˙ ˙ w w ˙˙ ˙ ˙ w wÓ ˙ ˙ ˙ ˙ ˙ ˙ ˙ wÓ˙˙ ˙ ˙ ˙ ˙ ˙w w ˙˙ ˙ ˙ w W 108 illustrated by using a “anticipated strong becomes weak” symbol at the initial rest going into the second verse of the chorale tune. Back in Example 2-4 (which described the chorale tune with respect to a prospective 4/1 metric structure), Horlacher’s metric reinterpretation method characterizes the end of each subphrase A as a low level written-out deceleration. That “4/1” conception could likewise point to a local measure-level elongation during melodic group Z, but only relative to expected four-beat measure, not as an elongated phrase-segment relative to the preceding melodic segments and P-series groups. I have shown that broader phrase-rhythmic structures (difficult to access via the meter-as-grid approach) become apparent through consideration of emergent intra-opus dynamic grouping interaction (in this case near periodic), but without explicit reference to “meter-as-grid.” For example, the interrelation of melodic group Z and prosodic group P7 (duration of 12) affords a sense of elongation when held against the kind of loosely parallel interrelationships between the melodic segments A, B, X, and Y (variable length) and their corresponding P-series groups (duration 10), whether or not one wishes to characterize the elongation as a specifically metric effect. Admittedly the “written out fermata” in Z/P7 is a modest instance of a phrase-rhythmic effect,60 especially in light of anecdotal evidence for how normal it is for a congregation to draw out the duration of the final note of one verse before the first note of the next whether or not something in the notation contributed to further to this slowing. But that fact offers support of the point rather than evidence against it; the reasons for 60 Some examples of the “composed out fermata” phrase-rhythmic effects in PRiTM are equally modest, consisting only of an augmented note value upon the ultimate note of a phrase. For instance, Rothstein (by way of Schenker) provides an example from the second movement of Beethoven Symphony no. 5 where the motive is sometimes written as (PRiTM, 81). 7233950& U&& ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑& ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑& ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑& ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑œ ™ œ œ œ œ œœ ™œ œ œ œ œ ™ œ7233950& U&& ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑& ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑& ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑& ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑œ ™ œ œ œ œ œœ ™œ œ œ œ œ ™ œ 109 drawing time-point equivalences between these events has to do with the metric qualia ascribed to timepoints on the basis of analogous interrelationships and parallelisms of P-groups and co-present melodic groups, and is not simply a result of equidistant spacing in time or “location” of an event relative an extrinsic metric grid.61 Some readers, especially those who subscribe to the kind of “continuous meter finding” process Mirka and Temperley describe, might prefer the dot-diagrams in examples Example 2-4 and Example 2-5 to the grouping-based descriptions in Example 2-6 (and in the examples in the last section of Chapter 1). But I hope that even these readers would concede that at a certain point, the designation of certain events in an irregular tune as being a “hyperbeat” becomes as much (or more) a qualitative attribute than a quantitative one. Once the determination of “hypermetric downbeat” is independent from the question of its designating as a beat within a well-formed, “entrainable” and periodic pulse layer, a Pandora’s box has been opened up. And it is at this point that the “wider” theories of rhythm and meter from 61 In an article on Louis Couperin’s Préludes non mesurés, IJzerman (2011) offers qualia-based definition of meter specifically for the curious un-measured examples he considers. (I thank Ed Klorman for pointing me toward IJzerman’s very interesting work, which at present remains not particularly well-known in North American music-theoretical scholarship.) While his definition of meter is more style-specific than the kind of “meter-as-grouping” I am seeking, it does offer an excellent example of a concept of “meter” as structure that can be very salient without any requirements for equidistant hierarchic pulse. IJzerman writes: …the toccata style, like the [unmeasured] prelude style of Louis Couperin, is not based on a regular pulse. The metric aspect of these works is basically a matter of harmony, as perceived in strict counterpoint. The regulation of consonances and dissonances causes a succession of accented and unaccented times. This leads us to a definition of metre, suitable for the unmeasured preludes: metre is the regular order of strong and weak times, regardless of their durations, and based on the relationship between consonances and dissonances. (IJzerman 2011, 116) 110 Benjamin, Berry, Hasty, and Zuckerkandl come rushing into the frame. The kind of structure being described in a Horlacher style dot diagram (with ample reinterpretations) is one that is already contingent on melodic, harmonic, and motivic factors (and their grouping parallelisms) as much if not more than on temporal/periodic factors. In the examples that follow in the latter part of this chapter I demonstrate a way in which Schachter’s “durational” and “tonal” rhythm, and Benjamin’s “theory of musical meter” offer ways to describe stable grouping structures—harmonic images, or durational reductions—that can co-exist and interact with melodic grouping structures to bring about Rothstein-esque phrase-rhythmic effects even in the absence of the kind of periodic structure that GTTM and PRiTM would recognize as metric. 2.2 Precedents for Phrase Rhythm without Meter in the Literature 2.2.1 Equivalences Among Timepoints: Quantitative or Qualitative? Many rhythmic analyses presume the relevance of equal partitioning of time. This is especially true in the currently predominating (cognitive-psychology informed) school of metric thinking which takes entrainability to be the primary feature of music that it calls metrical. But there do exist theories of meter that do not take equality of durations to be a “default” position for measuring music’s temporal design, and which instead argue implicitly or explicitly for ways in which equivalences among timepoints can be established for reasons other than their equidistance in time. However, because of the wide disparity between what kinds of things are meant by the term “meter,” misunderstandings and misreadings of important work on meter, rhythm, and phrase abound. Consider the rather drastic difference in conceptions of meter as articulated by Hauptmann and Zuckerkandl: Metrical determinations, as heretofore given, with all their manifoldness of accentuation, are nevertheless always made up of parts equal in time. (Hauptmann [1853] 1888, 272) 111 And, To talk of an equality of times, or of parts of time, has no reasonable meaning in the realm of meter. (Zuckerkandl [1956] 1969, 210) The stark difference in views expressed above leaves the reader with the feeling that either one is absolutely wrong and the other right—or that they mean quite different things by the “meter.” More recently, quite different categories of musical experience are described when William Benjamin and Justin London write about the role of “tapping” (or entrainment behaviour) in relation to “meter,” in their respective understandings of the term. London writes: Meter is entrainment in the special context of performing or listening to music. As entrainment more generally guides our coordinated actions and perceptions, meter serves as the temporal framework that guides our listening and performance. For the audience, the rhythms produced by the musician(s) serve as the driving oscillators, and its toe-tapping responses are the driven oscillations made manifest. (London 2013, 279) And Benjamin writes: One can always measure time while listening to music, by consulting a clock or by means of arbitrary kinetic or psychokinetic activity—beating a foot or thinking of beating a foot—but not without draining off attention from the listening proper. The special charm of listening to music that is metrically structured is that the instrument for measuring time—and, therefore, for making precise judgments concerning duration and evaluations of proportions—is given to us as a bonus for listening attentively to other aspects of the music’s structure. By way of contrast, clock watching and foot stomping, as substitutes for perceived meter, tend to obliterate some part of what is musical about the time being measured. (Benjamin 1984, 373) London concludes that audience members (or participants in meter-perception experiments) have received the communicated sense of meter when their tapping behaviour manifests the (correct/intended) sympathetic oscillation. Benjamin feels that musical meter—or at least, the 112 thing he finds musical about meter—is lost when attention is given to tapping (real or imagined) over and above the musical features that nominate certain varieties of time-point equivalences. Sometimes the impression of certain equivalences among timepoints will be reinforced by entrainable equidistant pulse—and other times, not. As is often the case, these theorists are using the same term for what are clearly different technical, aesthetic, and perceptual concepts. The nature of the thing(s) meant by “musical meter” will continue to change and will no doubt persist in opposing usages even among contemporaries. And many more things will be meant by meter over time, especially as the frame of reference for which repertoires are within the purview of analysis changes and expands, and as various clusters of theoretical writing on music’s modes of time organization likewise grow and bring new concepts into sharper view. The question of whether or not a given piece of music is or is not metrical (by this or that definition) has little to do with appreciating the beauties of different regions in the constellation of music’s many modes of “time-organization.”62 All agree that there is one very special kind of equivalence among near 62 Consider the very slow tempi and loose cycles of Gagaku and Korean Court Music, the widely varying gong-pattern cycles of contemporary Balinese Gamelan compositions, the timelines/bell patterns of various sub-Saharan African musics, or the talas of Carnatic music, to name just a few examples. None could or should be neatly mapped onto what is typically meant by the term “meter” in the conventional European common-practice sense, though none is an entirely heterogeneous or unrelated kind of time-organization either. This is what I mean by “constellations” of varieties of musics structured time-organization. I have shown that the repertoire need not be expanded very far beyond the centre of the European common practice (e.g., Luther, Debussy, Bartók) for the music to become irregular enough that describing their phrase rhythm in terms of the relationship between grouping and (hyper)metric structures becomes tenuous. I am advocating for a concept of phrase rhythm that, while not universally applicable, seeks to describe phrase rhythm in more basic terms (co-presence of different kinds of grouping) in order that phrase-rhythmic effects like “stretching out,” “compressing,” and “overlapping” might be described in a considerably wider corpus, and independently of a narrowly (and euro-centrically?) defined concept of “meter.” 113 uniform beats within a moderate tempo range, wherein listeners can "entrain" to the regularity and appreciate the beats as a stable structure that is felt on an automatic pyscho-kinaesthetic level of perception. And certainly, when motivic parallelisms coincide with isochronous pulses, the equivalences between those points can be felt all the more vividly because of that particular aptitude common to humans (and some non-human animals). But that is not to say that a sequence of aperiodic durational values precludes the mental act of drawing equivalences among time points. Indeed, there is much music in the world where this special mode of attending to near-uniformity of beats does not apply or is not especially important. No music should, as a default, be described primarily in terms of an aspect it is found to lack.63 In this section, I discuss some existing theories of rhythm wherein uniformity of pulsation is not a central part of the approach to a music’s time-organization. Zuckerkandl’s emphasis on the experiential aspect of meter points toward an approach where grouping criteria is as fundamental as uniformly “spaced” beats. He writes: … [automatic/sympathetic beating to pulse] is not the end of it. Meter in music achieves still more than binding the temporal succession of tones to a fixed proportion. A piece of music is not simply “in its Measure [im Takt].”… Metric organization not only divides a timespan [Zeitablauf] into segments 63 In some cases, over emphasizing a music’s “lack” of entrainable and hierarchic pulsation may contribute to unhelpful “other-ing” of musics whose time-organization does not resemble the metric structures of the European common practice, or of contemporary North American commercial music. That is, when a music is described in terms of its lack of some aspect music “ought” to have, the danger of exoticization looms. Edward Said identifies the very “impulse to classify and categorize nature and man into types”—for example among in enlightenment thinkers such as Diderot, Linnaeus, and Kant, et al.—as among the principal sources of modern Orientalism (Said, 1979, 119). Amiri Baraka nicely captures a more recent and explicitly musical instance of this tendency when he observes that descriptions of early Blues authored by White critics locate the expressive potential of a “blue note” in its distance from the “normal” version of that note in an equal tempered scale. Baraka advocates for describing the notes of the Blues scale in terms of what they are, instead of relying on description that emphasize what they are not (Baraka 1980, 24–25). 114 [Stückchen] of clearly equal length, but fastens the segments together into little groups. … And if we say that hearing music is always also beating along with its measures, then by measures we mean not only the ongoing salience [fortlaufende Sich-Abheben] of an always equal unit of time, but also the formation of groups [Gruppenbildung]. (Zuckerkandl [1956] 1969, 162)64 The dot-diagram conception of metric structure posits that beat is to duration as point is to line. This predominating view often brings with it the suggestion that the “beat” is best conceived as a durationless instant. Insisting on the “beat as durationless instant” should not bring with it a rejection of the relevance of groups for determining beat-hood. Rather, it is just that the “beat as durationless instant” view bestows a special status to the precise moment where a (small scale) event-affiliation group is begun. Proponents of the beat as timepoint view usually invoke geometric/spatial metaphors without also including an explicit endorsement of, or justification for that particular geometric/spatial metaphor for metric time. But detractors—most famously Hasty (1997)—have been quite explicit in rejection of the “beat as durationless instant” view. For the purposes of the theory of phrase rhythm I am proposing, the “beats as groups” view offers a considerable rhetorical advantage (even though, ultimately, I understand these perspectives as two sides of the same coin). The theoretical outlooks that take beats to be more like points than spans have typically also taken isochrony, entrainability, or some other 64 The quoted passage is my translation (with thanks to Alan Dodson): “Damit aber ist es nicht getan. Takt in der Musik leistet noch anderes, als die zeitliche Folge der Töne an ein festes Maß zu binden. Ein Musikstück ist ja nicht einfach <im Takt>…Die Taktordnung teilt da nicht nur den Zeitablauf in lauter gleich lange Stückchen, sondern faßt außerdem noch die Stückchen zu kleinen Gruppen zusammen.…Und wenn wir sagen, daß das Hören von Musik immer auch ein Mitschlagen mit ihrem Takte ist, so ist mit Takt nicht nur das fortlaufende Sich-Abheben der immer gleichen Zeiteinheit gemeint, sondern auch die Gruppenbildung” (Zuckerkandl, Die Wirklichkeit der Musik: der musikalische Begriff der Aussenwelt 1963, 154–55). Willard Trask’s published translation of the same passage can be found in Sound and Symbol ([1956] 1969, 162). 115 criteria for “normativity” or “well-formedness” of beat structures to be a principal requirement for meter (e.g., GTTM, Huron, Temperley, London). In that conception, the beat is a precise temporal “location.” In contrast, theories which take beats to be more like spans have de-emphasized isochrony as a principal requirement for temporal structures, instead determining (hyper)metric qualia according to a combination of other factors such as grouping parallelism, harmony, and accent. In this latter conception, the beat is a group that affiliates the events that occur internal to its span. To conferring beat-hood to a span is to recognize a way in which the events of that span are grouped, and to recognize a special kind of equivalence it holds to other such groups/beats. Consideration of hypermeter is an arena where temporal organization is more often detached from its strict equidistant pulse requirement, and so (hyper)meter is one place in the existing literature where “metric” structure tends to be described more qualitative than quantitative terms. Rothstein differentiates between meter in the conventional sense (low level) and hypermeter: the former being determined by regular nested pulsation, and the latter determined sometimes by regular pulsation, but elsewhere primarily according to tonal (and in their cases, Schenkerian) principles. Not so in GTTM, or in recent applications of GTTM’s metrical structure (e.g., Horlacher, Temperley), where no distinction is made between metrical structures at and below the bar, and (hyper)metrical structures well beyond the bar. For example: neither GTTM nor PRiTM specify that “elision” can only occur at certain levels of metrical depth (e.g., where the disturbance it causes interrupts an entrainable pulse): theoretically a reinterpretation (or metrical deletion) could occur on any metric level. Likewise, dot diagrams modified by Horlacher’s metric reinterpretation symbols use same orthography to describe the interruptions to (near) isochronony whether at level of the basic pulse, or at the 116 deepest levels of (hyper)meter.65 The psychological effect of an “anticipated weak becomes strong” (which compresses a time-span at the next slower level) or an “anticipated strong becomes weak” (which lengthens a time-span at the next slower level) is very different when it happens at a pulse level that is within the threshold of entrainability than when it happens at a pulse level beyond that threshold.66 I remain unconvinced that entrainability—or even more generally, any perception of different spans as approximately equal—is necessarily involved in the perception of elisions and other irregularities at the level of phrase rhythm. But there do exist theories of meter that do not take equality of durations to be a “default” position for measuring music’s temporal design. What interests me about the deep-level “metric” structures PRiTM describes as hypermeter, or that GTTM describes as deep-level meter, is that these designations are quite obviously made 65 For instance, in GTTM’s metrical analysis of the Schubert Valse (op. 50 no. 13), the slowest/deepest level of metric structure is eight bars in duration (GTTM, 250-3). The authors hedge a bit by putting the dots at this metrical level in parentheses, but Lerdahl and Jackendoff do suggest that a metrical deletion of some kind causes an irregularity in the metrical structure at the eight-measure level when the expected deepest-level beat does not occur at measure 27. Horlacher’s analytical examples typically keep metric reinterpretations at the levels closer to tempo-thresholds that would be entrainable—were they regular enough to afford entrainment. For example: the analysis of an excerpt of Les Noces show the irregularity of the tune as reshaping a prospective “3/4” structure, with reinterpretations occurring at the quarter, and dotted-half levels of pulsation (she does not show any levels of metric structure beyond the dotted-half) (Horlacher 1995, 299). Her analysis of Bartók’s Mikrokosmos no. 126 posits a modified “6/4” structure, where again reinterpretations occur at the quarter, and dotted-half levels of pulsation (Horlacher 2001 Example 3b). The fastest level of pulsation (sub-tactus) is perhaps the only level where reinterpretation would not occur. This is because if reinterpretation occurred at that level (i.e., an anticipated strong becomes weak) then that implies that the level presumed to be the fastest level of pulsation, is not the fastest since a reinterpretation at that level means the listener has instead chosen to place a beat between the pulses of what had previously been the fastest pulse level. 66 See Candace Brower’s “Memory and the Perception of Rhythm” which addresses the cognitive and aesthetic issues that arise from the assumption that metric structure can extended to deeper/slower levels and still be understood as “metric” (Brower 1993, 20–21). 117 on the basis of grouping conditions. Although the original authors do not describe it this way (reserving “grouping” for melodic segmentation) these deep level are “metric” qualia are equivalences among timepoints made on the basis of parallelism, harmony, or other tonal and motivic factors, not pulsation per se. Section 2.2.2 draws this theme out by juxtaposing GTTM’s grid-like view with metric/phrase structural outlooks from Schachter and Dahlhaus, who take tonal features to be more important for defining hypermeter-like structures independent of a presumed equidistant pulse. Then in section 2.3 I point to Berry, Benjamin, and Hasty’s theories of meter by which I show some ways that motivic and harmonic parallelisms contribute to the creation of phrase-rhythmic structure. These theories assign qualia to timepoints independent of the requirement for isochrony between points and independent of comparison of the sounding phrase to a prototypical template. 2.2.2 Three Analytical Views on Schubert’s “Irregular” Valse Sentimentale This section illustrates the ways in which different analytical approaches have looked to different criteria when making a determination of how regular or irregular a passage’s phrases are. Schubert’s Valse Sentimentale op. 50 no. 13 is the first example in the “Analyses” chapter toward the end of the book ("a complex rhythmic example" GTTM 10.1, 250–53). Lerdahl and Jackendoff base their analysis on aspects of Schachter’s analysis of the same piece (Schachter, 1999, 70–74).67 Consideration of the similarities and differences in their assessment of phrase rhythm in the Valse is instructive here because, although they agree on some basics, GTTM’s 67 Schachter’s Schubert analysis was published as the second installment of his “Rhythm and Linear Analysis” series, originally appearing in The Music Forum V (1980). It was later reprinted (along with parts 1 and 3) as a portion of Unfoldings (1999). 118 analysis proceeds from a view of meter as an extrinsic grid, which Schachter’s analysis does not share. Further, Carl Dahlhaus comments upon and problematizes Lerdahl and Jackendoff’s analysis in his review of GTTM (1987) in a way that sharply encapsulates differences between intrinsic and extrinsic conceptions of meter, hypermeter, and phrase structure. Dahlhaus’s review also points toward difference in analytical approaches with relevance for the approach to phrase taken in this dissertation: that is, whether an analysis proceeds according to an inter-opus frame of reference, or to an intra-opus frame. I will flesh out these differences via commentary on the various analyses of the Schubert Valse. My Example 2-7 presents a durational reduction of the Valse (after Schachter) that also illustrates GTTM’s analysis, and Dahlhaus’s critical response. In Example 2-7, the uppermost pair of staves show a slightly simplified version of Schubert’s score in a durational reduction (2:1). Above it, I use unlabelled brackets to reproduce Lerdahl and Jackendoff’s grouping structure; and immediately below that durational reduction I reproduce GTTM’s dot diagram representation of metrical structure. The bottom three staves provide a different reduction—my own, but informed by Schachter. Its lowest staff reproduces the complete bass line (with each “dotted quarter” representing one of Schubert’s 3/4 measures). The complete harmonies from the tenor-register accompaniment are included in small noteheads on the same staff. The two treble staves provide a reduction of the right-hand part conceived as a two-part contrapuntal texture. Counter to the GTTM approach to grouping (which understands the Valse to be monophonic), I provide separate melodic grouping analysis for the two upper voices in the contrapuntal texture: the leading (dux) voice is highlighted with blue noteheads and labeled with D-series brackets, while the following voice (comes) is highlighted with orange noteheads and labeled with C-series brackets. 119 Comparing my multilevel groupings in series C and D to GTTM’s unilevel melodic grouping structure, the reader will note that the GTTM groupings are determined according to the leading middle voice during the initial A section (i.e., “dux” groups D1.1–D1.4), but then determined according to the trailing upper voice during the B section (my comes groups C2.1–C2.2). In the final Aʹ section the question of which voice leads/follows is complicated by the added time after C2.2, and by the interruption of prospective D2.3, breaking the loose from its phrase-rhythmic parallelism it shared with the A material up to that point. And at the same time, octave doublings make the question of which voice is indeed “the upper” quite opaque. That there are at least two independent upper voices is clear, though Schubert’s writing makes keeping track of which is which a perplexing task. (I try to reflect this aspect of the composition in my reduction by exchanging the staves of blue/orange dux/comes parts at the boundary of sections B and Aʹ, and by extending certain (orange) C-series groups with dashed blue lines, and vice versa, when certain overlaps and parallelisms wind the threads round one another.) Lerdahl and Jackendoff’s analytical commentary does not provide detailed reasoning as to why they group according to the leading lower voice grouping during A section, but according to the upper voice during the B section. I suppose it is a result of their treating the composition as being essentially monophonic, which is for me, a confounding prospect. My P-series groupings represent a broader level grouping of phrase-level thematic parallelisms that take into consideration both D-series and C-series groupings. The P-series groupings correspond quite closely to the highest-level grouping structure in GTTM’s analysis. The lowest two levels of metrical structure in GTTM’s dot-diagram are conveyed in my durational reduction, where every two of Schubert’s notated measures are grouped as one measure in the reduction according to the harmonic rhythm and repeated bass pitches. The 120 third (and tentative fourth) level of metric structure from GTTM’s dot diagram is not reflected in my reduction because there is, for me, no intrinsic aspect of organization that points toward that level of metric structure. Finally, one last kind of grouping structure is added below the staves labeled with Z-series brackets. These communicate a phrase/grouping analysis distilled from Dahlhaus’s informal prose description of the piece’s phrase structure, in which he recognizes the composition as being made up three consecutive ten-measure phrase-groups (Z1, Z2, and Z3 in my example), the first and second of which are overlapped by two of Schubert’s notated measures. The dot-diagram from GTTM shows that Lerdahl and Jackendoff assume a regular quadruple hypermetric structure to underpin this excerpt, despite some unusual anti-parallel metric alignments this causes (especially noticeable in the A' section). They write: when groups become irregular in length, either a large-scale metrical adjustment of some kind occurs or large-scale metrical levels simply fail to apply. But Schubert … accomplishes the feat of maintaining hypermetrical beats four bars apart throughout, within the irregular grouping context of … 10+8+10+10 [bar groups]. (GTTM, 250) In their language it appears that they think of the nine- and ten-measure groups as distortions of an eight-measure norm (e.g., the phrases “become irregular in length,” emphasis mine). This framing is likely inherited from Schachter’s analysis, where he describes the B section “as resulting from the expansion of an eight-measure phrase” (Schachter 1999, 72). Lerdahl and Jackendoff also follow Schachter in assigning “strong beats” to mm. 3, 7, and 11, as opposed to mm. 1, 5 and 11. To paraphrase, they explain that this choice is preferred because [1] this way the hypermetric structure is better aligned with (their) grouping structure (under their metric preference rule 2); and [2] this reading allows for the “continuance of [metric] regularity into 121 the B section,” where they (and Schachter) feel it is more natural to hear the C♯-major sonority as strong relative to its intervening dominant seventh. Lerdahl and Jackendoff admit two features that present a challenge to their understanding of metrical structure. One is that their reading “results in a cadential 6/4 in a relatively weak metrical position” as in Schubert’s m. 5, for example (GTTM, 252).68 The other problem is more pertinent to the metrical/phrase-rhythmic issue at hand. That is, the continuance of the regular quadruple (hyper)metric structure from B into the Aʹ section means that the entirety of the material in Aʹ appears in the opposite metric orientation as the parallel material did in the initial A section. They remark with some astonishment that Schubert “accomplish[es] the feat of maintaining hypermetrical beats four bars apart” despite the significant irregularity caused by the nine- and ten-measure group lengths (GTTM 250–251). It remains unclear to me what—if anything—Schubert does to maintain hypermetrical beats four bars apart, but certainly Lerdahl and Jackendoff’s dot-diagram represents an extrinsic metric structure that accomplishes said feat. The only explanation that GTTM offers for their antiparallel alignment in Aʹ is that the “regularity [from out of the B section] then persists—… ambiguously—to the end of the piece … the metrical periodicity proceeds uniformly throughout, despite irregularities in grouping structure” (GTTM, 252). The same alignment 68This is not a problem for Schachter, who—citing precedent in Schenker’s examples of weak 6/4 chords—describes it as an “apparent” 6/4 that is in fact a bass voice anticipation of the subsequent dominant during continuation of a middleground predominant function b-minor (Schachter 1999, 73). One further complication that stems from this reading, is that it implies all of the A-major <V7, I> progressions in the A sections are “strong-weak.” This would not be a problem, except that the rest of the analysis hinges on the idea that it is “natural” to hear the V7/C♯ to C♯ (the local tonicized harmony within section B) as “weak-strong,” and so is the <V7 of IV, ii6> progression in mm. 29–31. This is a sort of second-level anti-parallelism problem not discussed in GTTM, nor in Schachter. 122 problem is not an issue for Schachter’s analysis. This is because his “weak” and “strong” are qualia that are attributed to events in the middleground harmonic rhythm on the basis of their harmonic function and role within the (Schenkerian) prolongational structure, meaning the hypermetric determinations are not contingent on equal measures of time. Schachter works out this position over the course of his three-part “Rhythm and Linear Analysis” articles, especially in the final “Aspects of Meter” (Schachter 1999, 79–117).69 (The influence of Schachter can been seen in the many examples where I use a sort of “fundamental bass” annotation to illustrate the way in which harmonic change—real or implied—can define grouping beginnings and nominate beat-hood for timepoints independent of equidistance in time, even if the language I more often use surrounding the examples is from Benjamin (1984)). By contrast, Lerdahl and Jackendoff’s “weak” and “strong” are qualia of positions within the imposed grid of their metrical structure, more-or-less independent of the action of the music.70 One wonders if a certain skepticism about their own conclusion made it into the text in the form of the curious formulation about the metric structure in the Schubert Valse whose regularity “persists … ambiguously” (GTTM, 252). 69 Particularly relevant subsections include “The Metrical Accent Accrues to the Compositional Event” (1999, 82), and “Unequal Partitioning: The Quasi-Metrical Effect” (idem, 92–93). Further, Schachter’s use of scare quotes in the sub-section headings “‘The’ Meter” and “‘The’ Hypermeter” highlight the independent status of prolongational structure (and the proportions among its durations) from metrical schemas in his analyses (idem, 97–99). 70 This conception of “strong” “weak” as properties related to positions in a well-formed metric grid stands in stark contrast to Cooper and Meyer, for whom accent and unaccent—together with some awareness of the number of pulses between accented events—constitutes meter (Cooper and Meyer 1966, 4). 123 Example 2-7: Schachter, GTTM, and Dahlhaus on Schubert’s Valse Sentimentale op. 50 no. 13 {{1 3 5 7 9 11 13 15 17. . .... . .. . ... . .. . .... . .. . ... . .. .& ###durational reduction 2:1GTTMgroupingstructure> > > >? ###GTTM metric structure ( ) ( )& ###reduction ofright-handpart as2(+) voicecounterpointAP1.1 C1.1 C1.2P1.2C1.3 C1.4& ###D1.1 D1.2 D1.3 D1.4 Cx/Dx//?###Zehntaktigkeit10-measuredness(after Dahlhaus)Z1.1Z1.2Œ Œ œ œ œ œ œœ œœ œœ œ œ œ œœ œœ œ œœ œ œ œ œ œœ œœ œ œ œ œœ œœ œ œœ œ œ œn œœ œœ œœ œ œ œ œœ œœ œ œœ œ œ œ œ œœ œ œ œœ œ œœ œœœ œ œ œ# œœ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™Ó œ ˙ œ œ ˙ ™ ˙ œ œ œ ˙ œ œ œ œ ˙˙ œ œ œ œ ˙ ™ œ œ œ œ œ ˙˙ œ œ œ ˙ œ œ œ œ œ ˙ ˙ œ œ œ ˙ œ œ œ œ œ œ œ œ œ œ ˙˙ Œœ œœ ™˙˙˙ ™™™œ ™ œ ™˙˙ ™™œ ™ ˙˙ ™™œ ™ œ ™ ˙˙˙ ™™™œ ™ œ ™ œ ™˙˙˙ ™™™œ ™ œ ™˙˙ ™™œ ™ ˙˙ ™™œ ™ œ ™ ˙˙˙ ™™™œ ™ œ ™ œ ™˙˙˙ ™™™œ ™{{19 21 23 25 27 29 31 33 35 37.... . .. . ... . .. . ... . .. . .... . .. . ... . .. .& ###durational reduction 2:1GTTMgroupingstructure> > #> # ## #Æ Æ >#> # ## #> n? ###( )GTTM metric structure [ ? ] ( )& ###reduction ofright-handpart as2(+) voicecounterpointBP2C2.1 C2.2A'P3D3.1 D3.2 D3.3 //& ###D2.1 D2.2# D2.3?#// C3.1 C3.2?###Zehntaktigkeit10-measuredness(after Dahlhaus)Z2# # #Z3œ œ œ œœ œ# œ# œ œœ œœœ œ œ œœœ œœ œœ œœœ œ œœ œ# œ# œ œœ œœœ œ œ œœœ œœ œœ œœœ œœœ œ œœœ œœœ œœœ œ œ œ œœ œœœ œœœ œ# œ œœ œœ œ œœ œ œ œ œ œœ œ œ œ™œ œ œœ ™™ œj ŒœœJœ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œ ™ œJ ŒÓ œ ˙ œ# œ# œ œ œ ˙ œ# œ# œ ˙ œ ˙n œ œ˙˙ œ œ œœ œ ˙ œ œ œ œ œ ˙ ™˙˙ ™ œ# œ ˙ ™œ œ# ˙œ œ# ˙ ™ œ# œ ˙ ™œ œ# ˙œ œ# ˙ œœ ˙˙ œœ œœ ˙˙ œœ œœ œœ œœ œ œ œ œ ˙ ™˙˙˙ ™™™œ ™ œ ™ œ ™˙˙˙# ™™™ œ ™ ˙˙˙ ™™™œ ™ œ ™ œ ™˙˙˙# ™™™ œ ™ ˙˙˙ ™™™œ ™ œ ™ œ ™˙˙˙<n> ™™™œ ™ œ ™˙˙ ™™œ ™ ˙˙ ™™œ ™ œ ™ ˙˙˙ ™™™œ ™ œ ™ œ ™˙˙˙ ™™™œ ™ 124 To summarize so far: GTTM follows Schachter in describing the ten-measure phrases of the Valse as expansions and distortions of would-be eight-measure phrases (recall also PRiTM’s “basic phrase”). They agree on the (hyper)metric interpretation of the A and B sections (i.e., regular strong-weak alternation at the two- and four-measure levels, with the first strong beat in m. 3), but differ on the (hyper)metric interpretation of the Aʹ section. This is because GTTM’s regular (hyper)metric structure is extrinsic, and the question of phrase structure pertains to the “fittedness” of the grouping/prolongational structure to the metric structure; while Schachter’s hypermetric structure is intrinsic to the fundamental harmonic structure, and so there is no question as to the music’s “fittedness” to a metric grid, since the middleground harmonic rhythm is the thing that creates the phrase-level meter structure. Mark Evan Bonds describes how schemas can fall into one of two categories: analysis according to conformational schemas, which are “based on the comparison of a specific work against an abstract, ideal type,” and according to generative schemas, which proceed by “[considering] how each individual work grows from within and how the various elements of a work coordinate to make a coherent whole” (Bonds 1991, 13–14). GTTM’s phrase/metric analysis takes a conformational approach when it evaluates the irregular groupings of the Valse according to their alignment with regular (i.e., periodic) metric structure. Schachter’s phrase/metric analysis is closer to a generative approach, in which metric and phrase-rhythmic structure arise out of Schenkerian fundamental structure and the temporal aspect of the harmonic middleground.71 71 Here, I am suggesting that the (hyper)metric outlook of Schachter is generative, since it is defined by the proportionality of middleground harmonic rhythm, not by fitting harmonic-prolongational 125 Carl Dahlhaus takes issue with Lerdahl and Jackendoff’s analysis of the Schubert Valse, and his objection illustrates the difference between analyses which assume an inter-opus frame and those which assume an intra-opus frame. He does not abide GTTM’s willingness to position the (apparent) cadential 6/4 as being in a weak metric position and does not accept structure to a regular grid. Whether analysis that makes use of the Schenkerian Ursatz represents a conformational or generative outlook is a difficult question (made even more complex by the divergent applications of prolongational analysis). There are ways in which details of the graph for a certain piece might be noteworthy for their differences from other more typical background structures, and in that sense, it is conformational: the piece is described in terms of conformity to “the [usual kind of] Ursatz” as a prototype. On the other hand, Bonds’ definition of the generative outlook as “how each individual work grows from within and how the various elements of a work coordinate to make a coherent whole” is very close to a workable definition of Schenkerian “organicism.” Eugene Narmour gets at this issue when he describes Schenker’s “assimilation-accommodation problem” where the Ursatz is “both origin and goal” (Narmour 1977, 41ff.). I tend to think of the Schenkerian approach as being more generative than conformational, though that depends on the practitioner. Critical evaluations of Schenkerianism in light of Schenker’s reprehensible racist views (which he understood as relevant to of his theory (Ewell 2020)) are pertinent to the question of the Ursatz and conformational vs. generative method. Suzannah Clark’s essay (2019, 144ff.) offers one account of how the commitment to a pre-existing idea of what the “correct” fundamental structure for all/most pieces ought to be, led Schenker (and some followers) to exclusionary positions that overlook important and potentially “structural” elements of compositions because of their non-conformity with what Schenker took to be prototypical fundamental structure. (Within the Schenkerian literature, the distinction between “salient” and “structural” features addresses this concern, to a degree). Alternatively, generative outlooks on prolongational analysis resemble Schenker’s approach in their basic tools and can sidestep the exclusionary trap so long as the notion of what a piece’s “fundamental structure” ought to be is not overly prescribed, and so long as the analysis seeks to describe the emergence of a middle-ground or background structure, whatever type of thing that turns out to be in the particular case. Examples of a generative outlook on prolongational analysis include David Loeb’s “Analytic Study of Japanese Koto Music” (The Music Forum IV, 1976), and more recently Somangshu Mukherji’s “From Bernstein to Bhatkhande” (AAWM Journal, 2023), which draws connections between the prolongational analytical methods of Schenker and of Vishnu Narayan Bhatkhande. The various attempts at using prolongational analysis for ‘post-tonal’ musics could also be put into this category. In the context of my phrase-rhythm project, the issue is pertinent because middleground prolongational structure often informs the attribution of temporal qualia to different points in the musical flow. Rothstein’s (extrinsic) hypermeter quantizes and simplifies this aspect of structure, filtering it through GTTM’s conception of metrical structure. The more flexible, more qualitative understanding of Rothstein’s hypermeter be glimpsed in instances where intrinsic to the basic phrase, which is to say, wherever he keeps closer to Schachter than to GTTM (see section 1.3.2). 126 the anti-parallel alignment of the Aʹ section. Most interesting of all, though, is that he does not accept the idea that the three ten-measure groups are to be understood as irregular distortions of an eight-measure norm. Instead, Dahlhaus describes how the “seemingly regular eight-measuredness forms an exception to the norm of ten-measuredness [Zehntaktigkeit]” (Dahlhaus 1987, 193).72 In other words, where both Schachter and GTTM offer analysis that takes the eight-measure group to be a prototype (from which most of the Valse’s phrase-groups depart), Dahlhaus understands ten-measuredness to be an intra-opus norm to which the single instance of apparent eight-measuredness is an exception. Schachter’s and GTTM’s idea of phrase-length norms is an inter-opus frame, distilling the norm from what “most pieces” (in whichever corpus) do “most of the time.” Conformational metric/phrase analysis fits under the umbrella of what Leonard Meyer calls “style analysis … [which] describes the patternings replicated in some group of works.” But as Meyer is careful to note: this invites a rather large question of precisely which group of works serves as the “set” from which the stylistic norms are to be determined (Meyer 1989, 38). Some possibilities he offers are: the works of a single composer, works from the same geographical area or national tradition in a certain time frame; works written over a considerable period of time in but in a single cultural group, the music of an entire human civilization or some segment thereof (ibidem). I do not object to the idea that there exist certain repertoires for which it is reasonable to say that eight-measuredness or other duple phrase lengths are bona fide norms. But the wider and more diverse the corpus, the less convincing 72 Dahlhaus’s GTTM review is written in German, the translation is mine (with thanks to Alan Dodson). Original German: “…die scheinbar reguläre Achttaktigkeit eine Ausnahme von der Norm der Zehntaktigkeit bildet.” 127 phrase-length normativity becomes as a point of analytical departure. Wherever one’s analytical approach assumes a conformational outlook default, the analyst risks losing sight of the ways in which an “irregular” and idiosyncratic piece creates its non-periodic structures in the very same manner by which other pieces deemed “regular” (according to whichever set) create theirs. While it is something of an extreme step, I am interested in exploring—via the generative outlook and intra-opus frame—what sorts of stories of phrase-rhythmic structure can be told when the “set” of pieces for the style analysis is restricted to a corpus of one.73 In this manner, the way in which a highly idiosyncratic piece creates its (irregular?) phrase rhythms can be described in same terms used to describe how an idiomatic piece creates its conventional ones. Dahlhaus’s analysis provides a model of this intra-opus/generative outlook: it proceeds according to what this piece does most of the time. With regard to the Luther example earlier in this chapter, the Horlacher-inspired metric-reinterpretation analyses were a conformational approach with an inter-opus frame, while the IFUNC analysis of the same offered a generative approach with an intra-opus frame. Conformational/inter-opus and generative/intra-opus are often coupled in this way, though it is not necessarily the case, since there are many ways in which an analysis might drift between conformation and generative outlooks, for instance the way that Rothstein and Schachter’s (generative) Schenkerian sensibilities combine with a 73 An exemplar of a music-theoretical approach that is quite literally developed for a “corpus of one” is offered by Benjamin’s “Pour les Sixtes: An Analysis” (1978), in which he outlines a theory of tetradic harmony and voice leading tailored specifically to the compositional constraint of a single Debussy Etude. Most stacked sixths could, of course, be described in terms of (extended) tertian harmony; but Benjamin’s analysis, crucially, establishes that the sixth-stacked tetrads “are not supercharged triads” (1978, 256). He goes on to characterizes the Etude’s explicitly tertian harmonies as “interruptive” (idem, 257) relative to the intra-opus norm of harmonic tetradicity. 128 conformational approach to phrase-length norms. Certain theories might be understood as coupling a generative method with an inter-opus frame: for example, GTTM’s metric well-formedness rules are determined generatively and according to wide inter-opus frame of reference, but then, the resulting metrical structures applied in a conformational way at the stage of analysis. The Z-series brackets in Example 2-7 illustrate Dahlhaus’s interpretation of the piece, which are largely based on harmonic and motivic parallelisms (though he does not go into detailed rationale). A five-element harmonic progression, <I, ii6, V6-/4-, V, I>, is the basis of the A sections—each element occupying two measures. The end of the first ten-measure group is overlapped or elided with the beginning of the second. Dahlhaus assumes regular strong-weak alternation beginning from first measure pair (mm. 1–2 as strong, mm. 3–4 as weak, and so on), whose pattern is not interrupted by the overlap at mm. 10–11, since the overlapped phrases were both odd in length (see also the question of elision in five-measure hypermeter in the context of Rothstein’s theory in Chapter 1 section 1.3.1, Example 1-8). For Dahlhaus, interruption to the measure-pairs occur only at mm. 18–19 and mm. 28–29 where he describes “the collision of two strong two-bar groups” as a metrical clash akin to the “abrupt modulation from A major to C♯ major and back again” occurring at those same locations.74 My Example 2-7 captures this aspect of Dahlhaus’s interpretation, but describes it in terms of grouping parallelisms instead of as metrical weak/strong alternation (as done for the analyses in section 1.4 of the previous chapter). Following Schachter’s cue to listen to the 74 “Und daß bei den Übergängen zu Takt 19 und Takt 29 zwei schwere Zweitaktgruppen aneinander stoßen, erscheint als metrisches Korrelat zu dem abrupten Tonartwechsel von A-Dur nach Cis-Dur und umgekehrt” (Dahlhaus 1987, 193). 129 middle voice as lead, the D-series melodic groupings are defined not simply by a gestalt segmentation of the surface melody, nor based on an extrinsic “norm” of two- and four-measure groups as default, but rather according to the melodic and motivic parallelisms revealed in the shallow-middleground durational reduction of the middle voice staff in Example 2-7. It is in this sense that the first group D1.1 can be imagined to begin from the first beat of the first measure (rather than from midway through Schubert’s m. 2 where the C♯5 first occurs), since, upon reflection—and especially after hearing it in relation to D2.1 and D3.1—my perception is that C♯5 as Kopfton corresponds to the supporting initial tonic harmony from beat 1 of the first measure. (In turn, this implies for D.1.1 the same <long, short, short, …> motivic rhythm that is more obviously stated by the groups C1.1 and C1.2.) I attribute the “collisions” of two-measure groups [Zweitaktgruppen aneinander stoßen] that Dahlhaus describes (Schubert’s mm. 18–19, and 28–29) to the dynamic interaction of co-present grouping criteria. Specifically, these are the moments where prospective two-measure groups—Cx/Dx which is presumed parallel to either C1.4 or D1.4 at end of A; and D2.3 which is presumed parallel to D2.1 and D2.2 at end of B—are cut short when new higher-level parallelisms are begun at the mid-point of their proposed lower-level groups. That is, the continuation of Cx/Dx interrupted and replaced by beginning of P2/D2.1 as strongly parallel to beginning of A (with D1.1 in its m. 1 beat 1 form); and the continuation of D2.3 interrupted replaced by A/Z3 as strongly parallel to B/Z2 and A/Z1. 2.2.3 A Way Forward: from Inter-opus to Intra-opus, and Conformational to Generative Schemas are useful for positioning a specific composition in relation to a larger body of work. As Leonard B. Meyer put it, “[schemas are] useful constants in terms of which style 130 differences, between contemporaries and over time, can be observed, identified, and classified” (Meyer 1989, 51). It is common to treat schemas as a “normative” or “natural” state—that is, “form” in the quasi-Platonic sense—and to describe the particularity of certain passage of music in terms of the extent of its difference from that “form” as of abstract ideal. I am suggesting here that (hyper)meter and phrase prototypes (e.g., “the sentence” “the period” etc.) are kinds of schemas, but ones that take the set of pieces from which the schemas are generated to be an enormous corpus something like “most notated European music from Bach to Brahms,” or similar. Yet the principles of normative phrase and hypermetric structure are not in fact the result of an enormous corpus study; but rather are rooted some combination of mid 20th-century studies of perception and 18th-century European ideas of “natural” proportion and numerology (recall the epigraph to Chapter 1, where Lester notes that “appeals to nature or cosmology have by used by many theories over the centuries to find a basis for then-current musical conventions. In all cases, changing styles … [revealing] the inadequacy of the appeal" (Lester 1986, 189)). A conformational, and (wide) inter-opus approach is not fundamentally flawed. But while that is one of the uses of schema thinking, it is also worth thinking seriously about how a prospective schema is formulated in the first place. Does each schema exist as a pure Ur-type somewhere out in the ether (as some imagined ultra-orthodox version of conformational view of phrase-types might have it)? No, schemas are built up little by little as a sort of residue, which are the trace aspects of kinds of family resemblance between discrete pieces in a body of work.75 It is more than likely a corpus study of phrase-unit lengths across, 75 Matthew Brown’s review/article “Rothstein’s Paradox and Neumeyer’s Fallacies” (Brown 1998) includes a discussion of the role and nature of “prototype” in Schenkerian thinking. Brown’s humorous 131 say, the first Viennese school, would reveal/confirm the four-bar norm. The issue is that reflexively appealing to that norm can render less conventional structures invisible. In theory, a most idiosyncratic phrase—whether from within, or without the corpus from which a set of “norms” is generated—ought to be described as generating its particularly “irregular” structure in exactly the way that other more conventional phrases generated their “regular” ones. The fact that the prospective schema suggested by a single piece—“intraopus style and intraopus structure” in Meyer’s terms (1989, 25)—does not closely resemble the schema arrived upon by some wide inter-opus generative approach should not invalidate the finding of an idiosyncratic form, or cause us to abandon an intra-opus approach. I have shown that prevailing approaches to phrase and meter such as GTTM and PRiTM generally assume an inter-opus frame, and understand meter/phrase structures as largely conformational schema.76 and helpful example of the defines “prototypical cat… a mammal with retractable claws, craving for canned tuna, fur, a tail, and a capacity to purr, [though] not all cats display these features or forms of behavior,” and reminds the reader: “perfect examples of a prototype may not “exist” in the world around us” (idem, 99). I would add: the fact that housecats are, for many of us, the most commonly encountered instantiation of the “cat” prototype does imply that “housecat” is somehow closer to Ur-cat-ness than other more “exotic” felines are. 76 This is not to say that GTTM or PRiTM is exclusively conformational in outlook. Balanced against the conformational analytical approach in much of Rothstein’s PRiTM is a generative aspect stemming from the Schenkerian roots of his project: each piece will create and establish an Ursatz (or basic phrase) in its own particular way. Note also that Rothstein’s many analyses outside of PRiTM are not constrained by a conformational approach: see especially “Ambiguity in the Themes of Chopin’s First, Second, and Fourth Ballades” (Rothstein, 1994). The generative aspect of Lerdahl and Jackendoff’s GTTM (promised in the book’s title) has to do with its nod to mid-20th-century Chomskian linguistics. Although they conceive of their project as generative, their “structures” (including well-formed meter) are established by characteristic features that recur across many pieces—the intra-opus frame. Within their corpus, emerging idiomatic (and as it happens, regular) metric situations become “schemas” that then become conceptually separable from the particular action of a given piece of music. Once separated and applied to a specific analysis such as the Schubert Valse, their approach becomes is overtly conformational. 132 Rothstein’s “basic phrase” for example—while nearer to a piece-specific model than other more abstract prototypes (like a content-less hypermetric norm of 4+4)—exists, conceptually, outside of the actual sounding phrase, where it to some degree informed by norms of “other pieces.” In many cases a sounding phrase that matches the proportion and hypermetric structure of the basic phrase may not occur at all, so, in that respect the sounding music is measured “from without” and according to norms from elsewhere. Similarly, for Lerdahl and Jackendoff, the idea of well-formed metric structure seems to be a purely abstract aspect of musical structure that is conceptually separable from the real metric structures present in an actual piece, where its regularity is subject to distorting manipulations caused by (more real?) aspects such as grouping and prolongational structure. Their widely known metric well-formedness rules might be likened to a set of precise instructions for how to construct a very accurate and well-made yardstick. The description of how to make one might even be understood as generative: [1] a survey of the “ways to measures stuff” corpus leads us to following definition of an imperial inch; [2] etching onto a true piece of wood, combine twelve inches for each foot [3] define each yard as three feet, and your well-formed yardstick is complete. The yardstick (like their metric structure) exists as a kind of measure even in the absence of a particular thing to be measured.77 However, the meter as grid approach inevitably adopts a conformational outlook when it comes time to be 77 Lerdahl and Jackendoff announce, at the end of their section of metric well-formedness that “[h]aving defined the possible metrical structures for tonal music, we [now] turn to the problem of relating these structures to a presented musical surface” (GTTM, 74). 133 used as an analytical tool.78 Conformational approaches typically understand the library of allowable metric/phrase structures (i.e., well-formed meters, basic phrase types) to be relatively small, but then go on to describe considerable diversity in the ways in which a given piece might interact with and refer to one (or more) of those structures. As argued in the preceding examples, this—at a certain point—causes a problem for the description of phrase rhythm in irregular musics when it becomes unclear as to why some particular regular schema should be relevant for the irregular passage. Earlier examples have shown that one of the unfortunate outcomes of the conformational approach to phrase is that ostensibly positive attributes of phrase (variety of forms, phrase-rhythmic effects, and interesting proportional relationships between spans) are described in overwhelmingly negative terms (i.e., phrase deformations, deletions, “weakly” metric, metrically “shallow”). The approaches described as an alternative in section 2.3 will expressly adopt an intra-opus frame and understand metric/phrase structure as generative schema. Counter to the conformational outlook, in this view the library of “allowable” metric/phrase structures invites comparison to Borges’s library of Babel. What is relatively limited is the library of strategies for meter/phrase creation. But the intra-opus/generative view observes considerable—perhaps infinite—diversity in the kinds of metric and phrase-rhythmic situation which can be brought about via this relatively modest set of strategies: melodic and harmonic grouping as conceptually separable, co-present, and interactive structures. 78 As useful as the well-formed yardstick might be to someone laying paving stones, it will have little to say about whether, say, a bowling ball is a perfect sphere. The musical examples in this dissertation I explore a musical equivalent to this kind of measuring problem. 134 Elements of Schachter’s approach—where (hyper)metric structure is defined principally by the harmonic rhythm of the middleground structure, not by an extrinsic grid—have already previewed an approach that is both intra-opus in frame, and generative in outlook. Especially once coupled with Dahlhaus’s commentary, Schachter’s approach to the Schubert Valse provides an example of a way to define phrase-rhythm regardless of how regular the hypermetric structure is. That aspect of Schacter’s theory of rhythm is overlooked by GTTM and PRiTM, which both default to emphasis on periodic structure that Schachter does not require.79 The limit of the Schachter approach does not have to do with aperiodicity, but atonality: that is, it can only define the (quasi)-hypermetric structures in pieces for which a Schenkerian/prolongational reading is plausible.80 In the section that follows I introduce William Benjamin’s concept of the “harmonic image” as one group that may serve a similar 79 It is true the hypermetric structures Schachter features as his analytical examples are fairly regular, but he points out that they would not have needed be so, writing: For this essay, I selected pieces in which groups of measures exhibit clearly defined metrical organization. To forestall possible misunderstandings, I should perhaps mention that there are pieces—indeed, whole categories of pieces—where the groupings are less regular. I have sometimes found that durational reduction can reveal interesting aspects of the free rhythmic structure of such pieces. But they are more difficult to understand than the ones that I have presented and, therefore, are less suitable for an introductory discussion. (Schachter 1999, 76) Ironically, his effort to forestall misunderstanding brought about some new and unforeseen misunderstandings: the mistaken idea that his theory expects any metric regularity at all. 80 While I cannot address each of them within this chapter are other noteworthy contributions to study of phrase-level rhythms of European common-practice tonal music that, like Schachter’s tonal and durational rhythm, operate mostly as a way of attending to patterned prosodic and rhetorical stresses, and proportionality among spans defined by harmonic rhythm and middleground prolongational structures. These include general Schenkerian approaches to tonal rhythm from Morgan (1978) and Lester (1986); as well as more recent work informed by a Schenkerian approach specifically dealing with questions of expansion and irregularity (independent of hypermeter), such as Samarotto (1999a; 1999b), Braunschweig (2004), Willner on “irregular” phrases (1995; 2007), and IJzerman on meter in Couperin’s Préludes non mesurés, and pattern-based approach to harmonic phrases (IJzerman 2011; 2017). 135 function as Schachter’s reductions and prolongational structures to a stable kind of structure which can co-exist and interact with melodic grouping to afford phrase rhythm in music that is neither periodic, or “tonal” in the conventional sense without require any certain harmonic syntax or variety of prolongational structure in order to be salient. 2.3 Meter as Grouping Zuckerkandl’s view of musical meter highlights qualitative rather than quantitative aspects of metric structure as being of foremost importance, arguing that the “flow” or wave-like oscillations of music’s metrical qualities ought to be foregrounded in discussions of music’s time-organization. He sees the more scientific, quantitative evaluations of music’s temporal design to be arbitrary, and indeed quite foreign to musicality, writing that “for the natural sciences, ‘time’ is merely another word for the fact that natural processes are susceptible of a certain kind of measurement, measurement by clocks. The second hand of our watch moves 498.7 steps forward while a light ray passes through the space from the sun to the earth: that is all” (Zuckerkandl [1956] 1969, 203–4). Whereas he describes music as being measured in a felt way, rather than in a chronometric way, writing: Hearing music, we oscillate with its metric wave. Each tone falls on a particular phase of this wave; each phase of the wave imparts to the tone that falls on it—and, through the tone, to the auditor—its particular directional impulse. Not because I count “one” to a tone (or because the tone was emphasized by an accent—for often it is not) do I know that I am at the beginning of a measure, but because I feel that, with this tone, I have reached the wave crest and at the same time have been carried beyond it, into a new wave cycle. (Zuckerkandl [1956] 1969, 204–5) The theme of “meter as flow” or “meter in flux” from Zuckerkandl’s rather poetic account of music’s wave-like temporal aspect is likewise explored by Wallace Berry, who describes musical rhythm as “playing an essential and telling role in the delineation of processes of growth and 136 decline, climax and subsidence, stability and flux” and speaks of metric rhythm “as the rate and pattern of metric change, or a tempo rhythm in music” (Berry [1976] 1987, 304). Most relevant to the topic of this chapter and section, Berry thinks of meter not as a (chronometric) grid, but rather as a special sub-variety of grouping, where [1] a metric unit is a group of events (or perhaps a single-element group) initiated by an accent of some kind, and [2] each group takes on an “impulse,” that is, a particular orientation toward surrounding groups— which he symbolizes using curved, and straight arrows that resemble conducting gestures ([1976] 1987, 317–18). With his “impulse symbols” Berry conveys a relational qualitative aspect accrued by grouping units that become metric, and they become metric precisely because of the interrelatedness that emerges for the passage (e.g., the ending previous unit as beginning of new unit; the oriented-ness of one unit toward the next (anacrustic), or oriented-ness of one unit toward the preceding (continuation)). Importantly, for Berry, the emerging metricality of a passage is not dependent on the durational equality among the units, but rather, is simply the feeling of “proportional interrelations of such groups at all levels” whether proportions among those groups are symmetrical, asymmetrical, ambiguous, or even “asymmetrical at one level, and symmetrical at another” (Berry [1976] 1987, 318). This is not to say that the units that become metric are not also equal, or near-equal in their duration; or that entrainment behaviour to isochronous pulse is not involved in sustaining metrical structure in a special way—indeed that is often the case. But in a section he titles “the concept of meter as, by definition, subject to fluctuation” Berry explicitly argues that differing styles of music explore fluctuant metrical organization to greater, and lesser degrees, but that it is a mistake to designate the more fluctuant musical contexts as being essentially different from the more-or-less regular ones; concluding instead, that “even much music commonly regarded … as regular 137 in metric grouping is indeed highly variable at least at certain levels. We thus avoid any suggestion of a terminological equation in which meter = regularity of unit grouping” (Berry [1976] 1987, 319). This tenet of Berry’s theory of meter is foundational to the theory of phrase-rhythm as-grouping-complex developed in this dissertation. Beats and measures are not (extrinsic) “container”-like structures that offer a way to measure whatever music happens within/against them. Instead beats and measures are among then aspects of organization that can result from grouping conditions, in turn becoming a kind grouping themselves. “Grouping” in this sense represents a much larger family of perceptual activities than in GTTM or PRiTM where the same term refers specifically to a certain kind of segmentation of mono- or homophonic texture that—in the end—settles on one “grouping structure” resolved of (nearly) all ambiguities.81 Adopting this analytical outlook on meter and grouping for the remainder of this project, I do not draw any fundamental distinction between music that proposes regular units (affording entrainment to pulse), and music which proposes units whose durations tend to be more widely varied. The phrase rhythms that I am interested in describing frequently present non-uniform 81 Lerdahl and Jackendoff provide an important caveat early in GTTM, explaining that they “are treating all music as essentially homophonic; that is, we assume that a single grouping analysis suffices for all voices of a piece. For the more contrapuntal varieties of tonal music, where this condition does not obtain, our theory is inadequate” (GTTM, 37). As argued in my chapters 1 and 2, I propose separate independent groupings for different strata of a composition: e.g., distinct (segmentation-like) groupings for accompaniment and melodic layers. This is totally antithetical to the concept of grouping as proposed in GTTM. However, according to Berry’s definition of grouping, not only are GTTM’s “groupings” groupings; so too are its prolongational structures, metric structures, time-span reductions. Thus—though Lerdahl and Jackendoff would object to calling all these things “grouping”—it cannot be said that the kind of “polyphonic” attending to different aspects of segmentation is prohibited within the GTTM outlook at large, even though it is expressly prohibited within their formalized notion of “grouping.” 138 grouping conditions which are not entrainable at some levels: not only because they are too irregular; but also because they exist at levels of metrical structure which may be “deep” (i.e., slow) enough that entrainment behaviour would not arise even if near-isochrony were present. Neither of these scenarios need present an obstacle to phrase-rhythm analysis when we replace the narrowly defined meter and grouping with the wider conceptions just described. Another precedent for the “widened” definition of musical meter adopted in this dissertation is found in Benjamin (1984), who follows Berry in taking meter to be a way of partitioning musical time by accent that involves attending to dynamic, and proportional relationships between those proposed units. To the basic tenets of Berry’s approach, Benjamin adds emphasis on meter’s cyclicity, describing beats as a special kind of cyclic “equivalence class” among time-points. This element of Benjamin’s outlook resembles the conventional “meter-as-grid” view so far as dot-diagrams, too, visually communicate a kind of equivalence class for “places” in a measure: e.g., given a “4/4” metric structure, beat ones are like all other beat ones in that they participate in the whole-note pulse stream and all of its isochronous subdivisions. However, Benjamin’s theory differs from the kind of equivalences among timepoints suggested by dot-diagrams in that, for him, the basis for their equivalence might result from factors other than their equidistance in time. The example from the opening phrases of the first of Stravinsky’s Three Pieces for Clarinet Solo [1913], which I use in the following section, to introduce Benjamin’s concept of the “harmonic image” and the accent by “image-shift” (Benjamin 1984, 379), which, along with other modes of grouping-by-event-affiliation, are among the ways that equivalences among time-points are proposed. 139 2.3.1 Benjamin’s (non-metric?) Time-span Structures Benjamin’s “Theory of Musical Meter” is a rather complex article with multiple aims. Benjamin recognizes that there are considerable disagreements in the theory literature as to the nature of musical meter, and with regard to how meter is determined. He asks whether these differing theoretical conclusions are “the results of real differences in musical perception” or whether “rather, they can be seen as incidental products of different terminologies and ways of speaking, arising perhaps from commitments to differing intellectual traditions” (Benjamin 1984, 357).82 For reasons already laid out earlier in this chapter the theory of phrase rhythm I develop here is agnostic when it comes to this or that definition of meter, since “meter” in the strict sense is something I have shown to not be required in order for phrase rhythm to exist. Accordingly, I will not go into the particulars of Benjamin’s theory of meter per se, though I do find it to be quite convincing: the article has, in general, shaped my view on music’s time-organization on a quite fundamental level (as those who know the article well will likely have already recognized during the preceding pages of this dissertation). There are two novel contributions of Benjamin’s article—neither which deals with “meter” per se—which need to 82 The difficulty of the Benjamin article lies partly in the fact that each of the article’s five sections “departs from a survey of prevalent views and proceeds to mediate among these by integrating them in a general framework in which they are more or less easily accommodated without mutual contradiction” (1984, 355). For the present-day reader it is somewhat difficult to glean the novel aspects of the article: first because some of the “prevalent views” have remained prevalent and taken on new significance (e.g., GTTM in light of recent cognitive and psychological approaches to meter as entrainment), while—in my estimation—others (namely Berry) are less familiar presently than they might have in 1984 (or to Benjamin). Second, so much time is spent on mediating and integrating discrete and sometimes apparently contradictory aspects of different existing theories, that Benjamin’s theory of meter is presented only gradually, in a process that resembles exposing then developing an image from its negative: little by little the picture of what meter is (and is not) comes into view by patiently distilling its basic terms, determining factors, and aesthetic and practical significances. 140 be formally introduced for the present analytical purpose. The first of these is a kind of stable grouping structure that Benjamin does not give an official name, but which he casually refers to as a “time-span structure” that is conceptually distinct from Lerdahl and Jackendoff’s metric structure, grouping, prolongational structure, and time-span segmentation (Benjamin 1984, 360). This type of structure is addressed, below, in Example 2-8 and surrounding discussion. The second novel insight from Benjamin particularly useful for my analytical purposes is the concept of “accent of image-shift” and specifically the “harmonic image” (Benjamin 1984, 379), which will be addressed toward the end of this chapter, where it is presented alongside a related example from Hasty’s Meter as Rhythm (1997). An excerpt from the opening of the first of Stravinsky’s Three Pieces for Clarinet Solo provides a useful example for illustrating the co-presence of, and differences between, grouping structure in the conventional sense (i.e., GTTM’s grouping structure/Tenney and Polansky’s melodic segmentation), and “meter-as-grouping” which, in this case, is determined by a recurring and semi-regular pattern of durations and rhythmic parallelisms, reified also by implied harmonic factors. In GTTM the same excerpt appears as an example of a type of music where consideration of grouping structure is germane, but where—in their view—there is no metric structure to speak of due to the irregularity/aperiodicity.83 The same Stravinsky excerpt appears also in Benjamin’s article, where he recognizes in it a quasi-metrical time-span structure that: [1] groups events and time-spans in a hierarchical way, [2] is independent of 83 Lerdahl and Jackendoff explain “…principles of grouping structure are more universal than those of metrical structure. In fact, though all music groups into units of various kinds, some music does not have metrical structure at all, in the specific sense that the listener is unable to extrapolate from the musical signal a hierarchy of beats. Examples that come immediately to mind are Gregorian chant, the alap (opening section) of a North Indian raga, and much contemporary music” (GTTM, 18). 141 GTTM’s metric/prolongational structures that [3] co-exists and interacts with the (conventional, melodic) grouping structure. Example 2-8A reproduces the information from Benjamin’s ‘figure 3’ (1984, 361). At levels 1 and 2 Benjamin’s grouping is exactly the same as the grouping structure proposed in GTTM, except that Lerdahl and Jackendoff’s excerpt ends with the D♯4 at end of group 2.5 (GTTM ex.3.34 and ex. 3.36b, 64–5) while Benjamin’s continues a few notes further. Their respective grouping analyses differ only at the highest level (my level 3): where Benjamin’s highest-level group “3.1” affiliates groups 2.1 and 2.2 on the basis of harmonic continuity, where the analysis in GTTM instead prefers to initiate a level-3 group at my 2.2. (on the basis of the parallelism between 2.1 and 2.2). While not crucial to my argument for phrase-rhythmic effects in this excerpt; this difference is indicative of the way in which GTTM does not consider harmonic or prolongational factors as part of their determination of grouping when treating examples that do not “have” prolongational or metrical structure in GTTM’s narrow sense.84 Contrasting their Stravinsky example with a tonally and metrically conventional Mozart example that appears in the same section Lerdahl and Jackendoff, write: “the difference between [Stravinsky’s] style and Mozart’s with respect to grouping is not in its grammar as such, but in what structures the composer choses to build using the grammar” (GTTM, 66). 84 In his review of GTTM, Richard Cohn independently comes to the same conclusion as Benjamin regarding the level-3 grouping, explaining that—counter to GTTM’s preference rules—he finds it “more appropriate [to initiate a level-3 group at “2.3” rather than at “2.2”] …[because] the written A strikes my ears as fresh. This impression of a new pitch-event is confirmed by the introduction of three new pitch-classes in the ensuing phrase. This factor far overrides any other preference parameter [i.e., parallelism between “2.1” and “2.2”] in my ears” (Cohn 1985, 43). 142 This is an interesting and revealing admission, where GTTM shows that it understands the emergence of grouping structure to be a flexible and open-ended way of creating structure in line with the generative outlook and intra-opus frame of reference I argue for earlier in this chapter. Famously, though, GTTM does not extend this same outlook to metrical structure, instead taking a decidedly conformational view of metric structure based on inter-opus metrical norms. Their reasons for this decision are clear enough given the repertoire to which they apply their theory and given the stock they put in metric well-formedness. But the choice results in a one-dimensional analysis of “irregular” passages such as the Stravinsky Clarinet Solo, where the analysis begins and ends with the grouping/segmentation. Benjamin’s analysis of the same passage offers an avenue for the consideration of co-present, conceptually separable, yet interacting groupings that are informed by different aspects of musical structure; and thus allowing for an account of emerging phrase-rhythmic effects whether or not the periodicities of one of those groupings arises to an extrinsic definition of metric or phrase-type norms (as was previewed in section 1.4). Introducing the example in terms of multi-faceted aspects of music’s grouping and time-span structures, Benjamin writes: Stravinsky and other composers of the early twentieth century are ametric in Lerdahl and Jackendoff’s sense of the term, yet an analysis of such works with reference only to the grouping of events hardly provides an adequate account of the ways in which the times of these pieces appear to us to be partitioned. This much can be said about their segmentation of part of the first of Stravinsky’s Three Pieces for Clarinet Solo … and it is true also of Tenney’s segmentations, which seem to start from the assumption that the ways in which we group musical events [i.e., on a Gestalt segmentation basis] exhaust the ways in which we partition musical time. (Benjamin 1984, 359–60) 143 The brackets below the staves in Example 2-8 represent a “time-span structure” which is not metric in the conventional sense. It is not an aspect of structure for which there is comparable concept in GTTM, PRiTM, or in Schachter. Benjamin describes this structure partly in terms of its (relative) regularity by demonstrating the recurring durational pattern <2, 2, 1, 2n> (where integers are notated durations measured in eighth notes, with variable multiplier n affecting only the last duration in the series). On the one hand, the variable length of the final element in the durational pattern is only a minor disturbance to its regularity—something like a “composed-out fermata” at the end of each phrase unit. But then there also are other irregularities—namely the break in the pattern around the end of melodic group 2.3—which, together with the variable duration, would prevent a listener from cognizing this recurring rhythmic pattern based on the durational values alone. Benjamin recognizes as much and, without providing a full account of his decisions regarding the determination of time-span structure for the excerpt, explains that it “has everything to do both with harmonic ways of grouping and (especially) with hearing certain time-points as accented relative to others” (Benjamin 1984, 361–62).85 85 I do not wish to unduly speculate as to the particulars of Benjamin’s harmonic understanding of this passage. However, I am quite convinced that the “harmonic grouping” to which Benjamin refers via his barring (my brackets below the staves) revolves around an understanding of F♯ as a stable and centric harmony in the first phrase of the excerpt, which is the basis for hearing the first <B♮, G♯, C♯> melodic segment as anacrusis. I can imagine those who instead understand G♯ as centric proposing a quasi-metrical time-span structure from the initial B♮ (as downbeat). I do not entirely discount the potential of such alternate understandings of the passage, though I do find the one I have diagramed easiest to hear. Chapter 4’s Mozart example will address phrase-rhythmic issues that arise specifically because of the co-presence of more than one viable interpretation of quasi-metrical time-span structures of the same passage—an effect that I call phrase-rhythmic “multivalence.” 144 Example 2-8: Grouping Complex in Stravinsky’s Three Pieces for Clarinet Solo mvt. I A) Benjamin’s melodic, and other time-span grouping (2+2+1+2n) with elision B) Modified melodic, and other grouping and with elision and internal expansion sempre p&1.12.13.12(n) e [n = 2] 2 eQ1.1 [n = 2] Q2.1 , 1.22 e-1 e 2(n) e2.21.32 eQ1.2 [n = 1] 2 e1.41 e 2(n) e,&Q2.2 Q1.3 [n = 2] 3.22.31.52 e 2 e 1 eQ1.4 [n = 4]2(n) e2 e1.6-2 eelision-,1 e2.42(n) e2.5Q2.3 Q1.5 [n = 2]2 e 2 e,1 e3.32.62(n) e- - -,Q1.6 [n = 1] 2.72 e 2 e 1 e 2(n) e-,œj œ œ#œ#œ#œ# œ#jœ# œj‰œj œ œ#œ#œ#œ# œ#œ œnœ œ# œ# œ# œ#œ#jœ#œnœ# œ#œ œ#œ#j œ# œ≈œnjœ#jœ# œ#j œ# œ# œœ#œ# œ#œn œ#sempre p&1.12.13.12(n) e [n = 2] 2 eQ1.1 [n = 2] Q2.1 , 1.22 e-1 e 2(n) e2.21.32 eQ1.2 [n = 1] 2 e1.41 e 2(n) e,&Q2.2 Q1.3 [n = 2] 3.22.31.52 e 2 e 1 eQ1.4 [n = 2]2(n) e2 e1.6-2 eelision-,1 e2.42(n) eQ1.5 [n = 2] 2 eQ2.3 2.5internal expansion2 e 2 e,1 e3.32.62(n) e- - -,Q1.6 [n = 2]2.72 e 2 e 1 e 2(n) e-,œj œ œ#œ#œ#œ# œ#jœ# œj‰œj œ œ#œ#œ#œ# œ#œ œnœ œ# œ# œ# œ#œ#jœ#œnœ# œ#œ œ#œ#j œ# œ≈œnjœ#jœ# œ#j œ# œ# œœ#œ# œ#œn œ# 145 Again, Benjamin’s purpose in the article is to work out some common ground between the apparent contradictory accounts of meter in the North American theory literature of the day, and so, ultimately decides against calling this time-span structure “metric” in the conventional sense. He writes: We have here a structure that is not metrical … [however,] we will want to consider, at some point how a structure of this kind differs from a metric structure in pragmatic terms, that is, in terms of the psychological purposes and effects of such structures. For the present, let us recall what the example shows: A time-span structure can be constructed which neither is metrical nor corresponds to that implied by motivic/melodic-phrase grouping. (Benjamin 1984, 362) Yet, consideration of this non-metrical time-span structure allows for description of elision in the excerpt, despite passage’s lack of meter in GTTM’s sense. Benjamin’s figure (and my adaption of it) shows an elision to occur when melodic group of segment 1.6 is understood to representing both the final element of time-span structure Q1.3 and also the beginning of time-span structure Q1.4. (I do not include, here, a detailed walkthrough of exactly which factors in the Stravinsky excerpt give rise to this elision, since I narrated a comparable quasi-tonal example in the Mussorgsky example from Chapter 1. The Bartók analysis in Chapter 5 will revisit the question of this type of time-span structure in less-tonal contexts, with another example of elision.) Example 2-8B offers a slightly different interpretation of the same passage where I show that a listener attending to phrase rhythm as the co-presence of and interaction between melodic grouping and time-span structure could hear an instance of internal phrase expansion during the excerpt in addition to the elision described by Benjamin. Differences between grouping and time-span structure in my interpretation and in Benjamin’s are highlighted via blue brackets/text. The most important difference has to do with the determination of the end 146 of Q1.4 and the beginning of Q1.5.86 My version takes the time-span pattern of Q1.4 to be <2, 2, 1, 2(2)> (i.e., ending after the tied D♯4) whereas Benjamin’s Q1.4 is <2, 2, 1, 2(4)>. I do not find Benjamin’s version difficult to hear, but tend to prefer mine for two principal reasons: one is that it means the variable lengths of the final element of each Q1 segment are more uniform (4, 4, 2, 4, 4, 4, and 4 eighth notes in duration in my version; versus 4, 4, 2, 4, 8, 4, and 2 in Benjamin’s). Second, it captures a sense in which Q1.5 creates a motivic/harmonic parallelism with Q1.3, as both leap upward from a harmonic “root” in the woody bottom-end of the clarinet’s range into the dulcet mid-register to form a consonant major 7th: Q1.3 {F♮3, [A♮4], E♮4}, and Q1.5 {E♮3, [G♯3], D♯4}. (For me, this recalls also Q1.1’s leaping ascent onto a consonant major 9th: Q1.1 {F♯3, G♯4}.) The instance of internal phrase expansion within Q1.5 hinges on that same intervallic/motivic parallelism, in combination with the local normative quasi-metrical patterning <2,2,1,2n>. Q1.5’s quarter note G♯4 is as an “extra” event both durationally—<2,[2],2,1,2n>—and also pitch-wise, since it inserted between the lower and upper pitches of the comparable ascending major 9th outlined in quarter-note rhythm at Q1.1 and major 7th within Q1.3. The beginnings for Q1-level segments and sub-segments are not determined solely by rhythmic parallelisms and patternings, but also informed by a special kind of accentuation which Benjamin calls “accent of image-shift”—often a shift from one “harmonic image” to another. 86 The other differences (Q1.6 and melodic groups 2.7/3.3) are not material to the issue of phrase-rhythmic effects in this passage. Those result from the fact Benjamin’s excerpt ends with the pitch A♮4, whereas I tend to think of the subsequent G♯4 (which is followed by a breath mark) as being included within those same groups. The harmonically contrasting material that follows the G♯4 complicate the matter but are not important for the discussion of the present topic. 147 2.3.2 Benjamin’s Accent of Image-shift and the Metric Role of “Harmonic Image” (or, what—besides duration—begins at Hasty’s “beginning”?) Benjamin’s approach to rhythm and meter in many ways takes on Berry’s view of meter as a special way of partitioning of musical time by accent. But Benjamin clarifies that different varieties of accent bear different relationships to grouping, and the way in which they partition time. Common types of accent include [1] accent of discontinuity: “caused by a contextually noticeable gain in one of music’s basic dimensions other than time (pitch, loudness, tone color, space),” [2] accent of climax: “ascribed to points at which the intensity peak of a shaped pattern of activity is heard to occur … (i.e., loudness, pitch height),” and [3] agogic accents “accents of length”). Of these three, accents of discontinuity often correspond to group boundaries (Benjamin 1984, 379); while accents of climax “have no necessary relationship with event groups” (idem, 381). Agogic accent may partition time by the accent that is attributed to the onset of its relatively long event/duration, but this kind of accent carries a more mercurial relationship to grouping. Benjamin explains, that … because the distance between an agogic accent (the attack of a long note) and the next attack is long in context, and because of a fundamental principle of grouping [i.e., melodic segmentation] that put a group boundary where there is a relatively long distance between adjacent attacks, we will tend to perceive notes beginning with agogic accents as the last event of groups. This results in the perfectly normal opposition between partitioning by grouping and partitioning by accent. (Benjamin 1984, 382–83) I am delving into typology of accent here because I want to show that recognizing accents of certain kinds will also bring with it associated groupings/partitions with possible relevance for the emerging metric sense of a passage. In the quoted definition the reader can see the way that Benjamin is thinking about partitioning time by accent as one manner in which musical time is 148 shaped and divided into spans, but that grouping by event affiliation is a separate and co-present way of partitioning time. To the three types of accent described above, Benjamin adds a fourth—“accent of image shift”—which, even more than accent of discontinuity, bears a strong relationship to grouping and (potentially metric) partitioning of time. Benjamin’s metaphor imagines tuning a television set to pick up a “stable image” on one of a variety of the “channels of perception.” An accent will always be attributed to the moment where some musical event is understood to represent the moment of change from one stable image to another. Each moment of image-shift, then, is both a group-initiating event (for one of multiple co-present kinds of grouping) while also a moment of accent, regardless of whether it receives a phenomenal accent. Benjamin writes: “there is no theoretical reason to think of areas of constant value—steady pictures—on some channel of perception (be it harmony, texture, timbre, or whatnot) other than as groups” (Benjamin 1984, 378). This is directly counter to Lerdahl and Jackendoff’s view, wherein prolongational structure is not a kind of grouping structure, nor a rhythmic aspect of the music (since the rhythmicized interpretation of prolongational structure in GTTM is appreciated only by filtering prolongational structure through metrical structure. Understanding harmonic rhythm (along with other kinds of accent-by-image shift) as being meter-creating (or at least stable-group-creating) factors, rather than features which “align,” “conform,” or otherwise participate in metrical/rhythmic structure is a critical aspect of the theory of phrase rhythm I am developing here. While it is not the prevailing view, this emphasis on the special role of harmonic change in meter creation is not unique only to Benjamin. Joseph Swain, in his monograph Harmonic Rhythm, characterizes the prevailing outlook (since Rameau) as one where harmonic changes 149 are thought of in terms of their coordination with strong and weak beats: for example, in instances where the composer/student is “advised against changing the harmony on the weak beat” (Swain 2002, 91). Swain advocates for a shift of perspective: harmonic changes are not actions or events to be coordinated with the patterns of a metric structure, but rather as an integral part of what “establish[es] and sustain[s] those patterns” (ibidem).87 That, today, such views feel out of the ordinary shows the extent to which the conceptually separable, extrinsic, grid-like view of meter (and its progeny, the “entrainment” view of meter as mode of attending to nested pulses) remains a resilient (and almost default) notion in the literature. Before concluding the chapter, I will introduce one other important theoretical outlook on meter and rhythm, and relate it to Benjamin’s accent of image shift and to my conception of phrase rhythm as grouping complex. Christopher Hasty’s Meter as Rhythm (1997) understands meter to be the in-time process of a listener’s assigning metric qualia (i.e., beginning (“|”); continuation (“\”), or its forward-oriented relative anacrusis (“/”); and deferral (“—”)) to 87 Swain cites, here, one of a number of relevant passages Lester’s The Rhythms of Tonal Music, where Lester describes harmonic change as “the most powerful meter-producing factor” (Lester 1986, 66). To Swain’s characterization of harmonic change as helping to establish and sustain meter, I would add earlier expressions of the same idea, such as Benjamin (1984), and Peter Westergaard who—rather than advising a composition student to put certain chords/suspensions on the correct beats—advises them to be mindful to write suspensions only in those moments they wish to become beats for the listener (Westergaard 1975, 238–41). Benjamin’s understanding of appoggiatura/suspension resembles Westergaard’s account of species counterpoint as “beat-making” rather than meter-conforming. Benjamin describes appoggiatura and suspension as an “accent-strengthening factor” which is a subsidiary intensifier of accent-of-image-shift on the harmony channel. “Individual dissonances are not in themselves factors of accentuation…what suspension and appoggiatura do is make us more keenly aware of the points of image shift on the harmony channel. When one voice “delays” its shift, producing a suspension, it makes us more ware that the others have [already] shifted. It thereby strengthens the accent of image shift at the point of delay” (Benjamin 1984, 379). 150 events.88 These designations are based on the way that the music’s events (or “becomings”) imply hierarchical durations that come into being simultaneously and in a mutually informing way. This represents the first of three ways that Hasty’s theory is important for this project: it represents one of a few well-known “wide” definitions of meter, wherein aperiodic structures need not—indeed should not—be described as less metric than regular ones. Whether or not someone accepts the aspect of music’s time organization that Hasty calls meter as bona fide meter (and many do not) is not important for whether the kinds of things Hasty describes as meter can be involved in creating phrase rhythm as grouping complex. Secondly, while Hasty’s “beginning” and “continuation” are often used to characterize in-time experiential aspects of meter, they also represent a qualitative rather than quantitative approach to emergent temporal structure.89 Finally—and most importantly for the purposes of the theory I am 88 In general, my reasons for incorporating Hasty do not require deep investment in Hasty’s specific concerns about process, philosophy of time, and “in-time” analysis (interesting as they may be). Hasty devotes considerable pages to working through these concerns and the ways they complicate theories of music meter. While this regularly results in thought-provoking claims, and novel framing, his commitment to this philosophical position necessitates rather complex past/future tense constructions that are especially noticeable (distracting even) when he is describes the re-evaluation and re-interpretation of recently past events as contributing to emergent metric particularity. While many analysts are quite happy to name this as a “retrospective” aspect to interpretation, Hasty works to eschew the concept (if not the term) on philosophical grounds, resulting in turns of phrase such as “there will never have been a time when [duration x] did not…” (1997, 128, 207). Further to this point, see my characterization of Hasty’s “wide ‘now’” in footnote 94. 89 It bears repeating that, for Hasty, a situation in which projected durations are not realized is not any “less metric” than a situation where they are. This critical distinction tends to get lost whenever a hybridized approach incorporates only certain aspects of Hasty’s theory (often the projective arrows) as a way to pepper “in-time/experiential” colour commentary amid “meter-as-grid” play by play. Analyses that contrast and juxtapose findings of qualitative/projective and quantitative/grid-based analyses are welcome as a way to build inter-connections between qualitative and quantitative theories of meter and their diverging literatures (e.g., Horlacher 2001; Roeder 2014, 103–05, Mirka 2009, 83, 147). My feeling is that hybridized approaches tend to overestimate the amount of common ground shared between the two theoretical positions, and thereby risk sustaining confusion and foster misunderstandings when the umbrella term “meter” covers distinct—even incompatible—concepts. 151 developing here—Hasty’s projective/projected duration might reasonably be understood as a close relative of the “meter-as-grouping” conceptions of Berry and Benjamin, and also to Horlacher’s qualitative model of meter as “[contextually determined] with respect to internal motivic organization” (1995, 302) (see discussion in footnotes 53 and 54). To determine that a duration is begun and continued (until something else begins on the same level of structure) is to understand the events that occur during the unfolding of that duration as being “grouped” by it (though Hasty does not describe it in such terms).90 To illustrate the relationship between Hasty’s theory, meter-as-group, and the special role of “harmonic image” in grouping, phrase rhythm and meter, I provide the following commentary on a figure from Hasty’s Meter as Rhythm (1997) where he describes a situation (London’s gloss on Horlacher’s (1995) Les Noces analysis is an overt example of one such a misunderstanding: London’s staunch commitment to quantitative definitions of “meter” and “cycle” caused him to overlook (or otherwise misunderstand) Horlacher’s nuanced position on “contextually-defined meter” as an emergent and qualitative aspect of irregular/aperiodic passages (London 2013, 284–85, cf. Horlacher 1995, 302ff.). 90 See footnote 2 which introduces my position on Hasty as related to “meter-as-grouping” after Berry and Benjamin. One passage in Benjamin’s article that anticipates Hasty’s position occurs where Benjamin describes metrical music as those instances where “the instrument for measuring time—and, therefore, for making precise judgments concerning duration and evaluation of proportions—is given to us as a bonus for listening attentively to other aspects of the music’s structure” (Benjamin 1984, 373). I characterize both as being more interested in meter as a kind of continuous measuring music by the various (fluctuating) durations the music itself proposes—rather than measuring music according to measures extrinsically defined (such as a grid, tactus, attentional-peaks, or foot-tapping). Likewise, I understand Berry’s approach to resemble Hasty in at least two ways. Both argue for a situation where awareness of proportion between “groups” (in Berry’s terms) or “durations” (in Hasty’s) are metrical whether or not they are periodic, since the segments measure one another when unequal just as when equal. Second, Berry’s metrical “impulses” ([1976] 1987, 326ff.) are qualia which describe the temporal orientations attributed certain moments in the musical flow. Berry’s impulses closely resemble Hasty’s concepts of “beginning” (cf. Berry’s initiative impulse), “continuation” and “deferral” (cf. Berry’s reactive impulse), and “anacrusis” (cf. Berry’s anticipative impulse). Berry is not cited in Hasty’s bibliography and appears to have been unknown to Hasty at of the time of writing Meter as Rhythm. 152 including a “silent beginning;” then commenting also on a similar example from Justin London (1993a) which pertains to a related issue of unarticulated strong beats, or “loud rests.” Neither of these authors provides a description of how/whether these metric effects arise as a result of pitch and harmonic-grouping factors. A gloss on their figures using Benjamin’s accent-of-image-shift will help clarify special role played by (implied) harmonic change in their examples along the lines of the harmonic approach to meter determination of which Lester (1986), Westergaard (1975), Swain (2001; 2002), Willner (1995), et al., are advocates. To start let us consider the way that information about conventional patterns of “strong and weak” beats are differently represented in a metric grid diagram, and by Hasty’s projective symbols beginning (“|”) continuation or anacrusis (“\” or “/”) and deferral (“—”). My Example 2-9 represents the conventional pattern of strong-weak alternation in measures of 4/4 and 6/8 using metric grid dot-diagrams and a somewhat equivalent illustration in Hasty’s projective symbols. While their usefulness goes considerably beyond illustrating the beat-patterns associated with time-signatures, it is more-or-less true to say that a dot-diagram offers a picture of hierarchic strong/weak patterning. Because the meter-as-grid view typically takes metric structure to be conceptually separable from the music itself, the grid is intelligible as a description of a metric “container” even without any music “in it.” The same does not hold for the approximation of this structure in Hasty’s symbols. It is easy enough to imagine idiomatic music with 4/4 patterning as hosting events that would define projective and projected durations at the quarter, half, and whole-note levels so that a symbolic projective analysis like the first of those illustrated in Example 2-9 would pertain. 153 Example 2-9: Conventional Metric Patterns as illustrated by Dot Diagram and by Hasty’s Symbols What “begins” at Hasty’s beginning is not “a beat” or “bar measure,” but a rather a duration that is necessarily defined by some action or process of becoming in the music.91 Put another way: for a measure of 6/8, the familiar framing of the meter-as-grid view might explain eighth-note beats 2 and 3 as the weak places in the pattern relative to the dotted-quarter-note pulse from 1 (as are beats 5 and 6 from 4). To translate this metrical situation into the language of Hasty’s projective view, it cannot be declared that “in 6/8” beats 2 and 3 simply are the continuation and continuation-deferred of the duration begun from beat 1; unless one is talking about a particular measure of music that is shown to host an event/duration begun at notated beat one and continued through beats 2 and 3. Relatedly, Hasty’s concept of “metric particularity” (1997, 148ff.) is an exemplar of the generative outlook and intra-opus 91 Hasty uses “bar” to refer to the notated measure. He uses “measure” to mean a duration that one measures by. So, in this sense, many, but not all “bars” are also “measures” and there are many “measures” which do not correspond to bars. “Bar measure” is Hasty’s term for a measure that corresponds to a notated bar (Hasty 1997, 107 fn1). | " " " "\ " " " "| " " " "\"| " "\ " "| " "\ " "| " "\ " "| " "\"| "\ "| "\ "| "\ "| "\ "| "\ "| "\ "| "\ "| "\""● "● "● "● "● "● "● "● "● "● "● "● "● "● "● "●"● " "● " "● " "● " "● " "● " "● " "●"● " " " "● " " " "● " " " "● " " ""● " " " " " " " "●""""""""
– "\ "| "\"""– "\ "| "\"""– "\""" "● "● "● "● "● "● "● "● "● "● "● "●"" "● " " "● " " "● " " "● """ "● " " " " " "● "" 154 frame, where “each measure or each metrical unit could be viewed as a unique projective situation in which uniqueness or particularity arises both from the measure’s internal constitution and from its assimilation of prior events (and its potential for being assimilated into future events)” (Hasty 1997, 149). While the projective aspect of Hasty’s account is not central to my project, his “particularity” is an important concept that serves as a precedent in the existing music theory literature for an approach to rhythm that adopts a decidedly generative (rather than conformational) outlook while also privileging an intra-opus frame of reference for the describing how a certain metrical situation arises for a particular passage. In Meter as Rhythm, Hasty rarely provides details for what aspect(s) of the music informed the decision to assign beginnings/continuations to a particular moment, instead devoting the bulk of discussion to a description of the nature of the time-organizing metrical processes (or environments) that result. I do not doubt that he holds very precise notions about exactly which aspects of the music inform the metric qualia his symbols represent—I am merely pointing out that the reader is seldom given the chance to peak behind the curtain. For example, in the context of Hasty’s explanation of a descending C-major scale whose first metric duration is understood to begin from a rest (his example 9.15), he tells the reader that this particular “silent beginning” arises because of the way that “tonal differentiation is employed to support distinctions between beginning and continuation” (Hasty 1997, 125), yet he offers almost no explanation at all as to the harmonic how-and-why of this claim.92 My Example 2-10 aims to 92 In the 1400-word commentary on the example the only sentences that include any specific pitch/harmonic details are the following: [1] “the pitch G is heard to begin a half-note-beat” (Hasty 1997, 127); and [2] “[an anacrustic] projective complex comprising seven notes…[leads] to the beginning of a second bar measure [begun] with the low C” (idem, 127-8). 155 provide an account of the “tonal differentiation” criteria (that Hasty takes as a given) by reproducing his example 9.15d (Hasty 1997, 126) on the uppermost staff, then providing an analysis of “harmonic image” on the lower staves where I have added a melodic reduction along with an implied functional/tonal bass line. The middle staff of Example 2-10 is a melodic reduction that shows a pitch hierarchy within the descending scale by using a combination of slurs and differently sized noteheads, with coloured circles to mark chord tones affiliated with their respective harmonic images. The lowest staff adds a (implied) bass line with coloured boxes to help represent the discrete harmonic images that correspond to Hasty’s metric interpretation of the passage as creating a fairly conventional 4/4 structure where the final C♮4 would initiate a new bar measure. Example 2-10: Hasty’s C-Major Scale Analysis (with added “harmonic images”) In Hasty’s symbolic representation of metric structure for this passage—with its quarter-note level projective symbols <|, /, |, \>—a reader could be forgiven for assuming that the projective analysis begins as a sort of “translation” of the “strong-weak” alternation familiar {{{{989898989898&( | )||/ |/\ ||/\ |\\( \ )|||&APT APT?&( | )||\ — \ |\ \ —— \|>/| \ — \( \ )||&?‰ œJœ œ œ œ œ œ œj‰ œ œ œ œ œ œ œ œœœ œœœœ œ œ œ œ œ œ œ œ œj Œ Ó™œ œ œ œ œ œ œ œ œ œŒ Ó™œ ™ œ ™œJœœ ™ Ó™"""""
{{{{&bLondon's string of pitches0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29?b ∑&b0||1 2\3 4\|5 6\7 8||9 10\11 12|\13 14\15 16||17 18\19 20|\21 22\23 24||25 26\27 28|\29\?b(or)&b0/1||2 3\4 5|\6 7 8/9||10 11\12 13|\14 15 16/17||18 19\20 21\|22 23 24/25||26 27\28 29\|?b&b0 1/2 3||4 5\6 7\|8 9/10 11||12 13\14 15\|16 17/18 19||20 21\22 23\|24 25/26 27||28 29\?bœjœjœjœjœjœj‰ ‰œjœjœjœjœjœJ‰ ‰ œnJœJœnJœJœnJœJ‰ ‰ œJœJœbJœjœjœjœ œ œœ œ œ‰ ‰œ œ œœ œ œ ‰ ‰ œnœ œn œ œn œ ‰ ‰ œ œ œb œ œ œŒww ww˙˙œjœ œ œœ œj‰ ‰œjœ œ œœ œJ‰ ‰ œnJœ œn œ œn œJ‰ ‰ œJœ œb œ œ œj ‰ Œ‰ ˙ ˙˙ ˙˙˙w˙˙œ œ œœ œ œ‰ ‰œ œ œœ œ œ ‰ ‰ œnœ œn œ œn œ ‰ ‰ œ œ œb œ œ œ‰ Ó‰ Œ ˙ ˙˙ ˙˙˙w3{{{{&bLondon's string of pitches0 1 2 3 4 5 6 7 8 9 10 11 12 1 25 26 27 28 29?b&b0||1 2\3 4\|5 6\7 8||9 10\11 12|\13 14\15 16||17 18\19 20|\21 22\23 24||25 26\27 28|\29\?b(or)&b0/1||2 3\4 5|\6 7 8/9||10 11\12 13|\14 15 16/17||18 19\20 21\|22 23 24/25||26 27\28 29\|?b&b0 1/2 3||4 5\6 7\|8 9/10 11||12 13\14 15\|16 17/18 19||20 21\22 23\|24 25/26 27||28 29\?bœjœjœjœjœjœj‰ ‰ j j jj j j jœ œ œœ œ œ‰ ‰œ œ œœ œ œ ‰ ‰ œnœ œn œ œn œ ‰ ‰ œ œ œb œ œ œŒww ww˙˙œjœ œ œœ œj‰ ‰œjœ œ œœ œJ‰ ‰ œnJœ œn œ œn œJ‰ ‰ œJœ œb œ œ œj ‰ Œ‰ ˙ ˙˙ ˙˙˙w˙˙œ œ œœ œ œ‰ ‰œ œ œœ œ œ ‰ ‰ œnœ œn œ œn œ ‰ ‰ œ œ œb œ œ œ‰ Ó‰ Œ ˙ ˙˙ ˙˙˙w3 167 meter as emergent: intrinsic aspects of the pitch string provide (incomplete) contextual clues that nominate this or that generative metric structure. Supposing some context ‘X’ establishes one metric understanding of a pitch-string, and some context ‘Y’ establishes another for the same-string, then what we have is two unique melodies, not one melody in two (extrinsic) metric orientations. On this, Lester agrees with “Heinrich Christoph Koch’s two-century old assertion that changing the placement of the beat changes the harmonic structure, and hence the nature of a phrase itself” (Lester 1993; Koch [1786] 1983, 19–22, 33).99 2.4 Summary This chapter has made the case that grouping structures—distinct and independent of melodic grouping—can fulfill the role of Rothstein's (hyper)metric structure toward analysis of phrase rhythm. These grouping structures are more flexible than (hyper)meter in the strict, conformational sense, and are defined by a combination of factors including rhythmic/motivic parallelisms (on multiple levels of structure), and other grouping-by-event-affiliations such as harmonic image. Conceiving of phrase rhythm as grouping complex rather than as the interaction of (melodic) grouping with (hyper)meter, my approach can describe the same phrase-rhythmic designs that others have observed and named according to strictly defined aspects of structure, whether metric well-formedness or according to Schenkerian prolongational structures or other harmonic/tonal prototypes (e.g., Schachter 1999; IJzerman 99 Recall the epigram to chapter 2 where Koch writes of “inner” accentuations of a tonal melody prior to its being notated in this or that meter. My sincere thanks to Joel Lester for discussing, this idea and helpfully pointing me to the relevant passages in Koch over email correspondence (2019-04-24). See also relevant sections of Lester’s Compositional Theory in the 18th Century where Lester argues “that Koch’s theories of harmony and harmonic structure of phrase mark a milestone in the development of harmonic theory” (1996, 275ff.). 168 2017). This is because hypermeter and prolongational structures can both be (re)conceptualized as special varieties of grouping. (For certain repertoires this brings my theory into alignment with much of GTTM, save for my de-emphasis on metrical structure.) In this way, my approach would in certain cases reproduce “textbook” interpretations of familiar examples, albeit with slightly different emphasis and terminology. More importantly though, the generative phrase-rhythm-as-grouping-complex method I develop—freed from conformational approaches to meter and phrase-prototype—can also account for idiosyncratic phrase-rhythmic situations in and outside of the common practice corpus, without resorting to describing their temporal organization in terms of the features it is found to lack. In the shorter chapters that follow I provide a series of analytical demonstrations to show that— without appealing to meter per se—co-present aspects of grouping can be used to describe phrase rhythm in musics from Bach to Bartók. One pillar in the foundation for the analytical methodology that I construct via these demonstrations is Hasty’s concept of metric particularity as an emergent property that is, in theory, unique to every distinct passage of music. The other indispensable part of the foundation is laid by Benjamin’s article on musical meter (1984). It is not an overexaggerating to say that this dissertation developed out of a years-long attempt to work out the many wide-ranging implications and insights sparked by his Theory of Musical Meter, and most especially its example of an elision (in the Stravinsky Piece for Clarinet Solo) which was described in terms of a (un-named) non-metric multi-aspect grouping complex. 169 Phrase-rhythmic Particularity in Bach’s Sarabande Double BWV 1002 (with special emphasis on “harmonic image”) As a temporal phenomenon, a measure is constituted gradually. At every stage of its evolution what will eventually become its parts are to an as yet undetermined composition. The quantitative or qualitative determination we ascribe to the complete measure may be described abstractly as something shared by all like measures; but an experience of a measure (which is, of course, always more than the experience of a measure pure and simple) is eminently particular, both in the sense that it is an experience of just this measure of just this piece and in the sense that it is the unique act of attention performed at this moment by this human being. Regarded as components of an act, the various determinations we would attempt to describe analytically may eventually become more or less fixed as the act is complete, but as components they must also bear traces of the process that led to an unpredetermined completion. Christopher Hasty, Meter as Rhythm (1997) 3.1 Introduction The purpose of this chapter is to provide an analytical demonstration of how metric structure, conceived as a generative, rather than a conformational schema for evaluating the time-sense of a passage of music, emerges out of co-present groupings. As in previous chapters, the term grouping refers not only to gestalt segmentations (or melodic grouping in GTTM’s narrow sense), but also to those timespans whose events are grouped together as a result of accents of various kinds, which demarcate boundaries of time-spans that may become metric (Berry [1976] 1987, 303; Benjamin 1984, 359–60). Examples in earlier chapters already pointed in this direction by demonstrating how co-present grouping structures contribute to phrase rhythm and phrase-rhythmic effects. In this chapter, the analytical demonstrations expand upon the sort of “naïve” generative approach demonstrated by the IFUNC-informed analysis of the Luther chorale tune (from Chapter 2 section 2.1.3). Referring only to very basic 170 accentual information, I propose various kinds of groups, some of which are periodic enough to establish regular pulse streams, or even meter in the conventional sense. But whether or not basic groups of this kind define durations that become “metric” in the usual sense, I will show that the complex of grouping structures are of analytical interest—just as they are in Benjamin’s Stravinsky analysis, where he identifies “elision” by referring to a kind of complex temporal-musical organization he ultimately elects not to call meter. Whereas the IFUNC example in Chapter 2 proposed the most salient metric durations using a brute-force tally of all inter-onset intervals, the examples in this chapter take a more multidimensional generative approach by considering contextual grouping criteria (that would be invisible to a rhythmic IFUNC tally of durations). To begin, I present a series of preliminary analytical demonstrations that use irregular series of durations without pitch content. The first unpitched example (adapted from Benjamin (1984)) shows how even with the most basic unpitched rhythmic elements there exist multiple kinds of accent and multiple affiliated grouping structures, any one of which may become syntactic or otherwise relevant for “measuring” the piece in one or more ways (toward eventual description of phrase-rhythm). The unpitched examples are metrically underdetermined in a way that pitched rhythms only seldom are, since pitch information—in most cases—introduces co-present grouping factors and accent types (e.g., by accent of harmonic-image shift, or according to contour) that tip the scale toward one emerging metrical potentiality or another, as was already seen in the concluding examples of the previous chapter.100 A second example of 100 Chapter 4 will deal with “metrical multivalence”—my term for special cases where multiple meter-like grouping structures remain viable, and where the different harmonic and motivic associations that result enter into a kind of dynamic and open-ended interrelation. 171 irregular unpitched rhythm is drawn from Longuet-Higgens and Lee’s “The Rhythmic Interpretation of Monophonic Music” (1984), where they propose their own algorithmic approach for selecting a “preferred rhythmic interpretation” for metrically ambiguous rhythms. Finally, the Sarabande Double from Bach’s unaccompanied Violin Sonata BWV 1002 is introduced, serving as the primary example in this chapter. Consisting almost entirely of uniform eighth notes, the Sarabande Double’s inter-onset intervals provide no basis for phrase and sub-phrase level groupings defined according to gestalt segmentation by temporal proximity/distance. Instead, I argue for ways that parallelism, contour, and harmonic image-shifts define multiple kinds of groupings for the monophonic Sarabande. By and large, these groups correlate to the metrical structure and grouping communicated by Bach’s notated time signature, but the analysis goes beyond merely confirming the appropriateness of the notated ‘9/8’ measures: it points to basic aspects of pitch organization and other groupings which confer the status of beat-hood onto certain events and not others. In so doing, the analysis paints a picture of how grouping conditions that contribute to conventional metric structure in one instance contribute to establishing more peculiar and distinct metric particularities in other instances. It is the kinds of grouping and accent, and not their isochrony per se, which informs and defines the particularity of its meter, and of its emergent phrase-rhythm more generally. 3.2 Grouping as Determinant of Generative Metric Structure(s) The analytical vignettes in this section, glossing on examples of irregular durational series from Benjamin (1984) and Longuet-Higgens and Lee (1984), will demonstrate how “grouping-by-event-affiliation-and-separation” and “grouping-by-accentuation” define groups/timespans that may become “measures” for their metrically ambiguous rhythms. The 172 spans proposed by these different groupings are conceptually distinct, yet often inextricably intertwined and mutually informing (Benjamin 1984, 368-69). In his article, Benjamin explains: in accent, we have a mode of partitioning … which is typically nonmetric if we understand meter to involve spans of uniform length. [But] while accent patterns, accordingly, cannot be equated with meter, accentuation is theoretically allied to meter in that both are organizations of time-points, not events. At the same time, both grouping [i.e., a group of events] and ‘grouping’ [i.e., the mental/conceptual act of gathering together timespans] … are potential bases of meter. ‘Grouping’ [i.e., partitioning by accent] might have a lot to do with the determination of meter at deeper levels, more than either large-scale [melodic] grouping or patterns of strong accent, since both of these normally result in partitioning into spans of variable length. (Benjamin 1984, 368) From the Clementi example in the first chapter, I have argued that groupings are involved in determining prospective metrical structure(s). But here, by focussing on very rudimentary examples, I will flesh out this approach in more detail. Example 3-1 shows an adaptation of Benjamin’s figure 7 (1984, 368–9), with additional annotations of my own. The brackets below the score are equivalent to those that result from a gestalt segmentation analysis based on temporal proximity (in the manner of Tenney and Polansky (1980)). Benjamin takes a different path to the same result by partitioning time according to “accents of image shift” (first introduced in section 2.3.2 of this project). In this very rudimentary unpitched example, the “image” is simply the articulation of a quarter-note pulse. There are two states: either the pulse on a given level is continually (re)articulated by sound, or it is not. While either state continues, it constitutes a group by event affiliation. In Example 3-1 the brackets directly under the rhythm show these grouping by basic state, labelling them with an ‘a’ or ‘u’ (for articulated and unarticulated pulse), and beside each bracket label the number of pulses in the group is given in parentheses. At the moment when an articulated pulse replaces an unarticulated pulse (or vice 173 versa) a new “image” begins, bringing with it an accent of image shift with potential metrical meaning.101 Example 3-1: Grouping According to Gestalt Segmentation (or accent of image shift) and Durational-accent-as-group-beginning in an Invented Rhythm from Benjamin’s Fig. 7 A quarter-note pulse is inferred from the first three events, which are affiliated as a group-image of steady pulses labelled a1 in Example 3-1. The absence of a fourth articulated event following the third consecutive quarter-note pulse is conspicuous, so a potential accent is attributed to the moment when the steady image of an “every-pulse-is-articulated” span is changed to a new “no-pulse-is-articulated” image. Another image-shift accent occurs two quarter-note pulses later when the stable “unarticulated” image (u1) switches back to the “articulated” image (a2), and so on. At a deeper level of structure, these relatively short a/u images can be grouped together into AU-series groups, defined by the alternating relationship <a, u> that creates a loose non-periodic parallelism within this rhythmic series. Groups AU-1, AU-2, and AU-3 are approximately equal in duration (5, 5, and 6 quarter-notes in duration, 101 Some might object to attributing an “accent” to an unarticulated event, since it does not fit with the usual definition of accent as a sound-event marked for attention in some way, usually by dynamic emphasis/stress. Benjamin (expanding on Berry’s typology of accent) defines accent in a considerably wider sense (Berry [1976] 1987, 339–345; Benjamin 1984, 379). °¢°¢°¢°¢/A AU-1 (5)a1 (3)G1 (6)u1 (2)AU-2 (5)a2 (4)G2 (4)u2 (1) AU-3 (6)a3 (3)G3 (?)u3 (3)AU-4 (?)a4 (?)G4 (?)u4 (?)/M1 M2 M3 M4 M5 M6/B AU-1 (5)a1 (3)G1 (6)u1 (2)AU-2 (5)2 (4)G2 (4)u2 (1) AU-3 (6)a3 ( )G3 (?)u3 (3)AU-4 (?)a4 (?)G4 (?)/M1 M2 M3 M4 M5 M6 M7/C AU-1 (5)G1 (6)AU-2 (5)G2 (4)AU-3 (6)G3 (?)AU-4 (?)G4 (?)/M1 M2 M3 M4/D AU-1 (5)a1 (3)G1 (6)u1 (2)AU-2 (5)a2 (4)G2 (4)u2 (1) AU-3 (6)a3 (3)G3 (?)u3 (3)AU-4 (?)a4 (?)G4 (?)/M1 M2 M3!M4œ œ œ Œ Œ œ œ œ œ Œ œ œ œ Œ Œ Œ œ œY™ Y ™ Y ™ Y ™ Y ™ Y ™œ œ œ Œ Œ œ œ œ œ Œ œ œ œ Œ Œ Œ œ œ œ ŒY™ Y ™ Y ™ Y ™ Y ™ Y ™ Y ™œ œ œ Œ Œ œ œ œ œ Œ œ œ œ Œ Œ Œ œ œY™ Y Y ™ Y Y ™ Y Y ™ YŒœ œ œ Œ Œ œ œ œ œ Œ œ œ œ Œ Œ Œ œ œY™ Y Y ™ Y Y ™ YŒY™ Y8 174 respectively). The length of a potential AU-4 (beyond the end of Benjamin’s example) is undetermined. The first-level a-, u-segments, and second-level AU-groups are not the only kind of grouping at play in this unpitched example. Another variety of grouping results from partitioning the timespan according to durational/agogic accent. In the rhythmic notation, no articulated event is any longer in duration than another, but by attending to inter-onset intervals (IOIs) instead of event durations, we can infer a pattern of shorts and longs (namely: the duration of the u-segments is heard, here, to extend the duration of the last articulated event of the preceding a-segment). According to a preference for grouping parallelism (in the spirit of GTTM’s parallelism preference rule, though not to the letter), we can then hear a G-series group beginning whenever an IOI exceeds the duration of the preceding IOI. This happens on the third (G1), ninth (G2), thirteenth (G3), and possibly eighteenth (G4?) pulse of the example. Assuming that the final quarter note does represent a group beginning, the G-group lengths for G1, G2, and G3 are 6, 4, and 5, respectively; slightly more varied than the AU-group lengths, yet of similar proportion. A second (lower) staff in Example 3-1 adds a dotted-quarter-note pulse stream initiating from the beginning of group G1. The stream of pulses is defined by (inferred from) the periodic distribution of accents between group beginnings G1, AU-2, and G2, which become the prospective “measures” of M-series groups M1, M2, and M3. The groups M4–M6 represent the abstract-yet-felt continuation of the dotted-quarter-note pulse stream, even in the absence of corresponding group beginnings in either the G-series or AU-series groupings after G2 (and until G4). The M-series groups are unlike the other series because they are not determined by a single variety of accent as partition but inferred from the perceived co-operation between the 175 two accent types. After M3, there is no coordination between M-series groups G-series/A-series accents, yet they may be felt to the extent that a listener projects the continuation of the pulse stream through the remainder of the rhythm. The reader is invited to imagine a performance rhythm from Example 3-1 where the dotted-quarter-note pulse becomes metric for the G-series and AU-series, as if the M-series timespans serve as a 3/4 measures. My Example 3-2A through 3-2D reproduces the same rhythm and same basic AU- and G-series groups as Example 3-1, four ways. Besides reproducing the same M-groups from the previous example (now reproduced as Example 3-2’s option “A”), Example 3-2 options B, C, and D posit three other possible metrical structures that could arise from very same accentual patterns and group boundaries, if experienced and conceptualized differently. As already discussed, the M-series in Example 3-2A depicts a dotted-half-note pulse that becomes salient when a listener attends to the regularly of the repeated duration between group-beginnings G1, AU-2, and G2, and then measures the subsequent unfolding of events relative to the imagined continuation of that dotted-half pulse even when the group beginnings of AU-3, G3, and AU-4 do not coincide with it. Example 3-2B imagines instead inferring the dotted-half note pulse as if initiating from the very first attack (the beginning of AU-1) rather than from the first durational accent (beginning of G1). For interpretation ‘B’ the beginnings of groups u1 and u2 are construed as unarticulated measure beginnings (M2 and M4). As a result, a motivic parallelism takes shape between <M1, M2> and <M3, M4> where the first measure in each of those measure-pairs is filled with straight quarter notes and the latter measure of the pair begins with a shift from an a-image to a u-image (something like a syncopated compound triple feel with a “rest” on beat 4). Then, in the second half of the rhythm, the durational accents that mark the beginning of 176 G3 (and potential G4) are what define option B’s metric groups M5 and M7. This draws out a potential rhythmic parallelism between a3 and a4 as anacrustic groups, relative to their proximate G-series beginnings102 Example 3-2: Four Possible Metrical Interpretations generated from Grouping Boundaries, Accents, and Parallelisms in an Invented Rhythm from Benjamin’s Fig. 7 102 Nota bene this “interpretation B” adds an extra downbeat arrival onto the end of Benjamin’s example, which a person attending to these pulses and parallelisms is likely to expect; with no “extra” attack, the interpretation is less viable since a prospective G4 would begin one pulse earlier, as in interpretations A, C, and D). °¢°¢°¢°¢/A AU-1 (5)a1 (3)G1 (6)u1 (2)AU-2 (5)a2 (4)G2 (4)u2 (1) AU-3 (6)a3 (3)G3 (?)u3 (3)AU-4 (?)a4 (?)G4 (?)u4 (?)/M1 M2 M3 M4 M5 M6/B AU-1 (5)a1 (3)G1 (6)u1 (2)AU-2 (5)a2 (4)G2 (4)u2 (1) AU-3 (6)a3 (3)G3 (?)u3 (3)AU-4 (?)a4 (?)G4 (?)/M1 M2 M3 M4 M5 M6 M7/C AU-1 (5)G1 (6)AU-2 (5)G2 (4)AU-3 (6)G3 (?)AU-4 (?)G4 (?)/M1 M2 M3 M4/D AU-1 (5)a1 (3)G1 (6)u1 (2)AU-2 (5)a2 (4)G2 (4)u2 (1) AU-3 (6)a3 (3)G3 (?)u3 (3)AU-4 (?)a4 (?)G4 (?)/M1 M2 M3!M4œ œ œ Œ Œ œ œ œ œ Œ œ œ œ Œ Œ Œ œ œY™ Y ™ Y ™ Y ™ Y ™ Y ™œ œ œ Œ Œ œ œ œ œ Œ œ œ œ Œ Œ Œ œ œ œ ŒY™ Y ™ Y ™ Y ™ Y ™ Y ™ Y ™œ œ œ Œ Œ œ œ œ œ Œ œ œ œ Œ Œ Œ œ œY™ Y Y ™ Y Y ™ Y Y ™ YŒœ œ œ Œ Œ œ œ œ œ Œ œ œ œ Œ Œ Œ œ œY™ Y Y ™ Y Y ™ YŒY™ Y8 177 Example 3-2C and 3-2D show two additional metrical situations that arise out of the accentual pattern suggested by the same G-series and AU-series groupings. Unlike the prior interpretations, where G-group beginnings determined the meter (with initial event of G either as downbeat, or as quarter-note anacrusis), the measure for Example 3-2C emerges when attending primarily to the equal (five quarter-note) durations between the beginnings of groups AU-1, AU-2, and AU-3. Relative to this quintuple pulse stream, a listener might understand the beginning AU-group 4 as “late,” and perceive a “loud rest” or “silent beginning” just before it. Example 3-2D involves the same quintuple pulse stream, and like Example 3-2C, associates AU-group beginnings with downbeat-qualia. But it interprets the “lateness” of AU-4 in a slightly different way: to the extent that AU-group beginnings are equivalent to each other, and to the extent they have accrued downbeat-qualia, the lateness of AU-4 is evidence for a composed-out deceleration during AU-3. In the absence of further context, this invented example does not and cannot establish any one of these possibilities as “the most likely” interpretation, or “the correct/best fit”—after all, the generative metric conception I favour is uninterested in the question of fit, or conformation to extrinsic norms. Rather, I look to these possibilities as a set of metric and phrase-rhythmic affordances, similar to those that are brought out in subtle ways in longer and more complex passages of music (especially the Schumann and Mozart examples to come, in Chapter 4). Expounding on which of multiple metrical possibilities for a series of events is the most “natural” has led some theorists into contradictory positions. For example, Benjamin points out that Riemann (1903, 8–10) and Westergaard (1975, 229ff.) take opposite positions as to which is the “natural” or “intuitive” way to infer a metric underpinning for a repeating series of one short duration, followed by a long one, as illustrated in Example 3-3. Westergaard favours 178 option a, and Riemann option b. In light of the preceding discussion, we can see that their disagreement hinges on whether one attends to grouping by durational accents, or to grouping by event-affiliation/proximity as defining the spans which correlate to the dotted-half pulse stream that becomes metrical for the pattern. The durational pattern itself, of course, establishes both of those pulse streams equally but by different criteria; and so deciding which one becomes a relevant measure for subsequent events has nothing to do with what is “natural” or the “best fit” (except to the extent that a listener believes their own biases to be “natural”), and everything to do with which affordances are drawn out by yet other kinds of grouping in surrounding music, which is the thing these elementary examples do not have, hence the indecision. Example 3-3: Benjamin’s ‘Figure 1’ (1984, 356–7), Riemann (b) and Westergaard (a) on “Natural” Metric Interpretation for Long-short Pattern One other approach to the rhythmic interpretation of durational series ought to be addressed here alongside these examples from Benjamin’s article—in fact, it is from an article that appeared in the very same issue of Music Perception—and that is Longuet-Higgins and Lee (1984). Like the Benjamin examples, above, Longuet-Higgins and Lee focus on an unpitched durational series as the testing ground for how their listener will reach a “rhythmic A Theory of Musical Meter 357 Fig. 1. Fig. 2. over that in Figure 2b. Or consider the assertion - by Cone (1968), Wester- gaard (1975, pp. 309-319) and others - that a phrase, in its accentual pattern, should not be thought of as a magnification of an ordinary measure, with its alternation of strong and weak beats, but as a kind of suspension bridge anchored in initial and terminal accents. At the same time, one must consider the opinions of many who hold that, typically, a phrase is precisely a measure writ large.2 What is one to make of such disagreements? Are they essential, that is, the results of real differences in musical perception? One hopes not, since their being so would seem to stamp the whole enterprise of metric analysis as hopelessly subjective in all but its simplest aspects. One hopes, rather, that they can be seen as incidental products of differing terminologies and ways of speaking, arising perhaps from commitments to differing intellec- tual traditions. Very probably, however, they are in part essential, and one wants to know to what extent. My excuse for persisting in thinking about meter is a need to sort these matters out. With so much of what has been written seeming indisputably true, what seems to be called for is theory that, in its generality, would clarify the sense in which earlier theories pertain to (often unacknowledged) special circumstances. This general theory would therefore embrace these earlier theories as special cases, preserving their truths while mediating among them. The goal of such a theory, or general framework, has signifi- cantly determined the form and content of this article: The discussion of 2. This opinion is basic to Riemann's analyses of phrases and periods as metric structures (see Riemann, 1903, pp. 198-213). It is also the view put forth in Wallace Berry's review of Cone (1968) in Perspectives of New Music 9/2 and 10/1 (1971), pp. 271-290. See, in particular, pp. 275-283. 179 interpretation” of an metrically ambiguous series of durations.103 Ostensibly their subject is “the criteria that might lead a listener to favour a particular rhythmic interpretation of a given sequence of notes [i.e., durations]” (idem, 425). Their hypothesis is that listeners resolve ambiguities by automatically and subconsciously seeking out the metric orientation that represents a rhythmic interpretation of the series resulting in the lowest possible degree of syncopation. Their syncopation value is calculated as follows (from Longuet-Higgins and Lee 1984, 430): [1] Define some metric structure, M, where the “heaviest” beat (i.e., downbeat) is given a metric weight of 0; with each successively “lighter” level of metric subdivision assigned a metric weight of -1, -2, -3, and so on. [2] Given a sounded event at timepoint N that is followed by a rest (or is prolonged by a tie) at timepoint R, (where WN is the metric weight at N, and WR is the metric weight at R) then the ordered pair <N, R> constitutes a syncopation relative to M if and only if WN ≤ WR. [3] The relative “strength” of a syncopation is WR – WN. Applied to the case of the repeating <quarter-note, half-note> from Example 3-3, a Longuet-Higgens and Lee rhythmic interpretation of the pattern <quarter-note, half-note> (where the quarter note is construed as the downbeat of a 3/4 measure, M=<0,-1,-1>), 103 Longuet-Higgins and Lee conclude that “any sequence of notes values [i.e., durations] is in principle infinitely ambiguous, but this ambiguity is seldom apparent to the listener” because of a preference to interpret the series as “realization of a regular [i.e., minimally syncopated] passage” and because of the role of “phrasing” by which a performer can convey simpler more regular rhythms beneath a syncopated surface rhythm (Longuet-Higgins and Lee, 1984, 440). The latter resembles London’s “metrically malleability,” as discussed in section 2.3.3 of the previous chapter. 180 understands the half-note duration on beats two and three to involve WN = -1 and WR= -1, and so it constitutes a syncopation (of weight 0). Comparatively, if the interpretation of <half-note, quarter-note> construes the half-note duration as the downbeat of a 3/4 measure (M=<0,-1,-1>) then the half-note duration involves WN = 0 and WR= -1, so it is not syncopated, and therefore preferred. Example 3-4: Three Rhythmic Interpretations of Duration Series <3, 1, 2, 3, 1, 2> assessed according to Longuet-Higgins and Lee’s Syncopation Getting any further into the nuts and bolts of their algorithmic approach is not necessary for the present purposes. The important takeaway is that their method draws on durational information (especially contextually “long” events) relative to a basic pulse and predicates the perception of meter in ambiguous/irregular contexts upon a bias against syncopation (as they define it). They describe the contents of their article as “a generative theory of musical rhythm”—but, crucially, this is not at all a “generative” approach in the sense of the rhythm x x x x x x
L-H&L def.WN = -1syncopation(strength 1)Metric Structure
L-H&L def. 181 conformational/generative binary I introduced in the previous chapter.104 In fact, their method is expressly and absolutely conformational, in the sense of Bonds’ binary: their rhythmic interpretations are only possible after the assessment of its level of syncopation, which in turn is only possible after the rhythmic signal is tested for fit against some pre-existing metrical structure, M. To conclude this section, consider the difference of conformational and generative approaches to metric structure and rhythmic interpretation in one last unpitched series of durations adapted from Longuet-Higgens and Lee’s examples 5 through 7 (1984, 426). The tables in Example 3-4, above, show a durational series (eighth-note value IOIs <3, 1, 2, 3, 1, 2>) imagined in three different metric orientations, A, B, and C. The rhythm is notated in TUBS notation, with an ‘x’ marking each attack/onset. Below the rhythm, a dot diagram illustrates the metrical structure according to which the “metric weight” for each pulse is calculated (using Longuet-Higgens and Lee’s formula). Where they occur, syncopations (with their relative “strength”) are shown in boxes above the rhythm. Placing the rhythm into the “2/4” metric structure (as in Example 3-4C) results in one syncopation of strength 1. Placing the rhythm into the “3/8” metric structure (as in Example 3-4B) results in two syncopations of strength 0.105 Longuet-Higgens and Lee prefer the rhythmic interpretation according to “3/4” metric structure (as in Example 3-4A) because it results in no syncopations at all. 104 For Longuet-Higgens and Lee, as for Lerdahl and Jackendoff, the term “generative” is an indication of certain academic allegiance to the structuralism of Chomskian linguistics. 105 In the 1984 article Longuet-Higgens and Lee do not indicate how they would reach a measure of “total syncopated-ness” for a passage. In certain cases, this would leave “least syncopated” undefined: for example, if sorted by greatest number of instances of syncopation, then B is more syncopated than C. If instead sorted by the summed “strength” (or median strength-per-instance), then B is less syncopated than C. 182 I am immediately skeptical of any method that would point to one interpretation as “more inevitable” or “more natural” than any other. Context (whether fleshed-out musical settings of the rhythms, or the “context” of competing metric theories and preference rules) could make possible any one of these rhythmic interpretations. My Example 3-5 represents the same rhythm (IOI <3, 1, 2, 3, 1, 2>) and the same three metric scenarios as in Longuet-Higgens and Lee’s examples (shown in my Example 3-4), but now illustrated in the same manner as my grouping-complex examples. With Example 3-5 I mean to show the way that prospective rhythmic interpretations arise out of the affordances of the grouping-complex, independently of any extrinsic measures or “metric weight.” Example 3-5: Three Rhythmic Interpretations of Duration Series <3, 1, 2, 3, 1, 2> as inferred from different Aspects of the same Grouping Complex °¢°¢°¢™™™™™™™™™™™™™™™™™™™™™™™™/ALonguet-Higgins & Lee's repeating durational pattern
Thesis/Dissertation
eng
Music - Theory
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Roeder, John Barlow
Phrase rhythm as grouping complex : hearing structure in “irregular” music
Text
Still Image