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UBC Theses and Dissertations

comparative study of selected Twentieth-century piano works involving the elements of chance and indeterminacy. Silvester , Trudy Helen 1971

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A COMPARATIVE STUDY OF SELECTED TWENTIETH-CENTURY PIANO WORKS INVOLVING THE ELEMENTS OF CHANCE AND INDETERMINACY by TRUDY HELEN SILVESTER B. Mus., University of B r i t i s h Columbia, 1968 A THESIS SUBMITTED. IN. PARTIAL. FULFILMENT OF THE REQUIREMENTS. FOR THE DEGREE OF MASTER. OF MUSIC i n the Department of MUSIC We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1971 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f ree l y ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes i s for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th i s thes i s f o r f inanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of lrYU^u^C_,  The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date (Xf^J- 7 , 1 ^ 7 / A study of selected chance and indeterminate piano pieces was made with a two-fold purpose: (1) to provide an intermediate stage between the composition and performance of the selected works and (2) to i l l u s t r a t e the diverse ways spontaneity may be invoked. The investigation centered on three points: (1) the problems the performer might encounter i n examining the pieces, (2) implications of the instructions and (3) possible r e a l i z a t i o n s of the score. While each score examined showed a s l i g h t l y d i f f e r e n t approach to chance or indeterminacy, the pieces within the two categories were seen to exhibit common features. Generalizations were made i n three broad areas: (1) the instructions i n the scores, (2) the notation and (3) possible solutions. The instructions i n the indeterminate scores were found to be r e l a t i v e l y straightforward; the performer i s made aware of his choices or alternatives and manner of performance i s discussed. Instructions i n chance works are less e x p l i c i t ; the performer i s not directed to any one solution. Some explanation of the nota-t i o n i s given and the performer i s led to discover how extensive his freedoms are. Indeterminate works on the whole are.found to use t r a d i t i o n a l notation. The notation may be altered i n some way, but retains a resemblance to i t s t r a d i t i o n a l source. Chance notation i s very diverse, but three general categories e x i s t : (1) works using t r a d i t i o n a l notation, (2) works using t r a d i t i o n a l and non-t r a d i t i o n a l (graphic) notation and (.3) works using only graphic notation. The possible solutions of indeterminate works are concerned with either the juxtaposition or the metric rel a t i o n s h i p of material. The degree to which the choices are guided varies i n the works discussed. Possible solutions of the chance works involve in t e r p r e t a t i o n of notation and various ideas presented i n the i n s t r u c t i o n s . The performer must respond i n an in d i v i d u a l way, drawing on his own ideas. In general, analyses of the works i l l u s t r a t e d some features that might be expected i n other chance and indeterminate works. INTRODUCTION 1 CHAPTER I. DEFINITION OF THE TERMS 2 II. THE MUSICAL PHILOSOPHY: the ideas underlying chance and indeterminacy 7 Assumptions of Chance 8 Assumptions of Indeterminacy . 12 II I . THE CHANCE PIECES 15 November 1952 (Earle Brown) . . . . . . 16 Octet '61 (Cornelius Cardew) 18 1953 (Earle Brown) 19 Cassiopeia (George Cacioppo) 21 Music for Piano #7 (Toshi Ichiyanagi) . . . . . . . 25 Music for Piano #2 (Toshi Ichiyanagi) 28 December 1952 (Earle Brown) . . . . . . . . . . . . 31 Four Systems (Earle Brown) 35 Timepiece (Udo Kasemets) . 36 IV. THE INDETERMINATE PIECES 41 Two Books of Study (Cornelius Cardew) 42 Pour Clavier (Sylvano Bussotti) 44 Caracteres (Henri Pousseur) 46 Nr. 7 Klavierstuck XI (Karlheinz Stockhausen) . . . 48 Duo for Pianists II (Christian Wolff) 49 Pianopiece II (George Cacioppo) . . . 53 March 1953 (Earle Brown) 57 Dance (Earle Brown) 58 Variations I (John Cage) 58 Variations II (John Cage) . . . 61 V. CONCLUSIONS 64 BIBLIOGRAPHY • 71 The main purpose of this study i s a di d a c t i c one, providing an intermediate stage between the composition and the performance of selected chance and indeterminate piano pieces. Investigation of the works centers on three points: (1) problems the performer might encounter i n examining the pieces, (2) implications of the instructions and (3) possible r e a l i z a t i o n s of the score. While certain ambiguities i n the works may be c l a r i f i e d and this may be useful to the performer, the attempt has been not so much to make the scores and t h e i r instructions e n t i r e l y e x p l i c i t as to help the performer orient his thinking i n terms of the musical philosophy underlying the works and to stimulate his i n t e r e s t and imagination along the lines of possible solutions. A secondary purpose i s the i l l u s t r a t i o n of the diverse means used to invoke spontaneous choice. Each score examined exhibits a d i f f e r e n t approach to chance and indeterminacy, and for this reason two chapters have been devoted to a discussion of the scores i n d i v i d u a l l y . Within the chapter on indeterminate works, pieces are grouped according to the type of indeterminacy present. In the following chapter the chance works are categor-ized according to notational s i m i l a r i t i e s . In a further chapter the results of the analyses w i l l be discussed and generalizations concerning the pieces made. Preceding the analyses, a background to the study i s provided by a discussion of the terminology and the musical philosophy associated with the works. DEFINITION OF THE TERMS In t h i s chapter e x i s t i n g d e f i n i t i o n s and applications of the terms chance, indeterminacy and aleatoricism w i l l be presented, noting the present confusion i n t h e i r usage. Following t h i s , the terms w i l l be defined as they are to be used i n t h i s study. The term chance by general d e f i n i t i o n implies that an event "happens unpredictably, without any di s c e r n i b l e human intention or d i r e c t i o n and i n d i s s o c i a t i o n from any observable pattern, causal r e l a t i o n , natural necessity, A chance event happens "unaccountably, without premeditation, pre-arrangement or any sign of motivation and without observable causal r e l a t i o n 2 to attendant circumstances." Indeterminacy suggests a vagueness or lack of preconceived end r e s u l t . Aleatory means "depending on 3 an uncertain event or contingency." By d e f i n i t i o n , chance and indeterminacy are close i n meaning; unlike chance, indeterminacy does not necessarily connote contingency, the undefined aspect may not be solved i n an unexpected manner. Confusion has arisen i n the application of these terms to Webster's Third New International Dictionary, S p r i n g f i e l d , Mass., G. & C. Merriam Co., 1969, p. 373. 2 Ibid., p. 373. Ibid., p. 51. music. Heinz-Klaus Metzger, noting t h i s confusion, claims that the musical application of the term al e a t o r i c evolves from two 4 i n d i v i d u a l s , Werner Meyer-Eppler and Pierre Boulez. The f i r s t volume of Die Reihe contains an a r t i c l e by Meyer-Eppler i n which he gives the following d e f i n i t i o n : "A process i s said to be a l e a t o r i c (from the Latin Alea = dice) i f i t s course i s determined 5 i n general but depends on chance i n d e t a i l . " In the f i r s t issue of the Darmstadter Beitrage a t r a n s l a t i o n of the a r t i c l e "Alea" by Boulez appears. In the a r t i c l e Boulez deals with chance as a compositional technique, r e f e r r i n g at one point to "a l e a t o r i c happenings." No reference i s made to al e a t o r i c form. Elsewhere, Boulez writes that the "notion of shunting does not belong to that of pure chance, but that of non-determinate choice, and t h i s 7 difference i s fundamental . . . ." Metzger i n f e r s from these sources that chance and aleatoricism are synonymous, both r e f e r r i n g to works i n which the d e t a i l i s unspecified, being f i n a l i z e d i n the performance. If the d e t a i l i s determined but there i s choice i n i t s ordering, the work may be considered indeterminate. Heinz-Klaus Metzger, "Abortive Concepts i n the Theory and C r i t i c i s m of Music," Die Reihe, 5 (1961) , p. 26. 5 Werner Meyer-Eppler, " S t a t i s t i c and Psychologic Problems of Sound," Die Reihe, 1 (1958) , p. 55. Pierre Boulez, "Alea," Darmstadter Beitrage, 1 (1958) , p. 53 . 7 Boulez, "Sonate, Que Me Veux-tu?" Perspectives of New Music, 1 (Spring 1 963 ) , p. 35 . A l e a t o r i c music i s defined i n the Harvard Dictionary as "music i n which the composer introduces elements of chance or u n p r e d i c t a b i l i t y with regard to.either the composition or i t s performance." This general statement aligns the terms chance and aleatoricism. The contributors noted the terms chance, indeterminate and a l e a t o r i c "have been applied to many works created since 1945 by composers who d i f f e r widely as to concepts, methods and r i g o r with which they employ procedures of random 9 selection.'. 1 The a r t i c l e gives the reader no basis for d i s -tinguishing indeterminacy from-chance and a l e a t o r i c ; the three terms are not c l e a r l y defined. Howard Riley states that a l e a t o r i c procedures are "those which are dependent on uncertain contingencies."'''^ Riley uses a l e a t o r i c i n a generic sense, to embrace compositions i n which d e t a i l s are s p e c i f i e d but which lack formal d e f i n i t i o n , as well as compositions i n which the performer must supply the d e t a i l s , being given a general course to follow. . In other words, Riley departs from the meaning of a l e a t o r i c set up by Metzger and Meyer Eppler by implying, that indeterminate and a l e a t o r i c are synonymou According to Roger Reynolds, "indeterminacy and chance are J.R. White and A. Boucourechliev, "Aleatoric Music," Harvard Dictionary, Harvard University Press, Cambridge, Mass., 1969, p. 26. 9 Ibxd., pp. 26 -27 . "^°Howard Riley, "Aleatoric Procedures i n Contemporary Piano Music," Musical Times, 107 (April 1 966 ) , p. 311. progressive degrees of'a tendency to leave detailed unspecified . . . . If . . . a composer wants an indetermined s i t u a t i o n , there can be no preferred solutions — and, ultimately, i n the case of chance, v i r t u a l l y no 'rules'.""^ He further submits that i n indeterminacy "categories of events" are determined, t h e i r ordering being the undetermined aspect. In chance, any event within c e r t a i n l i m i t s may occur. John Cage writes of works conceived by chance operations and works which are "indeterminate of performance." When the d e t a i l s of parts are.established but the form or sequence of the whole i s not, Cage would c l a s s i f y the work as "indeterminate of performance," that i s , f a l l i n g i n the category of indeterminacy. The sources above have been used as the basis for deriving the following categories to be used i n t h i s study: a. works i n which the d e t a i l s of sections or groups are determined, the ordering of these parts being undeter-mined; and works i n which s p e c i f i e d sounds are i n a given sequence, but t h e i r metric rel a t i o n s h i p to one another i s undetermined. b. works i n which the score exists as a stimulus to the performer, who must supply the d e t a i l s . A form may be given to some degree, or the form may evolve during performance. Roger Reynolds, "Indeterminacy: Some Considerations," Perspectives of New Music, 4 (Fall-Winter 1965), p. 136. The former category w i l l be referred to as indeterminate, the 12 l a t t e r as chance or a l e a t o r i c . In both categories a degree of spontaneity i s assumed. The terms w i l l be further c l a r i f i e d i n the discussion of the works. .Having defined chance and a l e a t o r i c as synonymous musical terms, the term chance rather than a l e a t o r i c w i l l be used here-after. THE MUSICAL PHILOSOPHY: the ideas underlying chance and indeterminacy The chance and indeterminate works are u n i f i e d i n concept by the denial of p r e v a i l i n g b e l i e f s or assumptions. H i s t o r i c a l l y t o n a l i t y had assumed an organization structured on a key system. Predictable relationships were a part of t o n a l i t y ; so was period-i c i t y . Not only were there root progressions related throughout a piece, but a piece was divided into related components; that i s , the whole was an integration.or causal i n t e r r e l a t i o n of parts. With the advent of atonality and non-periodic structure, p r e d i c t a b i l i t y was no longer a feature, although coherence or continuity was s t i l l a concern. Timbre came to be emphasized; pit c h and often other parameters were highly organized. Con-siderable s p e c i f i c a t i o n and exactness of notation exerted almost impossible demands upon the performer. By creating a s i t u a t i o n where the performer could not respond accurately, indeterminacy of a kind arose; the more numerous the demands, the more probable i t i s that unplanned actions w i l l occur. This sort of indeter-minacy, however, i s not desired. The philosophy of chance — and indeterminacy to a lesser extent — focused on new assumptions, searching for ways of circumventing the t r a d i t i o n a l ones. Since the assumptions of chance and indeterminacy do not e n t i r e l y coincide, the two categories w i l l be discussed separately. Composers advocating the chance philosophy f e e l that purposeful structuring or organization prevents, the perception of sounds as sensation. Habits of l i s t e n i n g , bound by theories about the organization of sound, are a hindrance. Sounds as discrete e n t i t i e s and t h e i r s i t u a t i o n i n time are major concerns of the new assumptions. The p o s s i b i l i t y of i s o l a t i n g any p a r t i c u l a r event as the cause of another event i s denied. Cage believes that "there are an incalculable i n f i n i t y of causes and e f f e c t s . . . . i n f a c t , each and every thing i n a l l of time and space 13 i s related to each and every thing i n a l l of time and space." There are no separable causes and e f f e c t s . Earle Brown refers to "the impenetrable i n f i n i t e complexities and connections of a l l II 14 things- Thus, events should be allowed to happen without d i r e c t i o n , t h e i r connection being too .in t r i c a t e to single out a one-to-one r e l a t i o n s h i p . Juxtaposition of sounds no longer indicates a d i r e c t r e l a t i o n s h i p . .It i s sounds as sensation, complete i n themselves and independent of structuring, that i s important, not t h e i r r e l a t i o n s h i p or t h e i r placement i n a progression. Cage states, "a sound does not view i t s e l f as thought, as ought, as needing another sound for i t s elucida-John Cage, Silence, Middletown, Conn., Wesleyan University Press, 1939, p.- 47. ^ 4 E a r l e Brown, "Form," Darmstadter Beitrage, 10 (1966), p. 66. 15 . t i o n . " Each sound should be uniquely perceived, heard for i t s own sake. The s i m p l i c i t y of, for example, being given A and from that expecting c e r t a i n alternatives i n B, does not e x i s t . The possi-b i l i t y of the performer envisioning the r e s u l t s of alternative courses of action and of making a r a t i o n a l choice on t h i s basis i s no longer acceptable. The composer may no longer predict a general response to what he has written. Prediction depends to a considerable degree upon the use of an established "grammar," a common ground, a "language" which when understood c a r r i e s with i t c e r t a i n meanings or implications; i n chance music t h i s language i s absent. The musical experience becomes a " t o t a l i t y of p o s s i b i l i t i e s " where "no knowing action i s commensurate, since the character of 16 the knowledge acted upon.prohibits a l l but some eventualities." If there i s s e l e c t i v i t y , the s i n g l i n g out of some p o s s i b i l i t i e s , there is'no longer a t o t a l f i e l d of possible r e s u l t s . The meaning of " t o t a l f i e l d " varies from score to score; the less there i s spe c i f i e d i n a score, the more manifold the r e s u l t s are l i k e l y to be. Within the l i m i t a t i o n s imposed by a " t o t a l f i e l d " the experience must be an experimental one, with no preconceptions about the musical r e s u l t ; any outcome w i l l be acceptable since . John Cage, "Experimental Music," Score, 12 (June 1955), p. 65. "^Ibid. , 66. there are no established r e s u l t s . With t h i s assumption, there can be no errors or approximations. Mistakes suggest a measuring up to expectations; for a mistake to occur there must be some knowledge beforehand of what should occur,.a r e s t r i c t i o n of what may occur. In chance music communication becomes involvement. The composer no longer ..determines exactly what w i l l be heard but rather instigates.a process, making the l i s t e n e r involve his senses i n response to a.unique.experience. The performer responds to a score which, exists primarily to e l i c i t t h i s response. Earle Brown stresses the idea of a momentary, spontaneous.response to a score, an immediate quality; the environment and circumstances are an influence on 17 thi s response. A work i s not re-created complete i n d e t a i l s and form but created from given ideas, the.score serving as a place to s t a r t , an occasion for incident, with.the degree and type of chance varying widely from score to score. The performer i s given.clues or suggested directions.but much i s l e f t free, allowing the performer to contribute responsively. The end r e s u l t goes beyond the suggested.directives, beyond what 'the composer had imagined. A purposeful.ambiguity impels the performer to become involved, to respond.to the ideas presented-by. the composer and The reader i s referred to Brown's a r t i c l e "Form," Darmstadter Beitrage,.10 (1966), pp. 57-69. d i r e c t i o n s . The solution arrived at by the i n d i v i d u a l performer should be one out of many possible responses to the materials at one moment in time. Some works contain a high degree of ambiguity; no s p e c i f i c directions are indicated, the clues being vague to the point of implying almost anything. However, a certain amount of pre-thinking i s needed to grasp the freedoms and l i m i t a t i o n s given in the score, and to possibly narrow down the alternatives from the t o t a l f i e l d . The d i f f i c u l t y , of performing d i r e c t l y from the score i s conceded; the performer may decide to respond to f i r s t impressions of a score, keeping i n mind alternatives for possible use during a performance. The work, however, must not be thought out or planned to the point of determination; room must be l e f t for spontaneity to operate during the performance. It i s the allowing for and accepting of contingencies that i s important. Cornelius Cardew feels that the "only c r i t e r i o n for a sound i s : was the player expecting (intending) to make i t ? If not, i t was a mistake, and makes a d i f f e r e n t sort of claim to beauty. As a mistake i t comes under c r i t e r i a for action: mistakes are the 18 only t r u l y spontaneous actions we are capable of." With the lack of determination of sounds and t h e i r ordering, form as i t i s t r a d i t i o n a l l y considered does not apply Cornelius Cardew, "Notation, Interpretation, Etc.," Tempo, 58 (Summer 1961), p. 26. to chance. Form, has t r a d i t i o n a l l y been a r e l a t i o n a l concept. Beginning-middle-end, antecedent-consequent and p e r i o d i c i t y were elements of t h i s form; i t was i m p l i c i t that.the end was brought about by what preceded i t . According to Brown, form i s "not a 19 receptacle but a fxeld of a c t i v i t y . " Brown claims that i t i s impossible for music to be formless, for i f . i t were, we would be unable to perceive i t s existence.. Accusations of formlessness usually are directed toward sounds having an unexpected form. Brown notes that the."form which the work takes each time i s a form of c o l l e c t i v e consciousness as i t moves through a labyrinth 20 of environmental influences." Thus, the form of a piece i s not a configuration determined by composer or performer; i t i s a process i n a constant state of flux , a l t e r i n g i n each new s i t u a t i o n . The Assumptions-of Indeterminacy T r a d i t i o n . i s not as completely denied i n indeterminacy as i t i s i n chance. By d e f i n i t i o n , indeterminacy admits the existence of d e f i n i t i v e , structured sections. The material within these-sections may be subject to t r a d i t i o n a l assumptions. The work i s not t o t a l l y determined, however. Choices must exi s t for the.performer, perhaps of the ordering of the sections Earle Brown, op. c i t . , p. 68. Ibid., pp. 60-61. or of the placement of sounds within a metrically unordered sequence. Choices may be guided to a c e r t a i n extent, or any juxtaposition or continuity may be allowed, but spontaneity i n making the choices i s usually assumed. Prediction of the r e a l i z a t i o n for any.one performance of a score i s not possible. The p o s s i b i l i t i e s are often more c l e a r l y established i n an indeterminate work than i n a chance work. Also, the performer, as a r e s u l t of his preparation of the work, might be aware that c e r t a i n alternatives may or may not occur i n the performance. Prediction might occur i n t h i s sense although the r e a l i z a t i o n w i l l be the r e s u l t of spontaneity. The composer may not predict the alternatives the performer w i l l choose, but depending upon the degree of choice given, he may predict possible, general solutions. In pieces where choices given are l i m i t e d , the composer may be aware of a l l possible choices. He may form the piece to allow only c e r t a i n c o n t i n u i t i e s . The " t o t a l i t y of p o s s i b i l i t i e s " w i l l be r e s t r i c t e d to c e r t a i n . p o s s i b i l i t i e s , even though these may be numerous. Within 21 these boundaries s e l e c t i v i t y i s denied. Ideally the performer should not predetermine any aspect of the continuity, "the f i n a l 22 • step of d e f i n i t i v e arrangement" should be " l e f t out." As i n See p. 9. Earle Brown, op. c i t . , p. 60. chance works, the response to a score w i l l be influenced by the circumstances of the performance. The form a work assumes i n each performance w i l l be the r e s u l t of the performer's spontaneous, momentary selections from the " t o t a l f i e l d " of p o s s i b i l i t i e s . The assumptions of chance and indeterminacy w i l l receive 23 further explication i n the following discussion of the works. It i s suggested because of the references made to the scores and t h e i r instructions that the reader of the following analyses consult the works. THE CHANCE PIECES Although c e r t a i n of the chance works discussed may be per-formed with any instrumentation, including two or more pianos, the problems considered are mainly those that would be encountered i f the works were performed by one p i a n i s t . In t h i s chapter the pieces are grouped according to notational s i m i l a r i t i e s . There i s d i v e r s i t y in.notation, but three c l a s s i f i c a t i o n s may be made: (1) works i n which only t r a d i t i o n a l symbols are used, pitches being m e t r i c a l l y notated on a s t a f f ; (2) works i n which t r a d i t i o n a l and non-traditional symbols are used, one or two parameters being represented by other than traditional.means; (3) works i n which "graphic" notation i s employed, -signs of varying shapes having ambiguous meaning, the performer assigning meanings to l i n e s , squares or shapes. Since each, piece d i f f e r s i n i t s means of occasioning chance, the pieces w i l l be discussed i n d i v i d u a l l y . B a s i c a l l y , each piece has been examined to discover what i s specified and by implication, what i s not s p e c i f i e d , with possible "interpreta-tions" of the score being given and the degree chance being noted. Generalizations concerning the chance pieces have been included in. the f i n a l chapter of the study. The "defined space" of November 1952 consists of fifty-l i n e s containing notes with metric values and dynamic signs. Clefs are.not s i g n i f i e d ; they are " f l o a t i n g . " While a note has a po s i t i o n on one of the l i n e s or spaces and i s preceded by an accidental, i t s pi t c h i s not established u n t i l a c l e f i s assigned. While at f i r s t glance the pitches appear determined, i n f a c t , any c l e f may be mentally assigned to any note being e f f e c t i v e only for that note; a c l e f i s not assigned consistently to any l i n e . If during.a performance each note is. used once and the curved connecting l i n e ( ) s i g n i f i e s a t i e , t h i r t y - f o u r d i f f e r e n t pitches are possible. The composer does not specify, however, that each note i s to be used only once. The question then arises whether, in.returning to a note, the note must maintain the same pit c h . T h e o r e t i c a l l y , one extreme p o s s i b i l i t y of c l e f assignment could result, i n a l l the notes being the same p i t c h . There are many possible interpretations of the defined space and f l o a t i n g c l e f s , however. The performer may make any d i v i s i o n of the f i f t y l i n e s , expanding or contracting the distance between them.or t h e i r length. He might a l t e r the space between the l i n e s depending upon his inte r p r e t a t i o n of the c l e f s . The layout of the page does, not necessarily imply an ordering of the pitches. The piece, may be "performed i n any d i r e c t i o n . " . No tempo i s s p e c i f i e d for any part of the piece. The suggested tempo i s "as fast as possible to as slow as possible, i n c l u s i v e . " The statement may imply that tempo v a r i a -tions e x i s t , the proportions of variance being at the performer's d i s c r e t i o n . The performer must then decide what meanings to assign to "as fa s t as ... ." and "as slow, as . . . ." The meaning may be variable according to the d i f f i c u l t i e s the performer sets for himself, or variable according to the context, the " s t y l e " or concept of the piece. The meaning may be "as fa s t as possible" while maintaining "accuracy." The notes are given durations, but within a widely fl u c t u a t i n g tempo i t i s debatable whether they would have .any actual meaning. Only.if the tempo remained constant for several, pitches would t h e i r metric.value be perceptible. Neither the manner nor the exact placement of attacks i s given. An attack may occur at any point during the length of time defined by the "performance." Since i t i s possible that two or more symbols w i l l be i d e n t i c a l i n p i t c h , dynamics and duration, and could occur simultaneously, two or more symbols might be taken care of with.one attack. In an extreme case a l l pitches might occur simultaneously; the performance would consist of one attack. Since the attack of any p i t c h may occur at any point i n the performance and within any order, combination,.range and tempo, a l l that has been determined i s the occasion for a performance. It i s presumed the composer wishes only to stimulate a response to the score. No mistakes are possible unless the performer has established c r i t e r i a for the performance. The performer needs to understand the instructions accompanying the score to be aware of his freedom, and he might conceive of possible r e a l i z a t i o n s , but the more spontaneous the performance of the score, the.truer it.would seem to be to Brown's intent. Cornelius Cardew: Octet '61 In Octet '61, six t y d i f f e r e n t "events" or composite symbols comprise the score. They are numbered, suggesting that although the performer may begin and.end anywhere, once the performance begins, the symbols are to be interpreted i n sequence. Most of the symbols contain a single s t a f f incorporating notes and/or dynamic signs and numbers as well as various other signs. Any part of any symbol, however, may be ignored. For example, a performer could heed dynamic indications of an event but apply them.to a p i t c h other than the suggested one. Black notes i n a symbol, when used, must be played i n the notated r e g i s t e r ; white notes are free of t h i s r e s t r i c t i o n . Durations of notated pitches, the "event" and the t o t a l performance are free. The score may be " f r e e l y " copied by hand. Since the performer may i n the end choose any notes for an event, an exact reproduction.of the score may be considered unnecessary. The score exists as a stimulation to bring ideas into existence. It may be.interpreted by following what appears to be obvious; i t may be used only as a s t a r t i n g point for a more spontaneous in t e r p r e t a t i o n . The pitches or various signs having t r a d i t i o n a l connotations might suggest some sound to the per-former, and the performance may evolve from that point on. To further spur on the performer, Cardew has written out i n the "notes" accompanying the score some possible r e a l i z a t i o n s . Cardew also recommends the performer work out the signs thus providing a "bloc of material." He may include alternative versions, of a sign, spontaneously choosing.one during performance or he may a l t e r or add to written-out material at the l a s t moment. Some pre-composed material.might, ensure.a reasonable continuity. An idea may be conceived spontaneously, but i t i s not necessary to leave i t unrefined. Ideas from which the i n t e r -preter w i l l , draw may.be well thought through. One possible "formal" suggestion i s given. The performer may include a sign which may be used as "punctuation." It would divide the piece into sections. If Octet '61 were being performed by an ensemble, one player might be assigned to playing only one sign. There are many p o s s i b i l i t i e s for v a r i a t i o n of an event i n the Octet, even when using the graphic suggestions; almost any-thing could happen. To r e a l i z e that nothing given i s necessary and that any notes, may.be f r e e l y added opens the s i t u a t i o n completely. In essence, any interpretation would meet the requirements of the score. Earle Brown: 1953 In 1953 there i s a gradual narrowing down, of p o s s i b i l i t i e s by a series of choices. The f i r s t choice consists of assigning either a bass or treble c l e f to each s t a f f of the four two-line systems. Once chosen, the c l e f presumably does not change during a performance of that l i n e . Each.notated event, therefore, has two possible pitches v i a one position of the score; p i t c h becomes precise once the c l e f is.determined. Since either side may be up, double sets of accidentals and attacks, which may appear confusing at f i r s t sight, are given. The dynamic signs are written i n such a f ashion . as.to be l e g i b l e either side up: or £lF9 dp . Duration of pitches i s indicated by the length of the "bars" denoting pitches. Decisions concerning time are made i n three stages. F i r s t , the t o t a l length of the piece must be chosen. Two basic choices are av a i l a b l e . The composer suggests the duration may be anywhere from twenty seconds to two.minutes. The performer may choose any time length, not necessarily one between these l i m i t s . In choosing the t o t a l time.length the performer might keep i n mind that his second stage of.determination w i l l be to decide the time i n seconds.of each two-line system. Here three choices e x i s t : "times pre-set by the composer, times obtained from the composer" 24 or times derived.spontaneously. Within t h i s framework the performer prescribes, durations for the i n d i v i d u a l notes, con-sidering t h e i r graphic length i n r e l a t i o n . t o the lengths of the systems and to each other. No notated rests appear. The gaps between "bars" presumably indicate time between events, that i s , Brown i s . u n s p e c i f i c concerning the meaning of "times pre-set by the composer" and "times obtained from the composer." This i s just one more puzzle for the performer. "rests" or silences. In t h i s case the performer must assign a r e l a t i v e value to the spaces. In making decisions concerning duration, many considerations a r i s e . For example, a.choice made i n the f i r s t category w i l l to some extent l i m i t the choice possible i n the second category. Durations chosen, the "tempo" w i l l i n part determine the simpli-c i t y or complexity of the piece. The faster the tempo, the more perceptually complicated the overlapping of sounds and time r e l a t i o n s become. Variable elements i n order of settlement are page and c l e f d i s p o s i t i o n , and time. The performer begins with an outline for a performance and by steps supplies the d e t a i l s , allowing for spontaneity i f desired during the performance s i t u a t i o n . George Cacioppo:.Cassiopeia On the one-page score there are four networks with an e l l i p s e p a r t i a l l y superimposed on two of the networks and "islands" interspersed i n or around a l l . networks. The e l l i p s e and i s l a n d symbols are considered fantasy forms. The networks consist of "paths" connecting pitches which are represented by black, and white noteheads. For the most, part the pitches are s p e c i f i c a l l y designated; a few are not. The s p a t i a l distance between the spe c i f i e d and the unspecified pitches may be used as a gauge for determining the unspecified pitches. The size of the notehead.is an in d i c a t i o n of the r e l a t i v e dynamic l e v e l of the pit c h . Two ways of choosing time values are suggested. The l i n e a r space between the pitches.may suggest a time value, or the values may evolve spontaneously, with no set c r i t e r i o n , during the performance. The performance may begin i n any one of.the four networks and may.or may not include fantasy elements. From one to four networks -- or as l i t t l e as a part of one — may be used. While the performance may include fantasy forms, at lea s t part of one network must be played as well. The performer proceeds from notehead to notehead.moving i n any d i r e c t i o n . When paths i n t e r -sect between noteheads the performer may change d i r e c t i o n . Not a l l notes on a path must necessarily be played before branching o f f or reversing d i r e c t i o n . The performer may choose to follow a c e r t a i n pattern, forming a "closed c i r c u i t . " To form a c i r c u i t the performer would have to make use of intersections and "superimposed" paths. The c i r c u i t may be repeated any number of times and may be altered at w i l l . To "exit" from a c i r c u i t either an in t e r s e c t i o n or " o r b i t a l jump" to any other part of the piece i s made. The piece may end. anywhere, spontaneously or at the end of a chosen time-length. While elements are either determined.or suggested i n the graphic score, there i s considerable room for spontaneity. The performer might be aware of ce r t a i n aspects such as. the general p i t c h layout, dynamics, harmonics, fantasy elements, a time value system and how to get from p i t c h to p i t c h or network to network; however, innumerable sound combinations exist.. The performer must become responsibly involved i n developing the material into a performance. In dealing with the networks the performer's choices center on ordering material, choosing routes and possibly-forming. patterns or "configurations.' 1 The forming of figures, which .may be repeated, or permutated. suggests choice i s operating with some t r a d i t i o n a l implications. Recurrence of patterns suggests a recognizable structure. The composer i s proposing i n essence that an "elementary form" may be set up; the piece i s not just a series of random sounds, complete i n themselves. The performer,.however,. may choose not to repeat material, not to thus.form ."figures." The performer must be fa m i l i a r with the material of the piece. In order to r e a d i l y locate a pit c h he must become accustomed to thinking of pitches and the keyboard.in terms of regi s t e r s rather than notes on a s t a f f . I t would help to notice the -general structuring of octave registers i n the. score. The reg i s t e r s are not mixed i n a random.fashion within the networks but are "layered".; octave seven i s higher than s i x , six higher than f i v e , and so on down the page. The performer might then gain some knowledge, of the p o s s i b i l i t i e s , perhaps deciding what w i l l be played spontaneously and what w i l l be considered or planned. Whatever decisions, general or s p e c i f i c , are made con-cerning, treatment of the networks, the area within the semi-enclosed dotted l i n e s need not be governed by them. This area may be considered anew, even played, by a d i f f e r e n t instrument from that used for the res t of the performance. Most subject to spontaneity are the fantasy forms. The performer reacts to a v i s u a l stimulation, t r a n s l a t i n g the reactions into a performance. A reaction may or may not lead to an aural.experience; the r e s u l t may be an action, some v i s u a l event or t h e a t r i c a l e f f e c t , not necessarily producing a sound. The reaction may produce a sound event but from a sound source other than the piano. There are no r e s t r i c t i o n s on.what the performer may derive from the fantasy forms. The e l l i p s e may merely serve .as a "cover"; any l i n e s going into i t may be imagined to pass behind i t , emerging into a new network. Two of the networks may be joined i n t h i s manner. Although some determinations are made by Cacioppo, the considerable freedom that exists i s at times of the indeterminate category, at other times chance. Use of terms such as network, configuration, permutation and in t e r s e c t i o n suggest defined meanings. Pitch i s mostly determined, dynamics are given i n a general way, suggestions concerning.time are made. Ordering of these elements, although subject to rather free rules of play, are open to spontaneity. The instructions are suggestive of possible pre-thought, planning, an awareness of p o s s i b i l i t i e s , the ordering of the materials being indeterminate. The fantasy forms, on the other hand, are t o t a l l y undetermined. Any meaning, action or sound i s possible. Details need.not.be pre-established but may evolve completely spontaneously, any r e s u l t being acceptable. The greater u n p r e d i c t a b i l i t y r e s u l t i n g .from the lack of deter-mination places the fantasy elements i n the chance category. The piece must include at least p a r t i a l networks, the indeterminate aspect, but may also involve fantasy forms, or the i n c l u s i o n of chance. Toshi Ichiyanagi: Music for Piano #7 The notation of Music for Piano #7 i s e n t i r e l y graphic. Patterns i n the centre of the score, contain very general i n d i c a -tions : 0 short sound, non-keyboard, but piano O long sound,.non-keyboard, but piano • short sound, non-piano Q . long sound, non-piano "t^ arpeggios Other symbols on either.side of the centre pattern represent a type of sound: 0 white and black keyboard used, tones chosen i n random order 0 only white keyboard used £ black keyboard only Certain t r a d i t i o n a l associations may be made from the composer's usage of black and white symbols. A white shape indicates a longer duration as a "white note" t r a d i t i o n a l l y receives more time than, a "black note!." Also, .the black and white centre patterns i l l u s t r a t e the black and white keys. Thin horizontal l i n e s also found on either side of the patterns indicate an approximate range. Lines to the r i g h t of the pattern designate higher sounds, from the upper keyboard; l i n e s to the l e f t desig-nate lower sounds. The notation has a f a m i l i a r connotation, suggestive.of the keyboard arrangement. Since the indications given.by the notation are general, decisions w i l l have to.be made at some point. The performer w i l l have to give the horizontal range l i n e s a more s p e c i f i c meaning. An approximate breakdown into areas covered by c e r t a i n l i n e s mightbe made. The placement of the patterns down the centre of the score might suggest a t r a d i t i o n a l d i v i s i o n . That i s , the pattern might divide the keyboard i n "half." When.the pattern shows a mixture of black and white keyboard, decisions w i l l have to be made concerning how much of each and in.what order the black and white keys w i l l be played. The performer could thus set l i m i t s , on various aspects of the material and prescribe spontaneity within.these.limits, so that a symbol consistently represents the same general idea. On the other hand, the meaning of a symbol could.change from context to context. The basic elements of ..Music for. Piano. #7 are. not .determined. While there may be r e l a t i v e meanings, as one sound i s "higher" or "longer" than another, there i s ambiguity as to t h e i r exact meaning. For instance, a longer l i n e may, but does not necessarily indicate a greater density of sound. A performer could.choose one or two notes from a shorter l i n e but s i m i l a r l y could choose one or two notes from a. longer l i n e . The exact p i t c h and even ..the number of pitches to be chosen i s not spe c i f i e d . It i s determined, however, that when two or more notes from a l i n e are chosen they-must be played as a chord or clus t e r . . Also.unspecified are attacks, dynamics and pedalling; overlapping of sounds i s free. The only i n d i c a t i o n for some sounds i s that they are not to be played on the keyboard, or i n some cases, not on the piano. In these instances any sounds may be chosen; a wide range and vari e t y of sounds are possible. It i s not.indicated that non-piano sounds need be limited to estab-li s h e d musical instruments. The imagination may be f r e e l y engaged; decisions might be a r b i t r a r y or quite spontaneous. The. nine pages are.numbered,. perhaps suggesting they be played i n order. Each page may be read either side, up, or both ways, and the.pages may be arranged so.that they overlap, the performer playing only the part of the page showing. Choices i n the arrangement of the pages in.the above ways i s another freedom. Since a l l pages are required to take the same time there w i l l be d i v e r s i t y of a c t i v i t y among pages. There are considerably fewer events on some pages than on others. No two pages are exactly a l i k e i n the type of content. Some pages contain only sounds produced at the keyboard, some .only harmonics. Other pages contain mixtures of keyboard, non-keyboard and non-piano sounds. . Only a general structure has been delineated i n t h i s piece. An. imprecise area of the keyboard.or q u a l i t y of sound i s desig-nated, but the treatment of the nine pages, t h e i r placement and th e i r . p o s i t i o n i n g w i l l produce d i f f e r i n g o v e r a l l structures. There are choices to be made i n .the-handling of a l l elements comprising the.structure. L i t t l e has been determined by the composer. With so few suggestions given, i t would be hard to say that any choice made, providing i t followed.the general requirements, was wrong. How fixed the d e t a i l s are before the performance w i l l depend upon the performer's conception of the piece. He might predetermine.details, providing alternative means of performance. He might decide upon general meanings for the graphic indications or he might allow the meanings to .change spontaneously during the performance. In any case, i n d i v i d u a l . r e a l i z a t i o n s of the score are bound to exhibit wide v a r i a t i o n . Toshi Ichiyanagi.:. Music for Piano #2 The score.for Music for. Piano #2 consists of four small sheets with graphic symbols. Symbols, except.those found i n the four corners of the sheets, indicate the following features: a. Register. This depends.upon the d i r e c t i o n of a branch i n r e l a t i o n to a c i r c l e . The d i r e c t i o n represents one of f i v e ranges into which the keyboard i s divided. b. Location of the sound. The type of c i r c l e indicates how the sound i s to be made. c. Relative duration of the sound. Length of the branch indicates t h i s . d. Simultaneity. Whether sounds are to be played singly or together i s indicated by the absence or presence of a " s l u r . " e. The number of sounds. This i s indicated by the number of branches. Although the graphic notation i s d i f f e r e n t , Music for  Piano #2 i s i n many respects s i m i l a r to Music for Piano #7. As i n the l a t t e r , no pitches are sp e c i f i e d ; only general ranges are given. In the former, however, the performer, i s directed to choose a s p e c i f i c number.of tones, .for example, one from the highest r e g i s t e r and one from the middle r e g i s t e r . . (Example 1.) Example 1. Music.for Piano #2, page 0. In example 1 the two notes.are to be played separately. The symbol does.not indicate which of the two tones i s to be played f i r s t . I t also does not indicate whether they are heard separately or whether.one sounds before the other.is released. As i n Music for Piano #7, no dynamic l e v e l of the sounds i s given. The performer, may choose i n d i v i d u a l dynamics, attacks, general dynamic l e v e l s , perhaps choosing d e l i b e r a t e l y , or spontaneously. Pedalling e f f e c t s are chosen at the.performer's d i s c r e t i o n . .The symbols i n the four.corners of.the sheets — or, on one sheet, the lack of symbols.in the four c o r n e r s . — are the cues or links, which j o i n the four pages, providing continuity i n the piece. Although not indicated, presumably the. performer may begin.anywhere, with any symbol from any of the four pages. As soon as he performs the requirements of a symbol he moves to the page whose cue matches .the c i r c l e of the symbol he has just played. For example, i f »• <c has just been played, the per-former would next choose a symbol from a sheet.with the black c i r c l e cues ( # ), and so forth.. The four sheets, thus would have to be arranged on the piano making a l l sheets v i s i b l e . Any of the four sides of each sheet may be up. As soon as a l l of the symbols of any one sheet have, been performed, the sheet i s rotated clockwise ninety degrees. The performer then continues playing as before. When any sheet has been rotated four times, the performance i s over. With r o t a t i o n , symbols a l t e r i n meaning. Whereas i n one po s i t i o n one would be directed to play inside the piano i n the "low" r e g i s t e r , with one r o t a t i o n one would be directed to play inside the piano i n the "high" r e g i s t e r . Thus, each symbol has four possible, general meanings. With the rotation of only one sheet at a time, many d i f f e r i n g c o n t i n u i t i e s w i l l r e s u l t . With each rotation the variable aspects of a symbol may receive new meanings. For instance, i f i n the symbol the two notes connected by,the slur and hence played together are performed before the single note, i t i s not r e q u i s i t e they maintain t h i s order i n any succeeding r o t a t i o n . Apart from general durations indicated by.the branch length, there is. no i n d i c a t i o n of either tempo.or chronometric time.of the.piece as.a whole. Nor i s there any i n d i c a t i o n of the amount of time a performer may take between sounds comprising the symbol or between symbols. Rests or silences, thus, have not been notated. In Music for Piano #2 the performer i s accorded si m i l a r r e s p o n s i b i l i t i e s to those i n Music.for Piano #7. As i n the l a t t e r , a general.continuity i s outlined with d e t a i l s to be completed. The performer w i l l doubtless become quite f a m i l i a r with the symbols and t h e i r general and possible meanings before playing.the.piece, but.this should.not prevent spontaneous choices from being made during a.performance. Again i t w i l l be the per-former's conception of the. piece.and how he handles the variable aspects that w i l l determine the f i n a l outcome. Earle Brown: December 1952 In.December 1952 the "defined space" Brown refers to consists of horizontal and v e r t i c a l l i n e s of varying lengths, width and positions. The score.has four positions; with one ninety-:degree rot a t i o n , a horizontal l i n e becomes a v e r t i c a l one, each l i n e having four.possible positions i n the score. It i s not spe c i f i e d that the performer must play all'events (lines) before ro t a t i n g the score or even that a l l events must be played before the performance i s complete. The performer may begin with.any one of the l i n e s and proceed i n any d i r e c t i o n . With.thirty-one events many possible orderings e x i s t ; presumably there i s no pre-selected ordering or interspersion. There appears to.be no r e s t r i c t i o n on returning to an event one or more times. Two broad choices are given for the inter p r e t a t i o n of the l i n e s . The performer may consider three or four "dimensions" as "active.". When three are active, they are v e r t i c a l , horizontal and "time"; the "thickness" of an event may be representative of r e l a t i v e i n t e n s i t y and/or c l u s t e r s . When four are active (horizontal, vertical,.depth, and time), the thickness (depth) i s not given a sp e c i f i e d meaning. A t r a n s l a t i o n of the l i n e s into.musical terms.will depend b a s i c a l l y upon the performer's conception of the terms "dimension" and "active." The l i n e s have three apparent proper-t i e s : (1) horizontal and v e r t i c a l o r i entation, (2) thickness and (3) pos i t i o n on the page. These properties are to be given musical meanings. They are to be associated with the character-i s t i c s .assigned to a sound: pi t c h (register, simultaneity . . . ) , duration, dynamics (intensity) and attack. When three dimensions are "active," the only, property with a suggested meaning i s thickness. Even here uncertainties a r i s e . If the performer chooses thickness to represent i n t e n s i t y only, may clust e r s be represented another way, or i f thickness . represents c l u s t e r s only, how w i l l dynamics be represented? Horizontal and v e r t i c a l might by t r a d i t i o n a l implication suggest p i t c h . Horizontal might r e f e r to a number of events i n succession, v e r t i c a l referring.to.simultaneity, while the length of the l i n e might indicate an approximate number of events. The question remains, then, how the dimension of time i s to be represented. A plane such as the score i s two-dimensional. In the i n s t r u c t i o n s , time i s conceptually represented i n a t h i r d dimension, perpendicular to the plane. The t h i r d dimensional l i n e might or might not be comparable i n size to one graphically representing an aspect of the event. Time i n t h i s work exists on various l e v e l s . There i s the time or duration of.an i n d i v i d u a l event, the time between events and the length of the piece. The performance length i s to be decided beforehand. Time as the t h i r d dimensions might presumably be the time of an event. The other aspects of time are not indicated and the composer gives no suggestions concerning tempo. Time between events might possibly be taken from the s p a t i a l distance between events. Silence then would not be just the time i t takes to go from one event to another, but a momentarily considered part of the composition. If the number of sounds contained within a simultaneity or a succession i s l e f t undetermined, the time of the event might be graphically represented. The length of the l i n e might repre-sent the duration during which the simultaneities or.successions occur. When the performer chooses to consider four dimensions as "active," horizontal and v e r t i c a l again may be represented graphically. The depth (thickness) of the event may or may not represent clusters and/or dynamics. Time again might possibly have a conceptual p o s i t i o n . B a s i c a l l y the same considerations a r i s e whether three or four dimensions are "active," the difference being that thickness i n the l a t t e r choice has an even The performer must decide how to interpret "active." One p o s s i b i l i t y would be to consider a dimension which i s not deter-mined, or given a suggested meaning, as active. It might further be considered that while the dimension i s graphically represented, the concept of i t s representation i s subject to transformation or.modification. The meaning assigned to a c e r t a i n . l i n e would not necessarily be consistent throughout the performance; i t could.be altered f r e e l y . If a performer did return to an event, he might possibly re-interpret i t . With the basic assumptions so undetermined, i t might be presumed that a po s i t i o n within the defined space i s not necessarily i n d i c a t i v e of a r e l a t i o n s h i p . One event need not be higher, lower than or i d e n t i c a l in., p i t c h with another according to i t s position within the defined space. The composer suggests performances be made d i r e c t l y from the graphic "implication."' If something.in the score or instructions appears i m p l i c i t to the performer, however, he may choose to incorporate t h i s idea into the performance. Should v e r t i c a l lines., for example, seem to. indicate simultaneity, the performer, may.decide.that he.will, do t h i s throughout the piece. While giving no expressed.meaning, the terms used i n the accompanying instructions to the. graphic score may suggest, even subconsciously, a working-out of the score, or may be enough at least to set a performance i n motion. Meanings assigned to the terms and solutions chosen w i l l be. multifarious. Almost anything could r e s u l t , depending upon the performer's conception, what seems implied to him, his response and imagination, and spontaneity. There i s no one solution; any alternative i s acceptable. Earle Brown: Four Systems For help i n understanding Four Systems the performer i s referred to the instructions, of December 1952. In each system the l i n e s of varying lengths and thicknesses are horizontal and are confined between two "continuous" horizontal l i n e s . Four such.systems comprise the one-page score. The continuous l i n e s bounding each system "define the outer l i m i t s of the keyboard." It i s not stipulated whether the l i n e s from, l e f t to r i g h t are to be translated."bottom to top" of the keyboard or whether the distance between the two l i n e s represents the outer l i m i t s . Apart from the general designation of. the continuous l i n e s , no further indications.are made. The performer must decide how the other, elements are to be represented. The length of the l i n e could possibly indicate.duration, the p i t c h being taken from the s t a r t i n g point of the l i n e . Or the length could indicate an approximate.number of-notes grouped i n succession or possibly heard.simultaneously. Dynamics, as in-December 1952, may be represented by. the thickness of the l i n e . No ordering either of the four systems or of the l i n e s within i s s p e c i f i e d . It i s not indicated i n the score whether l i n e s are to be heard separately or whether any two or more may be chosen to be heard simultaneously. Consequently, the performer presumably could play any sound (or combination of sounds) at any point during the performance of the piece. Sounds might be heard singly, simultaneously.or i n diverse overlapping arrangements. The profusion of l i n e s within each system might suggest a type of simultaneity. Although a possible solution, i t might be assumed, that the system would not be read from either l e f t to r i g h t or vice versa, but that events might be chosen spontaneously from any point within a system. It i s not determined whether a.performer must play every event within a system before moving to another system. It might be presumed since the entire performance time may be any length, that any event may be returned to any number of times. Some events might not be played at a l l . As in. December 1952 the graphic score i s ambiguously defined by.the composer. After becoming aware of how l i t t l e i s spe c i f i e d , the performer should respond to the score, evolving or creating a performance spontaneously. Anything.might happen and would.be acceptable. A very general outline for a.performance exi s t s ; a l l d e t a i l s must be supplied by the performer and they, i n turn, depend upon his reaction to the score. Udo Kasemets: Timepiece The graphic notation of Timepiece consists of .dots and dashes above and below, events. These dots and dashes indicate a r t i c u l a t i o n , denoting the difference between "short/long." "detached/sustained," "staccato/legato," "etcetera." Presumably the performer gives the dash a meaning, perhaps long or sustained or legato, depending upon the context. It might be supposed the dot would have.the opposite meaning. If an event has two dots, or two dashes, the "attack" i s uniform; i f not, both symbols must i n some way be expressed. When both must be used, the propor-tioning of the notes to one or the other sign i s not s p e c i f i e d . Dots and dashes appear to the l e f t and r i g h t of events, i n d i c a t i n g "same" and "d i f f e r e n t , " "colour.and/or playing technique." Again the player gives his interpretation of the dots and dashes. Many interpretations.are possible; the assigned meaning may vary from event to event, the meaning given being consistent only for that event. The t h i r t y - s i x events are represented i n two ways: t h i r t y consist of black or white "noteheads" while six events are comprised of larger, black in t e r l o c k i n g c i r c l e s . A white note-head indicates a sound i s "consonant" with the preceding or "closest neighbouring sound." A black notehead represents dissonance. Consonant and dissonant "need" not have t r a d i t i o n a l implications;.the performer defines the terms. The d e f i n i t i o n s set up must remain constant only throughout one performance or "cycle" of the t h i r t y - s i x events. The terms may be redefined from cycle to cycle. Kasemets uses the term "etc."; presumably the performer may give the dots and dashes a meaning other than those suggested. The large black c i r c l e s represent "clusters, glissando, extra-instrumental noises et a l . " Two c i r c l e s i s a small "cluster," three a large "cluster" and four a "very large, a l l -encompassing c l u s t e r . " The boundaries of the clusters are variable, apart from the general indications "small," "large" or "very large." I t . i s not sp e c i f i e d whether the c l u s t e r i s comprised.of black and white keys, a l l black or.white, or what proportions of each. Only the type of sound i s given. Each notehead.represents, a single note. Notes within an event (there w i l l be one to six) may be combined or juxtaposed i n any way. Pitch and range of the notes are subject only to general considerations. A movement from one event to a higher one on the score represents an upward movement on the keyboard, and vice versa. With respect to.horizontal movements, two choices e x i s t . .The event may stay i n the same range or be played i n any range. The performer must i n some way delimit the ranges. Spe c i f i c pitches within the ranges are. not designated; t h e i r choice may be l e f t to spontaneity, allowing for consonance-dissonance l i m i t a t i o n s . The p o s i t i o n of the events on the score need not represent exact distances or locations on the keyboard, representing instead only a general relationship.. Presumably the performer considers only the relationships between two events at a time. When .considering.events seven and eight, for example, the p o s i t i o n of events three and four i s not necessarily.relevant. Only i f the performer chooses to in t e r p r e t the "geographical" p o s i t i o n of "events" w i l l , t h e i r p o s i t i o n on the page indicate t h e i r range relationships to each.other. Position would then indicate a s p e c i f i c range rather than just "higher" or "lower." The horizontal "movements on the score . . . from one attack to the next" are made within given durations: Movements to the r i g h t are "slow": 1 column — 2 seconds 2 " — 4 - " ' 3 — 6 4 " — 8 " 5 " — 10 Movements to the l e f t are " f a s t " : 1 column — 1 second 2 " — 1/2 3 " ~ 1/3 4 " ~ 1/4 5 " ~ 1/5 The tables may be altered so long as the o r i g i n a l proportions remain constant throughout the cycle. It might be implied that the speed with which the event i s approached i s also the speed of the event. It i s not specified.; the duration of the event might be free. Two choices ex i s t for v e r t i c a l movements: they may "maintain the 'tempo' by which the column was approached" or they may be "free." Superimposed wedges indicate contrast between "softer" and "louder." The dynamic contrast between events depends upon t h e i r r e l a t i o n to the point of the wedge. The closer to the point, the quieter the event. There appears to be a discrepancy i n the i n s t r u c t i o n s . The performer i s f i r s t advised that he may move from event to event choosing any d i r e c t i o n and distance u n t i l a l l t h i r t y - s i x events have been played once. Later he i s t o l d that "any two successive events belong i n one common wedge." From th i s i t might be pre-sumed that although wedges overlap, allowing the performer to get from one wedge to another, movement i s somewhat r e s t r i c t e d ; c e r t a i n events cannot be performed successively. In Timepiece the performer i s given a graphic score with some general r e s t r i c t i o n s . Limitations are gradually imposed by the performer as he chooses from innumerable p o s s i b i l i t i e s . He might r e a l i z e the score a s . i t s layout suggests or he might impose almost any ideas.on c e r t a i n aspects. If the piece i s being played by more than. one ..performer, agreement on treatment of the elements must be made between the performers. Some choices w i l l be established beforehand; there w i l l be a degree of uniformity or consistency. The conception of the performer w i l l determine the degree of chance i n Timepiece. THE INDETERMINATE PIECES In t h i s chapter the pieces are grouped according to t h e i r type of indeterminacy. A l l pieces are comprised of s p e c i f i e d material, the organization of which i n some way i s not sp e c i f i e d . Although i t i s assumed that spontaneity.to some degree w i l l shape a l l works discussed, the choices are guided i n certain works. Spontaneity i s limited. The works f a l l into two basic categories: (1) works which are c l e a r l y divided into sections, the treatment of these sections involving choices. (2) works i n which the given material i s i n a deter-mined sequence, the metric r e l a t i o n s h i p of the sounds being unspecific. Works i n the f i r s t category vary; the ordering of complete sections may be f r e e l y chosen or guided, sections may have to be integrated, or. choices between determined material within sections may have to be made. Variations I and II_ by Cage do not coincide with either category.. It i s assumed that most choices w i l l be made while determining a part to be used i n performance, but that some aspects w i l l be l e f t for l a s t minute decisions. Cardew gives a stated purpose for the indeterminacy of Two Books of Study: the work i s "a study for the p i a n i s t s (two) not only i n the rhythmic presentation.of sounds i n time, i t i s also a study in. combining these sounds with, those presented by 2 6 the other p i a n i s t . " Pitch is.determined i n the.Study. No omissions or repeats are allowed and pitches must be played i n t h e i r written order; attacks and c l u s t e r s are also s p e c i f i e d . The duration of each page i s one minute. The piece i s divided into sections or "groups," the beginning and end.points being s t r i c t l y assigned by the minute and/or second they occur. Within the groups pitches are rhythmically notated. Although there are s p a t i a l gaps between the sounds, no rests are notated. Within designated time boundaries the d i s t r i b u t i o n of the notated sounds i s free, to be. decided by. the.performer. The performer also chooses a general dynamic l e v e l . f o r each.group with s p e c i f i c dynamic markings being indicated for i n d i v i d u a l sounds. Pedalling decisions are l e f t to the performer. When a chord which o r d i n a r i l y would be joined to other chords on a beam i s given a separate f l a g , the notes of that chord are to be played singly in.any order. Each p i a n i s t ' s part consists of "layers" or groups. It i s up to the.player to integrate the groups, superimposing the parts Notes., Two Books of Study. London, Hinrichsen, 1966. i n "counterpoint." Thus i t might be that two i n d i v i d u a l sounds are heard simultaneously, or, because the sounds of a l l groups may be f r e e l y d i s t r i b u t e d within time l i m i t s , sounds of one part may be inserted between sounds of another part. The two players must collaborate, integrating t h e i r parts into a two-piano work. Several alternatives.are given: material from the two parts may be heard simultaneously, there may be overlapping, or sounds from one.part may be interpolated between sounds.of.the other part. There are three ways .Study, may be approached. The per-formers may choose the version, which has been.already integrated by the composer. In t h i s case i t i s suggested the piece be t i t l e d Music for Two Pianos. Or the two performers may work out a.version beforehand, collaborating i n the integration, each having ideas about his part, but remaining f l e x i b l e . In t h i s case the piece becomes e s s e n t i a l l y determined. The piece also may be indeterminate i n performance. In- t h i s case, the performers, may be quite f a m i l i a r with the score, giving pre-thought to various p o s s i b i l i t i e s and even working out on the bottom (empty) score passages which, might be too d i f f i c u l t for a spontaneous approach. While the performer i s faced with decisions p r i o r to and possibly during, the.performance, the spontaneous aspect does not appear to be stressed i n Study. There, are a series of decisions for the performer to make — d i s p o s i t i o n of notes.within a.group, general dynamic l e v e l , integration of groups and parts, and degree of indeterminacy -- but the composer i s not concerned about the degree of predetermination the performer chooses. Although choices are l e f t for the performer,.the score i s worked over, even.in.cases where indeterminacy, w i l l be operating during the performance. The exact results may not be foreseen by the composer, but i t i s probable.that the performers w i l l have at leas t some expectations, regarding the performance. Sylvano.Bussotti: Pour Clavier Apart from some rhythmic f l e x i b i l i t y , the.details i n Pour  Clavier are s p e c i f i e d . The indeterminacies occur i n the ordering of the material.. On some pages of t h i s piece the performer plays exactly as written. On other pages there are limited choices. The. performer may not produce just any.juxtaposition, may not inser t material just anywhere, but i s .restricted to certa i n a l t e r n a t i v e s . There are to be no interruptions or pauses except those.marked i n the text.. The means of providing choices varies from page to page. The f i r s t choices i n ordering occur on pages 3, 4 and 5. The performer i s given three a l t e r n a t i v e s : 1. to play as written, ignoring dotted l i n e s and arrows accompanying numbers. 2. to follow the dotted l i n e s and numbers, omitting groups i n parentheses. 3. to play a mixture of the f i r s t two choices. Choices again are guided on pages 10 to 13. For l i v e performance, two alternatives are given: 1. to play as written, ignoring "boxes," or rectangles. 2. to segregate the material, playing the material i n the boxes before playing material outside the boxes or inside another box. The order of the material i s s l i g h t l y d i f f e r e n t from that suggested by the page layout. Numbers followed by.brackets re f e r the per-former to page 17 which contains fourteen .small groups. These groups.may be inserted at the indicated points. If the piece i s being recorded, the segregated materials may be separately recorded and l a t e r pieced together, t h e i r super-imposition and juxtaposition not being subject to a determined time scheme. Insertions of groups from page 17 may also occur on pages 14 to 16. Page 17 i s to be memorized. When the performer chooses not to play any of the page 17 groups at any point on pages 13 to 16, page 17 i s to be played "normally," group 1 through 14. The performer may consider pages 20 and 21 separately, or he may place them side by side so that they become one.page, the staves j o i n i n g . Page 22 may be performed "normally," ignoring dotted and.solid l i n k s , or by following the l i n e s and grouping the sounds together. The performer i s to aim for the greatest degree of simultaneity "possible." In the recorded version of page 22, e d i t i n g , as done for the recorded version of pages 10 to 13, may be done. The composer points out that i t i s possible to reduce the page to f i v e groups, played simultaneously. Dotted l i n e s and arrows guide the performer through the alternatives of page 23. Presumably each group may be inserted i n only one of i t s possible places. In Pour Cla v i e r choices•are.often between using the deter-mined order or a . s l i g h t l y . l e s s determined order where some al t e r n a t i v e s . e x i s t . Thus, while some material i s mobile, i t s movements are r e s t r i c t e d . The composer guides the choices allowing c e r t a i n juxtapositions of blocks of materials. Matters of simultaneity may be s l i g h t l y less predictable. The blocks to be heard simultaneously.are designated, but the composer does not control placement of in d i v i d u a l sounds within the blocks. The f a m i l i a r i t y . w i t h the score required for a performance of Pour. Clavier suggests the performer i s -probably cognizant of the results.of .the various a l t e r n a t i v e s . Last minute choices between possible, pre-planned r e s u l t s , could.make..the work indeterminate of performance. Henri Pousseur.: Caracteres Most aspects of Caracteres are determined and even the undetermined, aspects are guided. The composer has limited the choices to alternatives maintaining his determined harmonic and durational scheme. Spontaneity i s thus r e s t r i c t e d . There are.two parts — l a and lb — to the score. In l a two p o s s i b i l i t i e s for shaping the performance e x i s t . The f i r s t choice i s with which of the six double pages the performer i s to begin. During the performance of a double page further choices a r i s e . The performer chooses between one of the two upper groups, then plays a l l the remaining groups without r e p e t i t i o n . The performer may proceed from the f i r s t chosen group to any "neighbouring" group below, going from l e f t to r i g h t or v i a a r i s i n g diagonal. Performance of the double page ends with one of the two lower groups. Next, the performer chooses.one of two l e t t e r s from the. "index1,'' turning to the double page adjoined by that l e t t e r , performing i t , and so on through the piece. If i n the course of playing the piece one of the l e t t e r s of a double page has already been chosen, the performer must choose the other. If both have been.chosen, the piece ends. The second possible way of performing l a begins as the f i r s t p o s s i b i l i t y d i d. Groups, except groups acting as t r a n s i -tions from double page to double page, may be repeated. It i s not necessary to play a l l groups, although the page must end with a lower group choice. One may return to a page which has already been played. In lb the performer makes choices, but p r i o r to the performance. The work i s indeterminate only i n t h i s pre-performance stage,. juxtaposition of materials being decided at that time. The score consists of a double page written on both sides and four single pages with windows. The performer f i r s t chooses either side of the double page, which w i l l serve as an "envelope" or cover. The single pages are put i n any order and placed i n t h i s cover. Ib i s then played l i k e a normal volume. Whatever shows i n the windows i s performed. Some material on the single pages w i l l not appear.in the windows and not be played i n that performance. Not a l l material can be heard i n each perfor-mance . Karlheinz Stockhausen: Nr.. 7 Klavierstuck XI The score of Nr. 7 Klavierstuck XI consists of nineteen "groups" assembled on a large sheet backed by a wooden frame. This layout enables the performer to keep a l l groups r e a d i l y within view. Only the.ordering of the groups i s indeterminate. B a s i c a l l y the performance i s as follows: To begin, the performer chooses the f i r s t group that "catches his eye." He may then choose a tempo, dynamic l e v e l and a r t i c u l a t i o n for t h i s group. At the.end of t h i s group, however, there are.tempo, dynamic and a r t i c u l a t i o n indications. The performer must apply these to the next group that he at random chooses. This procedure continues.until a group i s played three times, whence the conclusion i s - s i g n i f i e d . Groups returned to.for the second time may be varied. For example, the instructions i n brackets w i l l vary the part of- a group over or under, which they appear. The v a r i a t i o n appears i n the form of s h i f t i n g of the material up or down an octave, or adding or deleting notes. Two i n i t i a l choices are.made: (1) a s t a r t i n g group and (2) i t s tempo, dynamic l e v e l and a r t i c u l a t i o n . . From then on only order i s unspecified. Ideally, the performer should look over the score with no preconceived notions about any of the groups, or about where to s t a r t . To perform the work i t would be r e q u i s i t e that the performer be."familiar" with the nineteen groups, s k i l l e d at performing them.so. that once his eye caught a group he could perform i t f l u e n t l y . He should have no hesita t i o n or preference with respect to any group. Since the performer may go from any group to any other group, innumerable r e a l i z a t i o n s e x i s t . .If i t i s possible for the performer to get away from any preconceptions about the various groups and from any.tendencies towards ce r t a i n routes, then i t i s possible that no two performances w i l l be a l i k e . It i s suggested that when possible the piece be performed at l e a s t twice during a program, thus showing at l e a s t two d i f f e r e n t r e a l i z a t i o n s of pre-formed materials. C h r i s t i a n Wolff: Duo for Pianis t s II The score of Duo for .Pianists I I , consisting of a part for each p i a n i s t , i s comprised of "blocks" and "pitch sources." The blocks may (1) contain reference to.a.pitch source outside the block, (2) designate the number of pitches to be played from the p i t c h source and (3) give suggestions for dynamics, durations, articulation.and block timings. However, i n no single block are a l l these s p e c i f i c a t i o n s made. The .pitch source i s incorporated i n some blocks while i n a few blocks there i s neither the i n c l u s i o n of a p i t c h source nor reference to one. In the l a t t e r case any p i t c h may be played. The p i t c h sources are notated i n the t r a d i t i o n a l manner. I l l u s t r a t i v e of the greatest degree of s p e c i f i c a t i o n i n Duo i s the following block: Following f i f t e e n seconds o f . s i l e n c e , two tones from p i t c h source "a" are played i n three seconds. There i s a choice of dynamics and durations. The measurement of time i s chronometric; i t i s indicated i n seconds rather than i n unit beats and measures. The note.values, also, represent a sp e c i f i e d number, of sounds: 1 = 1 second These are values of durations o f . i n d i v i d u a l tones, t h e i r spacing within the time l i m i t being unspecified. The plus sign (+) indicates the presence of durational p o s s i b i l i t i e s . At times dynamics are l e f t unspecified, at times, as i l l u s t r a t e d in.the preceding example, there are.choices to be made. The performer, for.example, may use one in d i c a t i o n for a l l notes or one in d i c a t i o n per note,.and so on. When indications are given at lea s t one must be used i n some way. At times two, for example fp_, may be applied to one note. Further s p e c i f i c a t i o n s e x i s t . The performer may be required to s h i f t a note up to any higher octave (* " l ), down to any lower octave (x—I ) or he may be asked to play i t i n any octave (*--). A r t i c u l a t i o n s such as mute, touch, snap and J = 1 second, also p i z z i c a t o are included i n some blocks. The following pertains to a s p e c i f i c means of attack and release: 9 Q 2) 3 Of nine tones, two (£2) are released simultaneously, t h e i r attack not being determined and two tones (2J 3) which are played three times are attacked but not necessarily released simultaneously. The ordering of.the blocks i s by response to cues. The cues, which are the same for both p i a n i s t ' s parts, consist of "low mute," "high except f f , " "low pp," "middle f f , " "high f f , " and so on. They are found at the beginning of each block. The performer l i s t e n s for a.cue as he i s f i n i s h i n g the r e a l i z a t i o n of a block. After he has recognized the cue he moves to a block preceded by that cue. Each performer must be aware of the other player's part. It i s . a n t i c i p a t e d by Wolff that a performer may miss a cue. In t h i s case the performer moves to "no cue" where two general p o s s i b i l i t i e s e x i s t : (1) 17 seconds of two tones from p i t c h source "g," or (2) 3 seconds of four-tones from "e." In the second.choice, the performer i s faced with.choosing four out of seventeen pitches and then.playing these pitches i n three seconds. It might.be presumed that the performer would have to be quite f a m i l i a r with the piece to keep the blocks anywhere near t h e i r designated time-lengths. It might also be presumed that by compressing much a c t i v i t y or many instructions into a short space of time, indeterminacy of an unintended type might r e s u l t . A cue may be misinterpreted i n some way, and the s t i p u l a t i o n of "no pause between sections" may cause a misjudgement. Also, requiring the. performer to measure f r a c t i o n a l seconds i s bound to lead to inaccuracies. The requirement appears to be an exactitude, but to c o r r e c t l y measure that amount of time would be impossible. The greater the r e s t r i c t i o n or s p e c i f i c a t i o n , the more l i k e l y unspecified indeterminacy w i l l occur. While there are aspects of the score which are undetermined or variable, p o s s i b i l i t i e s or a l t e r n a t i v e methods of handling the material can be almost pre-determined. That i s , the performer might, approach the piece from a t e l e o l o g i c a l point of view, for example choosing dynamics for a purpose.such as emphasizing some point i n the piece, or arranging.possible s p e c i f i c tones within a rhythmic pattern. .The performer may construct from the given material c e r t a i n pre-formed ideas about i t s usage. Despite t h i s , contingencies.may s t i l l occur i n the handling of the choices. There are, then, aspects of the score which are prone to indeter-minacy and even with, the most planned.solutions of the score, the demands could lead to either errors or last-minute choices. The degree to which the piece i s pre-determined w i l l depend upon the performer. He may decide how he w i l l perform the materials of each block, he may work out alternative versions to be chosen at the l a s t minute, or he may have only.very general ideas of how to work out the blocks i n performance. George Cacioppo.: Pianopiece II Pianopiece II i s comprised of four groups, each s l i g h t l y d i f f e r e n t from the others and designated by the Greek l e t t e r s Si (Omega), <^  (Phi), y (Chi) and ^ ( P s i ) . One of the four groups (Phi) consists of a network similar to.those of Cassiopeia. The other three groups . (Psi, Omega, Chi) contain networks which are located on a s t a f f . Pitch i s determined i n each of the four networks. Simultaneity at.times i s indicated (Phi), at times suggested, but when indicated the tones within the symbol s t i l l may be played singly. In Pianopiece II c e r t a i n aspects are determined, others undetermined. In groups.Phi, Chi and Omega, dynamics are pro-portional to the size of the notehead. The s t r i n g c l u s t e r i n Omega, however, i s marked f f . Dynamics are notated t r a d i t i o n a l l y i n P s i . Suggested time lengths for.the four groups (structures) and for.the time between groups i s given. The sounds within these groups.may be fr e e l y d i s t r i b u t e d . A measure of guidance e x i s t s , however. .In Chi, time may be proportional to the li n e a r distance between sounds and i n Omega the sounding of the.string c l u s t e r with the soft-headed drum mallet i s represented by a decay envelope, sounds notated below the envelope possible although not necessarily being performed.as.the cluster, dies. The curvature of the l i n e s joining sounds in.Omega represents r i t a r d and accelerando. In Phi no time suggestions are made, but the per-former might use li n e a r distance as a guide. For Psi i t i s suggested that the small-headed notes may be free i n tempo, to Further freedoms concerning time may be taken. The performer may disregard the above suggestions, developing a l l values according to.his own ideas. Thus, the performer has two alt e r n a t i v e s : he may. choose durations suggested i n the score or fre e l y develop a l l . v a l u e s . Omega consists-of ..five p i t c h groups, separated by a comma (.,) which indicates a b r i e f pause. Within the p i t c h groups further sub-groups may be made; these sub-groups may be played i n any order. It might be presumed that the pitches within the sub-groups follow an order suggested, by t h e i r layout and connecting l i n e s . Their, v e r t i c a l alignment, however, i s not necessarily i n d i c a t i v e of simultaneity. If a sub-group i s repeated, the previously.chosen order of pitches may be permutated Thus, the performer f i r s t decides on.an order of groups and then considers possible sub-groups.and t h e i r ordering. In P s i , six pitches are depressed s i l e n t l y and held while a chord i s played. While the chord i s struck s f z , the small notes above are.played at a dynamic l e v e l r e l a t i v e to t h e i r s i z e . The other half.of Psi i s a group of nine small-headed notes, played pp, i n any time. The Phi network contains open-ended, paths. The performer starts at any open end and follows the networks, choosing di r e c t i o n s at the intersections. If his choice leads him to an open end, he i s faced with four further choices: (1) the performance of the.structure may be ended, (2) he may turn back, and, covering the same path, either play i t exactly as before or permutate the p i t c h , order and time values, (3) he may turn back, moving v i a an.intersection to a new route, or (4) he may jump to any other open end and then follow a new route. Presumably the performance.of Phi could consists of two to four of these a l t e r -natives. Phi may be further structured, by segmenting the sound path linearly, into two or more sounds, the segments possibly being repeated l i t e r a l l y or permutated. Groups of.sounds may be played simultaneously., or l i n e a r and simultaneous sounds i n any mixture may be played. As ..in., the, networks, of ..Cassiopeia, the pitches are layered, the highest r e g i s t e r taking top po s i t i o n i n the. network and so. f o r t h down to the lowest r e g i s t e r . The performer follows the paths, moving from p i t c h (or p i t c h group) to pitch, (group), often using intersections. The composer's use of the terms structuring, developing and permutation would seem to indicate his point of view. Certain elements are l e f t unspecified for the performer to f r e e l y define, or the performer may choose c e r t a i n alternatives,.but i t might be presumed that forethought be given to the pieces. Some f a m i l i a r i t y with the pit c h designation, possible combinations, permutations of combinations or possible interpretations of the network.might be made. Details are given, the freedom e x i s t i n g i n t h e i r organization. In October 1952 the c l e f signs do not appear i n the score but the t r a d i t i o n a l treble-bass r e l a t i o n s h i p i s assumed. Pitch and dynamics are determined. Pitch events-are given a value, but the exact metrical.relationship of the events i s indeterminate. The score i s . t o be thought, of i n terms of a "discontinuous, spatial-coordinate." Events occur at a point i n space; they can be plotted i n terms of a point on the s t a f f . The performer appears to move through "space" (the score). The score might appear to indicate that the sounds are joined, but the composer has simply not inserted the res t s . The performer moves from chord (note) to chord without d i r e c t l y joining them. Not only the size of the rests but also, i n some cases,.their placement i s obscure. There are s p a t i a l gaps.between the sound symbols (single notes,. chords) i n most instances, suggestive of possible in t e r p o l a t i o n of res t s . Although.a gap might suggest a r e s t , the performer need not in s e r t one r e l a t i v e to the size of a gap. In f a c t , he need not even inse r t a res t , i t not being sp e c i f i e d whether there necessarily are rests between a l l events. Individual pitches are given a r e l a t i v e l y determined duration but the tempo of the piece i s undetermined. It may be constant or variable, chronological or i n t u i t i v e . It.might be supposed that t h i s would a f f e c t the perception of the assigned time values, that notated durations function .perceptibly only when the speed i s consistent. In October 1952 i t i s vague metrical relationships which give r i s e to indeterminacy. If rests between the events were supplied by the composer the piece would be determined. However, with the free i n s e r t i o n of frequent rests, and varying tempos, many d i f f e r e n t arrangements of material are possible. Earle Brown: March 1953 March 1953 i s s i m i l a r . i n concept, to October 1952. Pitch, dynamics and time values are again, determined. A rate of speed ( J = 87) i s also given. As.in October 1952 there are no notated rests; the exact position of.the otherwise determined events within the. composition i s not determined. While a group i s i n progress, the group i t s e l f w i l l be r e l a t i v e l y exact i n r e l a t i o n to the given tempo, but the exact simultaneity of or time between figures, chords and single notes i s to be determined by the performer. Similar considerations concerning the values given to the spaces between events a r i s e i n March 19 53 as aris e i n October 1952. Exact simultaneity of events i s s l i g h t l y more ambiguous.in.March 1953, but the o v e r a l l tempo i s more s p e c i f i c . In.both, performance could be made d i r e c t l y from the score with-out pre-planning or forethought apart from awareness of determined and undetermined aspects. The performer could spontaneously react to each performance s i t u a t i o n , producing many.versions of the same basic material. Dance i s a t r a n s c r i p t i o n into sound symbols of the f l o o r plan of a dance by Carolyn Brown. Forty notated pitches are sparsely located on four staves. The two types of duration e x i s t i n g i n Dance, determined by the odd and even numbers of the sequence chart of the dance, are: ^ ( " t i l l inaudible") and • ("shorter"). The l a t t e r duration i n p a r t i c u l a r i s not e x p l i c i t . Since dynamics are not given, the applied meaning of the two durations would i n part depend upon the forcefulness with which the note was struck. " T i l l inaudible" implies a sustained note; "shorter" may imply a sustained or detached note. The given duration of the piece i s the "length of the (original) dance." Presumably the t o t a l duration i s purposely ambiguous. I t i s possible that the s p a t i a l gaps between the notated sounds are suggestive of an approximate length between attacks. Rests, not being, notated, are presumably l e f t for the performer to i n s e r t . Duration.of i n d i v i d u a l notes and of the performance as a whole, as well as.dynamics and attack, are undetermined, but p i t c h and the general course of events are.determined. John Cage:. Variations I .The.score.of Variations I consists of six pieces of trans-parent p l a s t i c , f i v e containing f i v e l i n e s each and one with dots of four sizes. The dots, or points, represent events, the size of the point i n d i c a t i n g the number of sounds within an event. Multiple sound events, or " p l u r a l i t i e s " may be played together or as "constellations," that i s , groupings of sounds. The f i v e l i n e s of each square represent f i v e parameters: (1) lowest frequency, (2) simplest overtone structure, (3) greatest ampli-tude, (4) lea s t duration and. (5) e a r l i e s t occurrence within a decided upon time. Presumably the performer takes.five readings, making f i v e determinations for each event. For each sound within a p l u r a l i t y , a.different position of a square (there are four) or a d i f f e r e n t square must be used. The total.performance.length might be decided beforehand. With these instructions the r e s p o n s i b i l i t y of making a part devolves upon the performer. A l l aspects are."determined" by dropping perpendiculars from a point to a l i n e . When dropping perpendiculars, distance may be "measured or simply observed." The measurement may be. d e f i n i t e or approximate. Meanings w i l l have to be assigned.to r e s u l t i n g measurements; c r i t e r i a w i l l have to be set up. Some planning, even of a general type, might go into the taking of. measurements. That i s , the performer might i n t e n t i o n a l l y make a cer t a i n measurement large or small according to the aspect being determined or his.conception of the piece. Making measurements rather than, for example, simply deter-mining that the least duration w i l l be a certa i n value, suggests a taking.of the human w i l l out of.decisions. There i s some element of chance in. what the value w i l l be. For instance, the manner i n which the squares and dots are to be arranged i s not spe c i f i e d . I t could possibly be accomplished spontaneously. The performer's conception w i l l shape the piece.; i t w i l l determine how he feels i t should be put together, how capr i c i o u s l y the measurements w i l l be made. In designating the f i v e parameters of a sound only very general indications are given. A measurement taken from one l i n e , for example, might indicate the lowest frequency. "Lowest" suggests that a l i m i t or r e s t r i c t i o n i s going to be applied. The sound may be any p i t c h but no lower than a certain one. In these determinations, then, there i s the suggestion that variety exists on one side of a . l i m i t a t i o n . Aside from " e a r l i e s t occurrence" of sounds there i s no s p e c i f i c ordering of material. Certain events.presumably might appear any time from the beginning of the piece onward while the placement of other events, determined to appear towards the end, would be more s p e c i f i c . There i s no given form; a very general form arises a f t e r the determinations are made. The performer i s responsible for making his own part. By dropping perpendiculars and taking measurements, aspects of the piece are decided, cert a i n l i m i t a t i o n s are determined. At t h i s point there i s s t i l l considerable freedom. If only the greatest amplitude or least duration.is established, at some point the performer w i l l have.to further define the sounds. It i s not s p e c i f i e d whether t h i s d e f i n i t i o n occurs before or during per-formance, by further determinations or spontaneously. In any case, a n . i n f i n i t e number of r e a l i z a t i o n s of the score are possible. John Cage: Variations II This score consists of eleven.transparent sheets, six having a single l i n e each, f i v e with one point each. Sheets may be p a r t l y superimposed or e n t i r e l y separated. Perpendiculars are dropped from the points to the l i n e s and readings are taken to determine the following: frequency, amplitude, timbre, dura-t i o n , point of occurrence i n an established period of time and structure of the event (number of sounds making.up an aggregate or c o n s t e l l a t i o n ) . Thirty readings may be taken from one p o s i t i o n of the sheets. The p o s i t i o n of the sheets i s altered before taking further readings. I t i s not s p e c i f i e d that a l l t h i r t y readings must be taken before changing sheet positions. Any questions a r i s i n g are to be answered by dropping perpendiculars. The reading i s to be measured "by means of any r u l e . " C r i t e r i a , as .in Variation I, w i l l have to be "established"; a consistent system for measuring presumably being chosen. The r e s u l t s of the readings w i l l be more determined than i n Var i a t i o n I. For example, i n Var i a t i o n II a reading could determine.a s p e c i f i c frequency, whereas i n Variation I the frequency might.be anything above a determined lowest frequency. Thus, by assigning d e f i n i t e meanings to the measurements, the performer could come up with, s p e c i f i c answers. It could be possible by making enough measurements to completely determine the d e t a i l s . Or, the measurement could indicate a general range, or area, with f i n a l d e f i n i t i o n occurring spontaneously. Through calculations the performer must decide whether the event i s to be an "aggregate" or " c o n s t e l l a t i o n , " that i s , a simultaneity or a l i n e a r event. The number of sounds i n an event i s not graphically indicated. Ordering of the events also i s not s p e c i f i e d . The sounds might be heard i n the order they are determined, or the order might be decided by dropping perpendi-culars. The performance may be any length; any number of readings may be taken. Having to decide a l l factors by calculations c a l l s for many measurements i n order to produce even a r e l a t i v e l y short piece. It i s possible, but the performer might leave some aspect to be spontaneously decided during.the performance.. This might be a question for a dropped perpendicular. Whatever the p o s s i b i -l i t i e s , many solutions should a r i s e ; no two performances are l i k e l y to.be i d e n t i c a l . Again the performer's conception shapes the piece. Variations I and II Both scores are graphic; the materials given are to be further developed. Guidelines are given for the p a r t i c u l a r i z a t i o n of parameters. In both scores dropped perpendiculars es t a b l i s h a distance, the distance then being given sig n i f i c a n c e . It i s possible to determine Var i a t i o n II to a greater extent than Variation I. Both variations take form i n three basic, successive stages: 1. Materials ex i s t i n a graphic form; general instructions are given. There are innumerable choices at t h i s point. A more determined stage follows as some aspects are further defined, some or a l l d e t a i l s established. The performance i s one r e a l i z a t i o n of the work. CHAPTER V CONCLUSIONS D i f f i c u l t i e s a r i s e i n approaching chance and indeterminate scores from a t h e o r e t i c a l standpoint. The analyst can only point out some of the questions posed by the scores, noting possible implications.of. the instructions and possible solutions to various aspects of the score. It i s not possible to suggest that the performer choose any specified way of considering the score. A discussion of the instructions and suggested solutions may give r i s e to a .performer 1s spontaneity, possibly inducing his t r a i n of thought along c e r t a i n l i n e s . Some general features appear to be common to both chance and indeterminate works. In both.categories denial of intended t e l e o l o g i c a l o rientation exists to some extent. That i s , the performer i s given choices and i s responsible for the f i n a l d e f i n i t i o n of the.work. I t has been assumed that spontaneity exists to some degree i n . a l l the works discussed. In t h i s study the terms chance and indeterminacy were attached to two d i s t i n c t categories, the.pieces within a category exhibiting common features. The discussion of the features focused on three broad areas: the instructions of the scores, the notation, and.possible solutions. The instructions of the indeterminate scores discussed were found to be r e l a t i v e l y straightforward. The performer usually i s given clear "rules" for interpreting the scores. He i s made aware of his choices or a l t e r n a t i v e s . Any explanations of the notation are usually for the sake of c l a r i f i c a t i o n , not i n order to.point out purposeful ambiguity. The instructions deal primarily with the manner of. performance, how the score i s to be handled.. Nr. 7. Klavierstuck XI (Stockhausen) , Caracteres (Pousseur) , Two Books, of Study. (Cardew). and Pour Clavier (Bussotti) exemplify t h i s approach.. With March 1953 the performer needs to be aware that Brown's "discontinuous,, spatial-coordinate" simply implies a lack of rests; the value and position of the rests are to be spontaneously chosen.by the performer. In the chance category instructions are needed i n order to explain what i s given i n the score and how extensive the freedoms are. Instructions are often enigmatic; the composer i s c a r e f u l not to be too e x p l i c i t about any aspect of the notation. In Four Systems, for instance, Brown uses terms such as "active" and "dimension" but i s i n d e f i n i t e regarding t h e i r meaning. In Octet '61 Cardew gives possible solutions to several signs, but makes i t evident i n his "notes" that i t i s the r e s p o n s i b i l i t y of the performer to determine the d e t a i l s , to interpret the signs i n whatever manner he chooses. Explanations often appear to suggest meanings rather than to d e l i b e r a t e l y state them, leaving the performer to search out implications. For instance, the tempo d i r e c t i o n "as.fast as possible to as slow as possible" may be given, but without considering the statement i n terms of a conception of the piece, l i t t l e actual meaning i s obvious. Implications possibly w i l l assert t h e i r influence subconsciously on the performer, r e s u l t i n g i n c e r t a i n spontaneous reactions not pre-planned by him. For example, use of terms having a d e f i n i t e meaning i n another, non-musical.field.might immediately suggest possible musical meanings. The composer might have just provided a graphic score and noted that nothing i s determined except the occasion for.a performance. Instead there are often several pages of instructions noting what "may" be done and what i s l e f t for the performer to define. The reading through of the instructions i n conjunction with the score should influence the performer's response. Often, however, i t i s what i s not given i n the instructions that w i l l define what i s given. The oblique-ness of the instructions allows each performer to interpret the score i n d i v i d u a l l y ; many d i f f e r i n g solutions are l i k e l y to r e s u l t . The performer w i l l not be led to one p a r t i c u l a r solution; within cert a i n l i m i t s whatever comes to mind may be an interpretation. The notation of.works i n the indeterminate category on the whole i s " t r a d i t i o n a l . " The notation may be altered i n some way but retains a resemblance to i t s t r a d i t i o n a l source. In Duo (Wolff), material i s organized i n "blocks".but t r a d i t i o n a l symbols are used to represent dynamics, metric values and p i t c h . In Pianopiece II (Cacioppo) some of the pitches are connected by In November 1952 Brown gives the i n d i c a t i o n "as fast as possible to as slow as possible, i n c l u s i v e . " "networks" but they are s t i l l represented by "noteheads." In indeterminate works time may be measured chronometrically; a section i s sometimes bounded by time l i m i t s or a note may have a chronometric value, as ) = 1 second. Considerable d i v e r s i f i c a t i o n occurs i n the notation of the chance pieces, the degree of departure from t r a d i t i o n a l notation varying greatly. In November 1952 Brown r e l i e s b a s i c a l l y on t r a d i t i o n a l symbols; pitches with accidentals, metric.values and dynamic signs are located on a f i f t y - l i n e s t a f f . The f a m i l i a r i t y of t r a d i t i o n a l notation might r e a d i l y provoke an i n t e r p r e t a t i o n . Other scores such as 1953 (Brown) and.Octet '61 (Cardew) combine t r a d i t i o n a l and.non-traditional symbols. In these scores the t r a d i t i o n a l symbols might serve as a place to begin the perfor-mance, inducing, the performer's spontaneity. His imagination set i n motion, he may evolve possible meanings for the non-t r a d i t i o n a l symbols. With.the use of e n t i r e l y graphic (non-t r a d i t i o n a l ) notation often l i t t l e is.suggested to the performer. The notation of. Music for Piano #7 (Ichiyanagi) suggests very general ideas — representation of., a type of sound and general range, from which the sounds may be chosen constitutes the score. In Timepiece (Kasemets) the approximate number of sounds to be played i s indicated by.the symbols, but the nature of these sounds depends upon further r e s t r i c t i o n s the performer must necessarily make. He must, for example, define dots and dashes, "consonant" and "dissonant.'.' In the scores Four Systems and December 1952 (Brown) the notation does not provide even general solutions. Terms used i n describing the score have a defined meaning i n a mathematical context and might possibly provoke musical meanings; a l l symbols are to be defined by the performer. Spatial distance may be used in.graphic scores as a measurement specifying either a general.area of pitch., as i n one possible interpretation.of Timepiece, an approximate duration of events, or a r e l a t i v e duration between events, as possibly used i n December 1952. The possible solutions of indeterminate works w i l l involve either the juxtaposition or the metric r e l a t i o n s h i p of material. In Nr. 7 Klavierstuck XI (Stockhausen) the performer's spontaneity i d e a l l y could produce countless juxtapositions of i t s sections. In Caracteres (Pousseur) the ordering i s more li m i t e d . There are choices but the.composer wishes only c e r t a i n solutions. In lb of Caracteres choices are made before the performance; once the page ordering i s chosen.Ib becomes determined. In Two Books of  Study (Cardew) the determined material f i t s within time l i m i t s , but juxtaposition, i n t e r p o l a t i o n or superimposition of the material leads to innumerable solutions. Material too d i f f i c u l t to manage spontaneously might be worked out ahead of time. Owing to the complexity of the groups i n general and t h e i r necessary integration within a time l i m i t , i t would seem that spontaneity, while i t may be a factor, i s not stressed. In Pour Clavier (Bussotti) choices ex i s t only on c e r t a i n pages; other pages are determined. Juxtaposition of material i s guided by dotted l i n e s , arrows and other such means. The choices, rather than being decidedly affected by spontaneity, appear to be between (1) following the guided alternatives and (2) ignoring the alter n a t i v e s , playing the work a s . i t i s l a i d out. Indeterminate works might be considered i n terms of the v i s u a l analogy of viewing an object from d i f f e r e n t angles. D i f f e r i n g sequences of musical material may o f f e r varying e f f e c t s . With indeterminate works the analyst need only point out freedoms or .alternatives a v a i l a b l e , noting the degree of spontaneity l i k e l y . The performer must learn the determined material, applying pre-thought to that end, but allow spontaneity i n some measure to give the work i t s f i n a l shape. Solutions to the ambiguities of chance pieces may arise spontaneously, but for the sake of continuity i t i s l i k e l y that some pre-thought w i l l be given to these solutions. The performer may not r e a l i z e p r i o r to attempting a solution how l i t t l e i s given i n a chance score. Many.small choices.must at some point be made. The performer might define terms, impose meanings i n a general way, s t i l l allowing for l a s t minute choices•to be made during the performance. When the performer begins to shape the piece he w i l l l i k e l y discover how.one decision hinges on another. For instance, when ordering i s free the question of time between events may p a r t i a l l y depend upon the order, chosen. Solutions w i l l represent a.drawing out of the performer's ideas. The basic idea of the composition i s to make the performer become more involved. The performer faced with a score such as Four Systems or Timepiece cannot automatically or mechanically reproduce sounds according to the composer's r e l a t i v e l y clear-cut intentions. A response to the basic ideas i s needed to produce a solution to a chance work... The interpretation may be a f i n a l solution, that i s , a d i r e c t performance from the score, or, more l i k e l y , an intermediate stage where the performer becomes aware of possible solutions, the f i n a l choice occurring spontaneously i n perfor-mance. F i n a l l y , a t h e o r e t i c a l examination of chance and indeter-minate works for piano h a s . i l l u s t r a t e d - c e r t a i n of the diverse means used.to .provoke spontaneity. It has also shown how the underlying philosophy has been applied to the works. Explication of some of the steps the performer must make pr i o r to the performance and some of the problems encountered i n examining such-scores was.of primary consideration. The study exists to aid the.performer i n the r e a l i z a t i o n of chance and indeterminate works. Books and A r t i c l e s Anonymous. "Composing by Knucklebone." Time, 79: 82-83, A p r i l 13, 1962. Behrman, David. "What Indeterminate Notation Determines." 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