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A method for the computer analysis of music utilizing the IBM 360 digital computer Carr, Edwin Wayne 1971

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A METHOD FOR THE COMPUTER ANALYSIS OF MUSIC' UTILIZING- THE IBM 360 DIGITAL COMPUTER by? EDWIN WAYNE CARR 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' In the Department of MUSIC.: We accept t h i s thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1971 In present ing th i s thes i s in pa r t i a l f u l f i lmen t 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 ib ra ry sha l l make i t f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extensive copying of th i s thes i s f o r s cho la r l y purposes may be granted by the Head of my Department or by h i s representat ives . It is understood that copying or pub l i ca t i on o f th i s thes i s f o r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of MUSIC The Un iver s i t y of B r i t i s h Columbia Vancouver 8, Canada A p r i l 29, 1971 A METHOD FOR THE COMPUTER ANALYSIS OF MUSIC UTILIZING THE IBM 360 DIGITAL COMPUTER by EDWIE WAYNE CARR: B.Mus. A method for the computer analysis of certain melodic and rhythmic elements, such as i n t e r v a l s , range, and frequency of occurrence of elements, of selected musical works i s presented and re s u l t s are obtained. The method also u t i l i z e s new techniques of musical analysis which would otherwise be too lengthy or cumbersome to be p r a c t i c a l , i . e . , information-content, redundancy, and autocorrelation. The a n a l y t i c a l re-sul t s are discussed and both the method and the re s u l t s are evaluated with a view to alteration": and expansion leading to a more comprehensive analysis of music. Thesis Advisor THE UNIVERSITY- OF BRITISH COLUMBIA MARCH, 1971 I. Introduction: 1 I I . Method of Analysis 2 I I I . Analysis Results 6 1. Mbzart 2. Comparison of Bach and Webern 3. H i s t o r i c a l Trends IV. Evaluation 261 V/. Conclusion 31 APPENDIXES 1. Program 32 2. Results of Execution! 32 3. Numerical Representation of Music . 33 4. Examples 34 5. Information Theory 35 BIBLIOGRAPHY 49 The author wishes to acknowledge Mr. Wayne Fung for his assistance i n the programming of the method, and Professor Cortland Hultberg f o r his assistance i n the entire study. INTRODUCTION: The purpose of t h i s paper i s to present a method for the computer analysis of selected musical works. The research method used i s concerned only with certain melodic and rhythmic elements of a work, such as i n t e r v a l s , range, and frequency of occurrence of these elements. The method also allows for the exploration? of new techniques of musical analysis which would otherwise be too cumbersome or lengthy to be p r a c t i c a l , i . e . , entropy, redundancy, and autocorrelation:.^ The a n a l y t i c a l r e s u l t s are discussed in:> terms of single and comparative analyses, and i n terms of trends which appear i n a comparative h i s t o r i c a l sampling of musical works. In addition, both the method and the r e s u l t s are assessed with a view to possible corrections and expansions that could ultimately lead to a more comprehensive computer analysis of music. A d e t a i l e d description: of these aspects of information theory and t h e i r musical applications i s included i n Appendix 5. METHOD OF ANALYSIS The method of analysis described i n t h i s paper uses as input data the pitches and rhythms of each instrumental or vocal p l i n e of a musical work represented by numerical equivalents. M s input data i s subjected to a series of a n a l y t i c a l oper-ations which are now b r i e f l y described: 1. Sections Each l i n e of a work i s subdivided according to musical phrasing, allowing various sections of the work to be e f f e c t i v e -l y compared. The number of sections i n each musical l i n e and the 4 number of the l a s t note of each section- i s printed out. 2. Melodic and Rhythmic Range The melodic range i s determined by subtracting the value of the lowest note from- the highest i n each l i n e , while the rhythmic range i s found by subtracting the smallest from the largest duration:, value i n each l i n e . 3 . Melodic Intervals Two orders of in t e r v a l s are taken into consideration. Intervals of the f i r s t order occur between adjacent notes and are found by subtracting the numerical value of one note from the p See Appendix 3 ^See Appendix 2 4 I b i d . value of the note preceding i t . Intervals of the second order are found i n the same manner between every- other note. The numerical representation of the interv a l s i s : unison 0 minor second 1 major second 2 minor t h i r d 3 major t h i r d 4 perfect fourth ... 5 augmented fourth . 6 perfect f i f t h .... 7 minor s i x t h 8 major sixth 9 minor seventh .... 10 major seventh .... 11 octave 12 etc. No additional enharmonic spellings of intervals were found In: the selected works. 4 . Frequency of Occurrence The frequency of occurrence of p i t c h i s found by counting the number of times each d i f f e r e n t p i t c h i s found i n each l i n e , 5 and expressing the r e s u l t as a part of 1 .0. Frequency of occurrence i s also expressed as a computer holograph, i . e . , a v i s u a l representation- (graph) of the results obtained. The frequency of occurrence of the f i r s t and second-order i n t e r v a l s , and of duratiom values i s found i n the same way. 5. Autocorrelation: The application:of the autocorrelation function was an attempt to obtain: a graphic representation' of melodic and rhythmic re p e t i t i o n s i n each l i n e of the selected work, and thereby determine the formal structure of the work. However, 5 I b l d . since the autocorrelation function was not applied to the entire work the re s u l t s obtained have l i t t l e meaning. 6. Information Content Per Line^ Information content i s a measure of randomness, such that a completely ordered musical work would have no information and a completely random work would have a maximum amount of i n -formation. The information: content i s found for each d i f f e r e n t p i t c h , duration, and Interval i n each l i n e of the selected work, and i s expressed i n b i t s per symbol. 7. Information Content Per Section The information content per section i s the amount of i n -formation conveyed by each section of a work and i s found by summing the information content of the data analyzed for that section. 8 . Information Content of the Work and Maximum Information The Information content of the work i s amount of i n -formation conveyed by each musical l i n e i n the entire work. The maximum information i s the maximum amount of information that i t was possible f o r the work to convey. 9. Redundancy The redundancy i s a percentage figure which shows how much of the work is r e p e t i t i o u s . The method i s unique i n the following ways: the use of FORTRAN (a prevalent programming language), the application of autocorrelation to musical works, the application of information ^See Appendix 5 f o r more detailed d e s c r i p t i o n . theory to the selected works with a view to h i s t o r i c a l trend and the r e s u l t s obtained. Nine musical works, chosen as a h i s t o r i c a l sampling, were analyzed and three a n a l y t i c a l viewpoints are discussed: 1. the analysis of a single work: W. A. Mozart's "Ave Verum Corpus", K. 618, Waldhelm-Eberle Edition, Wlen, 1954, chosen as the central work i n the h i s t o r i c a l sample, 2. the comparative analysis of two works: J . S. Bach's "Jesu Meine Freude (V)", Anthology for Musical Analysis. Burk-hart, C., Holt, Rinehart, and Winston, Toronto, 1964, and Anton Webern's Op. 2, " E n t f l i e h t auf leichten Kahnen", Universal Edition. #6643, Anton Webern's Erben, 1948, 3. a discussion of the indications of h i s t o r i c a l trends which become apparent when nine examples from musical his t o r y are analyzed. See Appendix 4 g 1. The analysis of Mozart's "Ave Verum Corpus", K. 618. The r e s u l t s of the computer analysis found the range of the work to be two octaves and a minor sixth, while the ranges of the i n d i v i d u a l voices were: soprano, a major tenth; a l t o , a minor seventh; tenor, an octave; and bass, a major tenth, the ranges being representative of choral part-writing of t h i s and e a r l i e r periods. The frequency of occurrence calculations f o r the three most frequent pitches i n each l i n e resulted i n the following: Soprano Alto Tenor Bass A .219 E .373 A .284 A . 2 7 9 G-- .180 D .230 C# .215 D .158 F# .133 C# .131 D .159 GF .085 Therefore, A seems to be the most frequent p i t c h i n the \rork, accounting f o r 21.9$' of a l l pitches i n the soprano l i n e , 28.4$ i n the tenor, and 27.9$ in the bass. In the alto l i n e E i s very prominent, accounting for 37.3$ of a l l pitch occurrences. The frequency of occurrence of p i t c h A can be seen to reinforce the given t o n a l i t y of D major since i t i s the f i f t h of the tonic t r i a d and the root of the dominant, these two triads accounting for the bulk of the harmonic content. The p i t c h E also r e i n -forces the t o n a l i t y since i t i s the f i f t h of the dominant t r i a d and the root of the supertonic. The frequency of occurrence of the three most frequent melodic inte r v a l s was: °See Appendix 4. Soprano Alto Tenor Bass unison: .346: unison: .466; unison. . 4 5 9 unison .382 maj. 2 n d . 2 6 9 min. 2nd.288 maj. 2nd.275 min. 2nd.271 min; 2nd.211 maj. 2 n d . 1 5 5 min. 2nd.126 maj. 2nd.123 The unison was the most frequent melodic i n t e r v a l i n a l l the l i n e s , i n d i c a t i n g the use of many repeated notes. The unison was also used s i g n i f i c a n t l y more often i n the tenor and alto than i n the soprano and bass, i . e . , i n the alto and tenor l i n e s the unison occurred approximately 46^ of the time, while In the soprano and bass the unison averaged approximately 36% of the i n t e r v a l s . Another Interesting r e s u l t was the a l t e r a t i o n of the frequencies of occurrence of major second and minor second between the l i n e s , i . e . , the major second was the more frequent i n t e r v a l of the two i n the soprano and tenor, while the minor second was the more frequent i n the alto and bass. Such s i m i l a r i t i e s between the soprano and tenor, and between the alto and bass, are also found i n other aspects of t h i s analysis. The second-order Interval calculations were found to be essen-t i a l l y the same as the f i r s t - o r d e r , so there was no need to consider them. As i n most choral music of this period the alto and tenor were melodically subordinate, with the soprano and bass having fewer repeated pitches. The most frequently used rests i n a l l lines were the whole and half r e s t s : Soprano Alto Tenor Bass Whole .888 .666 . 6 9 2 .642 Half .111 .333 . 3 0 7 . 3 5 7 It should be noted that the frequency of occurrence of the whole rest generally decreases from the soprano through to the bass while the half rest generally increases. The quarter-note was the most frequent note value with s i g n i f i c a n t occurrences of the whole-note, dotted half-note, half-note, and eighth-note: Soprano Alto - Tenor Bass Quarter .641 .510 .574 .530 Half .160 .239 .241 .289 Dotted-Half .056 .108 .114 .120 The quarter-note was most prominent i n the soprano with 64 . 1 $ of a l l occurrences and 57.4$ i n the tenor. The half and dotted half-notes increased i n - frequency of occurrence from soprano to bass. The frequency of occurrence of rests and notes combined shows quarter, half, and whole-note durations to be the most frequent: Soprano Alto Tenor Bass Quarter- . 5 9 6 . 4 5 6 . 5 0 5 . 4 5 8 Half .157 . 2 5 2 . 2 5 2 .302 Whole . 122 .116^ .121 . 1 3 5 Therefore, i n a l l voices the quarter-note value, for both notes and rests, accounted for approximately 50$ of a l l occurrences and the half-note for another 2 5 $ , i . e . , Mozart's rhythmic a c t i v i t y was generally constant from voice to voice. Again, the second-order re s u l t s were e s s e n t i a l l y the same and were not considered here. It should be noted that i n a l l frequency of 9 occurrence calculations the holograph restates a l l r e s u l t s i n graphic form. Since the information content calculations f o r pitches, 9 See Appendix 2 . i n t e r v a l s , and rhythms were based on t h e i r frequency of occur-rence, the r e s u l t s were merely a restatement of the frequency of occurrence r e s u l t s , i . e . , the most frequent elements had the highest information content. However, the results did reveal that throughout the range of each voice the most information was found in notes i n the center of the range with a gradual de-crease i n information on each side of thi s central area. The information content of the int e r v a l s showed a gradual decrease i n the amount of information content as the intervals became larg e r . The information content per section was found to be: Pitches Sectiom Soprano Alto Tenor Bass 1 mm.1-23 2.365 2.008' .953 1 .722 2 mm.24-44 2.355 2.052 1.739 1.486 3 mm.45-67 2.712 1.533 2.915 3.046 4 mm.68-88 2.321 2.353 2.413 2.744 5 mm.89-105 3.127 2.074 2 . 6 2 5 2.749 TJhitervals Sectiom Soprano Alto Tenor Bass 1 mra..1-23 1.782 1.180 .565 1.340 2 mm.24-44 1.712 1.327 1.345 1.486 3 mm.45-67 1.404 1.069 1.587 1.717 4 mm.68-88 1.379 1.307 1 .192 1.826 5 mm.89-105 1.765 1.214 1.342 1.441 Rhythm Section Soprano Alto Tenor Bass 1 mm.1-23 .974 1.162 1.650 1.668 2 mm.24-44 1.280 1.728 1.673 1.445 3 mm.45-67 1.6.11 1 .727 1.067 1.284 4 ram.68-88 1.549 1.549 1.271 1.233 5 mm.89-105 1.954 1 .913 1.774 1.665 Each section - of the piece has one dominant voice i n terms of the highest Information content f o r that section: Sectioni Pitches Intervals Rhythm 1 Soprano Soprano Bass 2 Soprano Soprano Alto 3 Tenor Bass Alto 4 Bass Bass Alto & Soprano 5 Soprano Soprano Soprano These r e s u l t s indicate the kind of part-writing used i n the work, i . e . , in.section one melodic Information was mainly con-veyed by the soprano and rhythmic information: by the bass, i n section two rhythmic a c t i v i t y shifted to the alto while the soprano maintained the melody, i n section three the tenor and bass took over melodic dominance and the alto continued with rhythmic dominance, In section four the bass dominated a l l me-lo d i c information and the soprano joined the al t o in rhythmic dominance, and in section f i v e the soprano dominated both melodic-a l l y and rhythmically. In addition, each voice has a section with the highest information'! content for each voice: Voice Pitches Intervals Rhythms Section- Section Section Soprano 5 1 5 Alto 4 2 5 Tenor 3 3 5 Bass 3 4 1 Most noticeably, these r e s u l t s show that In the l a s t section of the piece the soprano, a l t o , and tenor a l l have t h e i r highest amount of rhythmic information as well as the soprano having i t s highest amount of pitch information. Calculations f o r the information content of the work, the maximum Information:- possible, and the redundancy produced the following r e s u l t s : Pitches Soprano Alto Tenor Bass INF .323 .249 .267 .354 MAX .671 .650 .645 .635 RED 51 # 6\% 58% 44$ Intervals INF .231 .189 .208 .235 MAX .670 .649 .644 .633 RED 65$ 70$ 67% 62.% Rhythm INF .178" .205 .183 . 176r MAX .683 .668' .662 .658 RED 73$" 69% 72% 73% (INF i s the information- content of the work, MAX i s the maximum-possible information, and RED i s the redundancy). The maximum information of the pitches t e l l s us how much information: the pitches i n each voice could contain In: t h i s work, i . e . , i t was possible f o r the soprano pitches to have .671 b i t s of information: content (and .670 i n in t e r v a l s , .683 i n rhythm). However, the soprano pitches contained only .323 b i t s of a possible .671 and was therefore 51$ redundant i n pitches. The information- content of a l l voices was higher i n t h e i r pitches than i n interv a l s or rhythm. The bass pitches had the highest'- information content, .354 b i t s , as well as low maximum information content, r e s u l t i n g in the lowest redundancy of any voice, 44$. In a l l voices the maximum possible information of the Intervals was almost the same as the pitches, and since the i n t e r v a l s were o r i g i n a l l y calculated from the pitches, the possible information- of each should be the same. The informa-t i o n content of the in t e r v a l s was lower than the pitches, re-s u l t i n g i n redundancy values approximately 10$ higher, i . e . , the soprano pitches were 51$ redundant while the soprano i n t e r -vals were 65$. The rhythm i n a l l voices had low information content, high maximum information, and therefore higher redun-dancy than the pitches or i n t e r v a l s . Rhythm in the soprano and bass had the highest redundancy of the work, 73$, as well as the lowest information content, .178 b i t s in the soprano and .176 in the bass. The rhythm of the soprano also had the highest maxi-mum information i n the work, .683 b i t s . Therefore, i n a l l voices generally one-half of a l l musical information considered was melodically conveyed, one-third conveyed by intervale, and one-sixth rhythmically conveyed. The holographs 1 0 of pitches, i n t e r v a l s , and rhythm simply restate i n another form the results which have been already discussed and w i l l therefore not be considered in th i s discuss-ion. In terms of the fl u c t u a t i o n of information content, when a f u l l e r use i s made of the musical vocabulary the r e s u l t i n g work becomes more complex and less predictable, while with less of the musical vocabulary in use the opposite i s true. More generally, the fl u c t u a t i o n of the elements creating a musical structure can be judged from the basis of any musical 1 0 I b i d . s t y l e and r e s u l t i n meaningful quantitative measures, i . e . , Information content. In addition, information theory analysi as used iw t h i s paper, expresses a viewpoint independent of h i s t o r i c a l content and without q u a l i t a t i v e connotations. 2. The comparative analysis of two works: J . S. Bach's motet, "Jesu Meine Freude", Section V, and Anton Webern's Op. 2, " E n t f l i e h t auf leichten Kilhnen", Universal E d . 1 1 Since the comparative analysis of two works i s often a practise in t r a d i t i o n a l analysis, the Bach and Webern examples are compared i n terms of the results of an information theory analysis. Both examples are vocal works with three sections and si m i l a r three octave ranges. In t h i s discussion results are considered i n terms of a l l voices, but the s p e c i f i c examples of res u l t s are only given f o r the soprano. Also, only three re-sul t s are considered i n the examples. This is to allow a clear and concise discussion of the more s i g n i f i c a n t r e s u l t s . The three most frequent pitches In each work were: Bach B .190, E .119, C .097 Webern ... D .108, B - b . 1 0 2 , B . 0 9 5 Since B i s much more frequent than any other p i t c h i n the Bach i t i s the tonic or dominant of the key, but i n the Webern the pitches are very s i m i l a r i n frequency of occurrence, i n d i c a t i n g a less prominent tonal center, although several analysts have concluded that G serves a tonic function i n the work and i t i s i n t e r e s t i n g that D , the dominant of G, should be the most frequent p i t c h . The three most frequently occurring intervals i n each were: Bach Major Second .316, Minor Second .218, Unison .114 Webern ... Minor Second .424, Major Second .205, Minor Third .102 1 1See Appendix 4. As indicated, both works are dominated by major and minor seconds although Bach concentrated on smaller intervals and Webern tended toward larger ones. In both examples the most frequently used rhythmic value was the eighth-note, although Webern generally tended to use smaller values: Bach Webern Eighth 614 Eighth 650 Quarter 266 Sixteenth 205 Dotted Quarter .032 Quarter 136 It should be noted that the unit beat d i f f e r s i n these two works (Bach: quarter-note, Webern: eighth-note), and must be taken into consideration. In t h e i r use of rests Bach used only eighth and quarter rests, while Webern used sixteenth, eighth, and quarter r e s t s : Bach Webern Quarter 529 Eighth' 411 Eighth 470 Sixteenth 352 Quarter 176 The information content results were: Bach Webern Pitches B E C .455 .366 .328 .348T .335 .323 Intervals Maj. Second Min. Second Unison .525 .479 .358 Min. Second Maj. Second Min. Third .524 .469 .337 Rhythm Quarter 523 Eighth 447 Dotted Quarter .142 Sixteenth Eighth .. Quarter . .480 .426 .397 The information content of the pitches i n each work indicates that Webern employed a more equiprobable choice of pitches since the pitches had a more even information content than those i n the Bach. Both works had very s i m i l a r information content for the dominant i n t e r v a l of a second. Concerning the information content of rhythm, eighth and quarter-notes dominate the Bach, but in the Webern the sixteenth, eighth, and quarter-notes are dominant at a l e v e l of Information content similar to the Bach. The r e s u l t s of the information per section calculations were again similar i n both pieces. Concerning the information content of t h e i r pitches, both have a center section with the lowest information content, but i n the Bach the f i r s t section has the highest while i n the Webern the l a s t i s highest: Section Bach Webern Section; Pitches 1 mm.1-54 3.54 3.66 1 mm.1-43 2 mm.55-91 3.29 3.53 2 mm.44-95 3 mm.92-184 3.43 3.80 3 mm.96-147 Intervals 1 mm.1-54 2.66 2.11 1 mm.1-43 2 mm.55-91 2.73 2.25 2 mm.44-95 3 mm.92-184 2.18 2.42 3 mm.96-147 Rhythm 1 mm.1-54 1.31 .70 1 mm.1-43 2 mm.55-91 1.39 1.28 2 mm.44-95 3 mm.92-184 1.58 .89 3 mm.96-147 The Information content of the interv a l s i n the Webern shows a gradual increase as the work progresses, while in the Bach the opposite i s true. In the rhythm Information content the Bach shows a gradual increase, but the Webern begins very low, i n -creases, and then decreases to a sim i l a r low section of i n -formation content. Again In both works, the amount of Infor-mation i n d i f f e r e n t parameters i s used to balance or strengthen one another; when one parameter, such as melody or rhythm, i s i n the foreground i n terms of information being conveyed, another is i n the background and vice versa. The results of the calculations f o r the information content of the works, the maximum possible information, and the re-dundancy were: Pitches Bach Webern Information Content .396 . 3 6 0 Maximum Information .719 .752 Redundancy 44 . 9 $ 5 2 . 0 $ Intervals Information Content . . 2 6 0 . 2 8 7 Maximum Information .718 .751 Redundancy 6 3 . 7 $ 6 1 . 7 $ Rhythm Information Content .140 . 153 Maximum Information . 7 3 5 .776 Redundancy 8 0 . 8 $ 8 0 . 2 $ In both works Information content was highest in the pitches, with less Information i n the i n t e r v a l s , and the least i n the rhythm. There was more Information content i n the pitches of the Bach than i n the Webern, but i n terms of in t e r v a l s and rhythm the Webern had more information-content; the maximum possible information r e s u l t s were higher i n the Webern in a l l parameters although the difference was a small one. In fa c t , a l l parameters were very similar i n maximum possible information. The redundancy percentages for both were very s i m i l a r i n i n t e r -vals and rhythm, however, the Bach was somewhat less redundant, espe c i a l l y i n pitches. Lastly, i n both works rhythm was the most redundant, the inter v a l s less redundant, and the pitches the least redundant. 3. H i s t o r i c a l Trends The nine selected works were: 1. Perotlnus, "Dominus" for the Christmas Gradual, "From St. Martial to Notre Dame," Journal of the American Muslcological  Society. F a l l , 1949. 2. Machaut, Guillaume De, "Dame Je suis o i l s , " Works. Guillaume De Machaut, ed. L. Schrade, Les Ramparts, Monaco, 1956. 3. V i c t o r i a , Tomas Luis de, "0 Vos Omnes," Opera Omnia. 8 vols., ed. F. P e d r e l l , Leipzig, 1902-13. 4. Bach, J . S., "Jesu Meine Freude (V)," Anthology for Musical  Analysis. Burkhart, C , Holt, Rlnehart, and Winston, Toronto, 1964. 5. Mozart, W. A., "Ave Verum Corpus," K. 618, Waldheim-Eberle, Wien, 1954. 6. Beethoven, Ludwig van, String Quartet, Op. 135, H I , Anthology f o r Musical Analysis. Burkhart, C , Holt, Rlnehart, and Winston, Toronto, 1964. 7. Schubert, Franz, "Der Doppelganger" from "Schwanengesang," Anthology for Musical Analysis. Burkhart, C., Holt, Rlnehart, and Winston, Toronto, 1964. 8. Chopin, F., Prelude No. 4 i n E Minor, Anthology f o r Musical  Analysis. Burkhart, C., Holt, Rinehart, and Winston, Toronto, 1964. 9. Webern, Anton, Op. 2, " E n t f l i e h t auf leichten Kahnen," Uni-v e r s a l E d i t i o n , #6643, Anton Webern's Erben, 1948. Calculations were made for ranges and rhythmic values i n each of the selected works, however, since several works were transcriptions and sinoe both instrumental and vocal works were included, no v a l i d comparisons could be made regarding t h e i r ranges or t h e i r rhythms. Even though autocorrelation r e s u l t s were generally found to be i n s i g n i f i c a n t , some i n t e r e s t i n g trends were found I n the autocorrelations of the beginning of each work. In a l l examples the autocorrelation graph of the pitches gradually decreased to a point between . 7 6 0 for Chopin to . 9 1 6 for Beethoven, that i s , i n a l l works the r e l a t i v e dependence of one note on the next i s quite high. The Beethoven autocorrelation f e l l the least of any of the examples i n d i c a t i n g the highest degree of dependence between pitches, even though the range of the r e s u l t s covered only 1 5 . 6 $ " of the possible range. The autocorrelation: graph of the i n t e r v a l s was more e r r a t i c but always began with a drop to a point between . 2 5 2 i n the Schubert and . 6 6 8 i n the Bach, i n -d i c a t i n g that i n t e r v a l s have low to medium influence over the interv a l s which follow them. The rhythmic values also began t h e i r autocorrelation: graph with a drop, though not as severe as the int e r v a l s , to a point between: .400 in: the Schubert and . 7 2 9 i n the Machaut, Indicating that rhythmic values are somewhat more predictable. The Information of the piece, maximum information 1 possible, and redundancy re s u l t s f o r a l l parameters were: Example Pitches Intervals Rhythm INF MAX RED INF MAX RED IiW MAX RED 1 .264 .714 63$ .202 .713 72$ .151 .730 79$ 2 .297 .772 61$ .187 .771 75$ .147 .780 81$ 3 .286 .682 58$' .219 .680 68$ .201 .707 71$ 4 .360 .752 52$ .287 .751 61$ .153 .776 80$ 5 .323 .671 51$ .231 .670 65$ .178 .683 74$ 6 .451 .824 45$ .284 .824 65$ .201 .843 76$ 7 .312 .702 55$ .238 .701 65$ .258 .751 8 .352 .626 43$ .265 .624 57$ .257 .632 59$ 9 .396 .719 44$ .260 .718 63$ .140 .735 80$ Examples i : 1 Perotinus 5 Mozart 2 Machaut 6" Beethoven 3 V i c t o r i a 7 Schubert 4 Bach 8 Chopin: 9 Webern The p i t c h information content tends to increase gradually as the example becomes more recent, with the highest information! i n pitches and corresponding decrease i n redundancy found In the Beethoven. The Schubert work shows a noticeable increase i n redundancy and decrease i n information, contrary to the trends found i n chronological comparison. Maximum information content shows no p a r t i c u l a r trends. However, i t i s i n t e r e s t i n g that the Perotinus and Webern examples should share such simi-l a r maximum information r e s u l t s . The Beethoven had the highest maximum information while the Chopin had the lowest. The Intervals also have a gradual increase In information content and a corresponding decrease i n redundancy as the examples become more recent, although the amount of increase i s approxi-mately h a l f that of the increase in pit c h Information content. The i n t e r v a l s ' maximum information again shows no p a r t i c u l a r trend and the f i r s t and l a s t examples are again very close 1m t h e i r maximum possible information. The rhythmic values gradually tend toward higher information content except f o r noticeable increases i n the V i c t o r i a example and a marked de-crease i n the Webern. The Webern example again reveals a reversal i n trends. As found in the pitch calculations the maximum information content of the rhythm shows no trend, with the lowest value i n the Chopin and the highest i n the Beethoven. The redundancy percentages gradually drop in magnitude, but revert to a higher value i n the Webern example. The Machaut was found to be the most rhythmically redundant, followed by the Bach, Webern, and Perotinus, while the Chopin and Schubert were found to be the least redundant. There i s therefore a general increase in the information content of pitches, i n t e r v a l s , and rhythm as the selected works become more recent. There i s also a corresponding decrease in redundancy, but the maximum possible information f o r the se-lected works showed no trends but varied from work to work. The information r e s u l t s per section showed that i n most examples the l a s t section had the highest information content i n terms of pitches. There was also a general tendency f o r a l l sections to contain more information as the example became more recent. Beethoven, Webern, and Bach had the highest Information content i n a l l sections while the Machaut and V i c t o r i a had the lowest pitch Information. The information content of the pitches was found to be the highest of a l l parameters, i n d i c a t i n g that i n a l l the works the composers placed great stress on the pitches they chose. The information content of the intervals was found to be the highest i n the Bach and Webern, and lowest in the Perotinus and Machaut. The following graphs are based on the information content per section f o r various parameters i n two works (arbitrarily-chosen), showing how d i f f e r e n t parameters are given more or less emphasis from section to section i n the amount of i n f o r -mation conveyed. V i c t o r i a : "0 Vos Omnes," motet-Information Oontent 4 3 2 1 Pitch Rhythm Interval Mozart: Information Content Pitch - - Rhythm Interval 4 5 In the V i c t o r i a work the information content for the rhythm remains quite s i m i l a r for section one through four but shows a marked increase at section f i v e to balance the inform-ation content of the pitches and intervals which remains sim i l a r f o r sections four and f i v e . Also, the rhythmic inform-ation decreases i n section six as the information of the pitches and Intervals increases. In the Mozart example the Interval Information content tends to decrease as the Information of the pitches and rhythm increases. A l l parameters Increase i n information i n the l a s t section, Indicating a climax. The decrease In information of pitches i n section four would tend to place more emphasis on the pitches as t h e i r Information increased i n section f i v e f o r the climax. The information: content of rhythmic values i n a l l nine examples revealed o v e r a l l lower values than the pitch and i n -t e r v a l values, except i n the Schubert and Chopin where parameter importance i n terms of information conveyed would be ordered pi t c h , rhythm, and i n t e r v a l . These two examples also had the highest rhythmic information content while the Webern had the lowest. The method of analysis used in t h i s study could be Im-proved by a more d i r e c t handling of the input data. Possibly the best approach to the handling of Input data would be to have the piece of music read d i r e c t l y into the computer, v i a the appropriate interface, from tape or disc recordings. Ne-c e s s a r i l y , the method of analysis would have to be modified to accept the music as i t sounds rather than as i t is notated, and obtain the same r e s u l t s . One obvious advantage to t h i s method Is that pieces without t r a d i t i o n a l scores could be computer analyzed. There i s also the p o s s i b i l i t y that a more accurate analysis of any work could r e s u l t since the actual input data to the computer would be s i m i l a r to a l i s t e n e r ' s perception of a work. The analysis r e s u l t s and t h e i r presentation could be modified to be more concise and thorough i n the following ways: 1. simultaneously p r i n t out a l l voices of a work f o r each parameter (pitch, i n t e r v a l , or rhythm) i n a graph, 2. add harmony as another parameter of the analysis, 3. p r i n t i n a table the highest and lowest elements and t h e i r range f o r a l l parameters and a l l voices, 4. print the frequency of occurrence, autocorrelation, information measure, information measure per section, and maximum information i n one graph f o r a l l voices, 5. take great care i n the s e l e c t i o n of works for the analysis of h i s t o r i c a l trends, i . e . , the works should be written f o r the same performance media and should tend to be " t y p i c a l " of the composer, e s p e c i a l l y i f only several of his works are represented, 6. take the autocorrelation of a l l points of a work rather than only the f i r s t few, 7. present redundancy results as a table of percentages including the redundancy of a l l parameters for each voice as well as cumulative r e s u l t s , 8 . include information rate per second to r e f l e c t the tempo, texture, and harmonic density of the music, 9. include a table of the average rates of information transmission per second f o r each section of a work, 10. delete second-order calculations since they merely repeat the results of the f i r s t - o r d e r , 11. give a l l graphs the same axis d i v i s i o n s , 12. include with the frequency of occurrence c a l c u l a t i o n an additional set of calculations which would assume octave equivalence, 13. restate various r e s u l t s in the form of comparative graphs to make results as concise and usable as possible, 14. include provision f o r the simultaneous computer com-parison of several d i f f e r e n t musical works, 15. include i n each graph the average of the r e s u l t s , 16. analyze larger pieces, e s p e c i a l l y orchestral works, by the consideration of parameters rather than single l i n e s , 17. Include calculations to analyze timbre, dynamics, and orchestration, and 18. compare the information content of chords and one 12 aspect of Hindemith's compositional theory. Hindemith as-signed r e l a t i v e values to chords according to t h e i r r e l a t i v e harmonic tension, such that motion from a high to a lower value chord w i l l cause a decrease i n tension, with the reverse also being true. As a r e s u l t , harmonic f l u c t u a t i o n becomes a change of tension, making tension approximately analogous to information trends. For example, when a chord has a low tension value in a harmonic sequence i t w i l l also have been a r e l a t i v e l y certain choice in terms of the l i s t e n e r ' s expectations, and w i l l therefore have low information. On the other hand, chords with high information content would have high tension values. An aesthetic problem arises when a computer analysis of music i s attempted since i t i s d i f f i c u l t to attach l i t e r a l meanings to music even though an aim of analysis i s to deter-mine the significance of a musical composition as well as to understand the mental and technical processes involved i n creating and responding to a musical composition. Meyer 1^ states that musical meaning depends upon learned responses to musical stimuli and he proposed an "a f f e c t theory of music" based on the concept that emotion is generated when a tendency; 12 P. Hindemith, The Graft of Musical Composition. Associated Music Publishers, Inc., London, 1945. 1^L. B. Meyer, Emotion and Meaning In Music. University of Chicago Press, Chicago, 1956. to respond i s i n h i b i t e d , and musical meaning i s a product of 14 expectation. T i s c h l e r f e l t that an aesthetic appreciation of music must be based on. a f a m i l i a r i t y with the medium and i t s technical p o s s i b i l i t i e s . T i s c h l e r defines two types of re-lationships which characterize musical aesthetics: 1. .internal ... these change with the medium and i n music consist of rhythm, melody, harmony, counterpoint, tone color, expression, and form or contour, 2. external ... these are true of a l l the arts and consist of gesture, program, ethics, technical mastery, psychological drives of the a r t i s t , function, relevant h i s t o r i c a l and s o c i o l o g i c a l data, and performance. ^ He proposed that the greater the number of relationships a work of art reveals, the greater aesthetic significance we must attach to i t . This may or may not be true, but his d e f i n i t i o n of Internal and external relations i s h e l p f u l i n determining what can be r e a d i l y extracted from a musical score and per-formance by the computer, and what must be read into a score to determine i t s general s i g n i f i c a n c e . Several possible future musical applications of the com-puter have become evident as a r e s u l t of this ana|rsis. However, the investigation of s p e c i f i c aspects of musical works rather than general concepts seems a better application of the computer f o r the present time i n seeking more precise descriptions of musical concepts. One such application i s information theory analysis of music of the type presented i n t h i s paper, which bases i t s analysis on the premise that most musical works 1Z*H. T i s c h l e r , "The Aesthetic Experience", The Music  Review. 17:189, 1956. ^ I b i d . r e f l e c t a balance between the extremes of order and disorder, and that s t y l i s t i c differences can depend to a large degree on fluctuations between these two extremes. Included i n t h i s a p p l i c a t i o n could be comparative s t y l i s t i c analyses which might be of i n t e r e s t to musicologists i n determining c h a r a c t e r i s t i c s of p a r t i c u l a r h i s t o r i c a l s t y l e s . Another application might be the analysis of musical sounds themselves as sound waves. Acoustical applications are also possible, such as spectrum analyses of musical compositions to study orchestration, dissonance, density, and texture as well as performance. A comparative study of instrumental timbres would also be possible using spectrum analysis. The method presented i n t h i s paper was applied to various musical parameters, and s i g n i f i c a n t results regarding Informa-t i o n content and redundancy were obtained, i n addition to de-termining i n t e r v a l s , ranges, and frequency of occurrence of elements i n the selected works. Frequency of occurrence r e s u l t s were made possible by the use of the d i g i t a l computer to perform operations on musical parameters which would be too time consum-ing to be considered by any other method. This computer analysis of music Is unique i n several ways. The method uses FORTRAN, a prevalent programming language available at most computer i n s t a l l a t i o n s . The autocorrelation function was applied to the selected works and was found to be a useful a n a l y t i c a l t o o l for determining forms and contours. Also, the selected works had not previously been subjected to an information theory analysis. It was found that the musical analyst can increase his understanding of h i s t o r i c a l trends i n music as well as in d i v i d u a l works by adding information content and redundancy calculations to his a n a l y t i c a l method. A P P E N D I X 1 A P P E N D I X 2 These appendixes, the computer program, printouts, and Input cards, have not been included here because of t h e i r size but are available from the Library, Music Department, University of B.C. The numerical representation of musical pitch was based on the range of the normal piano keyboard and assumed enhar-monic equivalency, f o r example: 4¥. : #-6 •3 [ 0 & 0 1 © c 8va-' 1 etc 16 17 18 19 20 21 22 23 24 25 26 27 28 etc. Rhythmic representation was as follows: 1 2 3 4 5 6 7 8 etc 12 16 24 32 etc. rests were indicated as negative durations. 1. Perotlmis, "Dominus" f o r the Christmas Gradual, "From St. Martial to Notre Dame," Journal of the American Muslcologlcal  Society. F a l l , 1949. 2. Machaut, Guillaume De, "Dame je suis c i l s , " Works. Guillaume De Machaut, ed. L. Schrade, Les Ramparts, Monaco, 1956. 3. V i c t o r i a , Tomas Luis de, "0 Vos Omnes," Opera Omnia. 8 vols., ed. F. P e d r e l l , Leipzig, 1902-13. 4. Bach, J . S., "Jesu Melne Freude (V)," Anthology for  Musical Analysis. Burkhart, C , Holt, Rinehart, and Winston, Toronto, 1964. 5. Mozart, W. A., "Ave Verum Corpus," K. 618, Waldheim-Eberle, Wien, 1954. 6. Beethoven, Ludwig van, St r i n g Quartet, Op. 135, H I , Anthology f o r Musical Analysis. Burkhart, C , Holt, Rlnehart, and Winston, Toronto, 1964. 7. Schubert, Franz, "Der Doppelganger" from "Schwanen-gesang," Anthology for Musical Analysis. Burkhart, C , Holt, Rlnehart, and Winston, Toronto, 1964. 8. Chopin, F., Prelude No. 4 l n E Minor, Anthology for  Musical Analysis. Burkhart, C , Holt, Rlnehart, and Winston, Toronto, 1964. 9. Webern, Anton, Op. 2, " E n t f l l e h t auf leichten Kahnen," Universal E d i t i o n #6643, Anton Webern1s Erben, 1948. Information theory i s a s t a t i s t i c a l theory of communi-cation. This theory, originated i n 194-8 by 0 . E. Shannon,10" i s an outgrowth of applied p r o b a b i l i t y theory with extensive applications to communications systems. The fundamental problem of communication i s to reproduce at one point a message transmitted from another point. Whether the message has meaning or not i s i r r e l e v a n t ; what i s s i g n i f -icant is that the actual message i s one selected from a large number of p o s s i b i l i t i e s , and the communication system i s de-signed to operate for each possible message. Three levels of communication determine a message: the technical l e v e l to transmit symbols accurately, the semantic l e v e l to make symbols convey the desired meaning, and the effectiveness l e v e l to determine how the received message affects conduct i n the desired way. The message and network of a communications system may be of any nature, but the entire system has certain fundamental c r i t e r i a : 1. any communication system can be considered to be i n -dependent of the human receiver and sender, 2. the message is a sequence of signals or some physical representation of signals, 3. there i s no concern for the meaning of the information, 16 C. E. Shannon, W. Weaver, The Mathematical Theory of  Communication. University of I l l i n o i s Press, Urbana, 111., 194-9. but the number of symbols and how fast they are trans-ferred i s of great concern. Since information theory grew out of p r o b a b i l i t y theory many aspects of p r o b a b i l i t y theory are carried over. Proba-b i l i t y can be defined as an event's r e l a t i v e frequency of occurrence, and the information content of a system r e f l e c t s the p r o b a b i l i t y of occurrence of events. If the system has no defined properties or r e s t r i c t i o n s , the choice i s random and the information content i s at a maxi-mum, but when a r e s t r i c t i o n i s applied the Information content i s reduced. A completely random message consists of a sequence of symbols chosen independently from those available with equi-p r o b a b i l i t y ; any conceivable message might be generated. Entropy can be thought of as a measure of missing i n f o r -mation, so that when an outcome i s certain, entropy w i l l d i s -appear. Entropy can also be a measure of randomness, i . e . , the tendency of a system to become more and more unorganized as entropy increases, so that when the entropy i s said to be low there i s l i t t l e choice within the system. When the entropy of an information source has been calculated, i t can be compared to the maximum value the entropy could have to give the r e l a t i v e entropy. For example, a . 8 r e l a t i v e entropy means the source i s 8 0 $ free i n i t s choice of symbols from a given set to form a message. Conversely, redundancy i s the f r a c t i o n of the message determined by rules governing the use of the elements i n question; therefore, the r e l a t i v e entropy i s a measure of the free r e s u l t s while the redundancy i s a measure of the governed r e s u l t s . Shannon developed the basic equation for information con-tent computations derived from p r o b a b i l i t i e s p ( i ) where 1, which i s equal to 1 , 2 , ,N, i s associated with each symbol i n an alphabet of N symbols and the p( i ) are not equal i n value. Therefore, information content (H^) i s : N 1 7 % =T n(1) loggpd) bits/symbol. ' Because the magnitude of the information* content i n a message depends not only on the size of the alphabet, but also on the pr o b a b i l i t y d i s t r i b u t i o n ) o f symbols drawn from the alphabet, the information content i n messages i s analogous to the entropy. Actually, entropy? and information are often interchangeable. The formula for information rate becomes: N Hrp = - M > -p(l) l o g 2 p ( i ) b i t s / second, and the t o t a l information conveyed becomes: HP = - M T Z Z P U ) l o g 2 p ( i ) b i t s . The scale pf information goes from maximum information 1 atL t o t a l disorder to zero Information at t o t a l order. Conversely, i f redundancy i s defined as R = Hmax - H x 100$ where Hmax = Hmax loggN bits/symbol, or as R = 1 - % x 100$, then the predict-a b i l i t y goes from 100$ f o r t o t a l order to 0$ f o r t o t a l disorder. It should be noted that Shannon's measure of Information 1^C. E. Shannon, W. Weaver, The Mathematical Theory of Com- munication. University of I l l i n o i s Press, Urbana, 1 1 1 . , 1950. content through his HJJ, formula cannot be considered as a measure of meaning because of the quantitative nature of the r e s u l t s . Also, the % formula applies to ensembles of events or messages whereas meaning applies to a single message. In f a c t , the theory-relates to what one could say, rather than what one does say. Information i s a measure of one's freedom of choice when one selects a sequence of symbols to form a message. 1 ft A. A. Moles has worked out a general approach to r e l a t i n g music and information theory. He relates ambiguity, form, and entropy variances, and thereby shows the importance of sequential choice processes i n b u i l d i n g up musical structures. MEMORY PERCEPTION; 1. Instantaneous Memory 2. Dated Memory 1. Semantic Mode 3. Undated Memorization 2. Aesthetic Mode MUSICAL SOUNDS-MESSAGE 1 9 The semantic mode i s a system of symbols which can be coded, i . e . , translated into another language (a score for example), and the aesthetic mode i s any element of sensory appeal. The aesthetic and semantic modes are related by acoustical quantities which are symbols on a given scale of duration and information rate which can be computed. It should be noted that the aver-age person can grasp 10 to 20 b i t s of Information per second with a maximum capacity of perhaps 100 b i t s per second; therefore, 1^A. Moles, Information and Esthetic Perception, trans. J . E. Cohen, University of I l l i n o i s Press, Urbana, 111., 1966. 1 9 l b i d . perception i s usually a selection of d e f i n i t e symbols from the whole message. In terms of perception the timbre, density, and amount of r e p e t i t i o n should also be considered. Memory Is simply divided in terms of span. More s p e c i f i c a l l y , music can be seen as a discrete system with the most di f f u s e music being a successive note selection which i s completely random, and which forms the basis from which to fashion more c h a r a c t e r i s t i c structures. In most cases the main methods considered f o r r e s t r i c t i n g the note selection are combinatorial p r i n c i p l e s and s t a t i s t i c a l methods with the computation of t r a n s l s t i o n a l p r o b a b i l i t i e s of information theory. Other methods of r e s t r i c t -ion are possible, but these concepts receive the main usage i n r e l a t i n g information theory and music. To apply information theory a n a l y t i c a l l y to music certain aspects and Ideas of information theory are d i r e c t l y trans-ferred or assumed to be d i r e c t l y transferable. Two general 20 types of assumptions are made: 1. mathematical assumptions of the musical system being applicable to the analytic c a l c u l a t i o n of information theory: a. no element not already known to be i n a sequence can occur; i . e . , unused elements have zero pr o b a b i l i t y , b. a s u f f i c i e n t l y large sample from an i n f i n i t e se-quence has the same s t a t i s t i c a l structure as the i n f i n i t e sequence, c. the s t a t i s t i c a l structure of a sequence i s i n -2 o J . Cohen, "Information Theory and Music", Behaviour  Science. Vol. 7, 1962. d. there i s the same order of patterning throughout the sample, e. i n f i n i t e memory i s available for storing music as numbers. 2. aesthetic assumptions of the s u i t a b i l i t y of applying information theory to music. Aesthetic assumptions can be reduced to one basic question, i . e . , do the s t a t i s t i c a l p r o b a b i l i t i e s correspond to the l i s t e n -er's expectations. However, f o r t h i s assumption to be e n t i r e l y true the l i s t e n e r would necessarily completely store the music l i k e a computer; the l i s t e n e r ' s experience i s much more l i k e a constantly changing p r o b a b i l i t y system affected by such things as past experience, averaging, attitude, comprehension, etc. W. Meyer-Eppler s approach to the information theory analysis of music i s based on the premise that music i s made up of definable discrete elements which can be described, a l -though the sum of these elements may not necessarily r e s u l t In music. Information theory serves as a c r i t e r i o n of form by the observation of the frequency of occurrence of elements. For example, p i i s the frequency of occurrence of the element " i " i n the works to be analyzed and therefore, f i r s t - o r d e r information entropy i s the s t a t i s t i c a l d i s t r i b u t i o n of elements 'W. Meyer-Eppler, " S t a t i s t i c and Psychological Problems of Sound", Die Relhe. Vol. 1, 1955. with no reference to mutual r e l a t i o n s . In c a l c u l a t i n g second-order frequencies becomes p/. . suggesting pairs of elements. The second and higher orders begin to deal with Markov chains and the frequency of transfer from one element to another, making considerable mathematical calculations necessary. o p Professor W. Fuchs of Germany has applied a s t a t i s t i c a l method to the analysis of a representative group of works from the Renaissance period through the modern period. His method i s based on the computation of frequency d i s t r i b u t i o n s and re-lated parameters such as mean values, variances, higher orders, skewness (the state of assymetry or symmetry as shown by the frequency d i s t r i b u t i o n ) , and kurtosis (the state of peaked or f l a t graphic representations of a s t a t i s t i c a l d i s t r i b u t i o n ) . Also, his method has an entropy basis, i . e . , information values as obtained from Shannon's formula. Professor Fuchs applied these methods to the analysis of note and i n t e r v a l counts, and when he plotted the various parameters against the h i s t o r i c a l periods he found d e f i n i t e trends i n , for example, pi t c h d i s -t r i b u t i o n which seemed to correlate with the h i s t o r i c a l devel-opment of t o n a l i t y . Typical r e s u l t s show that s e r i a l compos-i t i o n s indicate a reversal i n the trend to more uniform d i s t r i -bution. J . G. Brawley applied information theory to the analysis of rhythm, being concerned "not so much with the intent of describ-ing i n d i v i d u a l styles as with i n v e s t i g a t i n g the p o s s i b i l i t i e s p p Described i n J . V. Cohen, "Information Theory and Music," Behaviour Science. Vol. 7, 1962. of a method for using information theory as a t o o l i n describing styles."2-5 He makes two preliminary assumptions: rhythm i n music i s a discrete system of communication and, s t y l e in music i s a stochastic process with the structure of a stationary Markov chain. Brawley's rhythmic analysis described the elements of this discrete system such that . . i n perception of occurrences i n time, these occur-rences are grouped into patterns," and "whether these groupings are regular or not, we s h a l l adopt this group-ing of any number of unaccented beats with a single ac-cented beat as one of the basic p r i n c i p l e s of t h i s study. More s p e c i f i c a l l y , t h i s w i l l be the basis f o r determining the length of the primary elementary symbols of our d i s -crete system, i . e . , the length of a single rhythmic unit, or rhythm."24 Tempo was r e s t r i c t e d within the range of one to two beats per second. I n i t i a l l y Brawley analyzed Bach's Two-Part Invention #14 and l i s t e d each rhythmic pattern, i t s frequency of occurrence, and i t s r e l a t i v e frequency. Both parts of the invention were considered but preference was given to the upper part. The eight rhythmic patterns that were found had an average i n f o r -mation content of 2.6 b i t s and a redundancy of 13.1$. Because of the limited nature of t h i s application no generalizations were made u n t i l other samples had been analyzed, with emphasis being given to redundancy calculations for comparison: 2 3 I b l d . , p. 153. Hec dies (Perotinus, 12c.) 25 . 8 $ Acun Motet (P. de Cruice) 15.0$ Two ballads of M. dePerusio 8 . 9 $ Schoenberg, 4th S t r i n g Quartet 6.4$ Schoenberg, Verklarte Nacht 18.8$ Schubert, Three songs 5.7$ Mendelssohn, Three songs 19.6$ Brahms, Three songs 16 . 3 $ F i n a l l y Brawley analyzed a large closed body of material, the minuets of Mozart's s t r i n g quartets. Each measure was con-sidered a rhythmic group and the redundancy of these was found to be 1 9 . 4 $ . Dr. J . E. Youngblood 2^ attempted "to explore the usefulness of information theory as a method of i d e n t i f y i n g and defining musical s t y l e s . " "Musical s t y l e may be considered a probabil-i t y system which must be s t a t i o n a r y . " 2 ^ To begin th i s study Youngblood selected a number of Romantic songs i n major keys: eight songs from Schubert's "Die Schone Mullerln," six arias from Mendelssohn's "St. Paul," and six songs from Schumann's "Frauen-lieb e und Leben." After f i n d i n g an approximation of the pro-b a b i l i t y of each of the twelve notes of the scale f o r each composer he calculated f i r s t - o r d e r information content and re-dundancy. Secondly, he found the frequency of occurrence of a l l pairs of tones and calculated the second-order frequency as well as the information gained by one note from another, 2 5 l b i d . , p. 151. 2 6 I b i d . , p. 151. and the redundancy of t h i s order: $R1 $R2 $R2c Schubert 1 2 . 5 2 0 . 4 3 5 . 6 Mendelssohn 14 .9 24 . 0 4 3 . 5 Schumann 14 ,7 2 2 . 4 3 7 , 3 Cumulative 13.4 2 0 . 5 2 9 . 2 Some conclusions could be reached from these figures, espec-i a l l y i s a harmonic analysis were also considered ; f o r example, Mendelssohn's use of chromaticism was less frequent than Schubert's and Schubert i s less redundant than Mendelssohn. To compare this data on Romantic music to that of another st y l e , Youngblood analyzed the Gloria, Sanctus, and Agnus Dei from the F i r s t Mass for Solemn Feasts (Liber Usualis pp. 19-22) and the Kyrie from the mass Orbis Factor (Liber Usualis p. 46). Using seven and twelve note systems the results were:2''' J&1 $R2 %R2c Seven Note 3.2 23 . 9 28 . 8 Twelve Note 23 . 9 41.7 44.0 Gregorian chant as a twelve-note system i s therefore s l i g h t l y more redundant than any of the Romantic composers and much more redundant than the three of them combined. Although these re-sults are very i n t e r e s t i n g as they stand, Youngblood concluded that to obtain precise a n a l y t i c a l results analysis of at least twice the number analyzed here, or another type of analysis, would be necessary. 2 7 I b i d . , p. 151. L. A. H i l l e r and C. Bean used information theory to study the st r u c t u r a l d e t a i l s of musical compositions. Four sonata expositions were analyzed and compared using elementary information theory ana l y s i s : 1. Mozart ... Sonata i n C Major. K. 54-5, 1788 2. Beethoven ... Sonata i n E Minor. Op. 90, 1814 3. Berg ... Sonata i n B Minor. Op. 1, 1908 4. Hindemith ... Sonata i n G- Minor. 1936 The analysis was limited to the study of pit c h d i s t r i b u t i o n s i n the expositions of the f i r s t movements of each sonata. The expositions were subdivided at phrase endings or other natural d i v i s i o n points so that changes i n information content could be seen as each exposition progressed. Two basic element sets were used, a twelve-note set of the chromatic scale assuming octave equivalence and a twenty-one-note set to include d i f -ferences i n enharmonic content. Two counts were made for each phrase (both using the twelve and twenty-one-note sets), count A based on the number of note attacks only and count B which included durations of notes where a sixteenth note i s equal to one unit, with smaller durations disregarded. Eight calculations were made fo r each exposition: note occurrences, the p r o b a b i l i t i e s of note occurrences, the i n -formation content, the t o t a l Information content of the symbols i n each of the subsections, the average information value 2 ^ L . A. H i l l e r , C. Bean, "information Theory Analysis f o r Four Sonata Expositions," Journal of Music Theory. Spring, 1966. of the symbols In each exposition, redundancy, rates of i n -formation transmission; and the average rate of note trans-mission for each subsection. Generally, i n counts A and B, there i s l i t t l e sharing of information content or redundancy values. However, chronologically there i s an Increase i n average information and a decrease i n redundancy. For example, a comparison of the extreme high and low values i n the pitch calculations shows section V of the Mozart to have an i n f o r -mation content of 2 .19 and a redundancy of 50 .1$, and section IV of the Berg to have an information content of 3.97 and a redundancy of 9 .5$ . Differences between count A and count Bl-are a r e s u l t of a general tendency of durational values to cause lower Information content. This information theory analysis of four sonata expos-i t i o n s i s based on zero and f i r s t - o r d e r stochastic processes. A zero-order approximation to some message i s that order gen-erated by the random choice process with equiprobable choices, while a f i r s t - o r d e r approximation i s based on a p r o b a b i l i t y d i s t r i b u t i o n derived from the frequency of occurrence of sym-bols and does not allow for the possible ways each choice may a f f e c t a l l subsequent choices. The problem seems to be how one might consider the long-range structure of music, i . e . , o v e r a l l form in i n d i v i d u a l compositions as well as the h i s t o r i c a l continuity or lack of continuity of a l l music. Conditional p r o b a b i l i t i e s have been used i n the analysis of Webern's Symphonle. Op. 21. L. A. H i l l e r and R. F u l l e r 2 9 carried out a comparison of two analyses of the f i r s t move-ment of Webern's Symphonie: s t r u c t u r a l analysis and i n f o r -mation theory analysis. Their purpose was to show any com-plementary aspects which the two methods might have and to int e n s i v e l y analyze the movement using information theory, showing the procedures used, the types of results obtained, and r e s t r i c t i o n s , such as sample size, which aff e c t the f i n a l r e s u l t s . The Webern Symphonle was chosen because i t is a " c l a s s i c " example of highly ordered s e r i a l music, and there-fore the s t r u c t u r a l organization would be d i r e c t l y r e f l e c t e d i n information content measurements. The f i r s t movement was chosen because certain recent tendencies occur (such as a r i g i d l y controlled p i t c h structure), and the three main sections of the movement are a l l long enough for v a l i d s t a t i s t i c a l analysis. The purpose of the s t r u c t u r a l analysis was to reconstruct precompositlonal procedures presumably used by Webern himself, i . e . , affected by the composer's logic and not by any indep-endent evaluation of his plan. Seven aspects were considered: tone row, sectional plan, canonic structure, v e r t i c a l struc-ture, transpositions of the row, instrumentation, and rhythm. Four parameters of analysis by information theory were considered: pitch, four types of i n t e r v a l l i c r e l a t i o n s bet-2 9 L . A. H i l l e r , R. F u l l e r , "Structure and Information i n Webern's Symphonie Op. 21", Journal of Music Theory. Spring, 1967. ween pitches, rhythm, and pitch and attack intervals combined. Redundancy plots were found f o r each section and information content plots were found f o r each section f o r one, two, and three element groups; one element determined by another, a two element group determined by one element, and one element determined by a two element group. The pitches i n each section were analyzed according to t h e i r own symbol alphabet. For example, the exposition section was analyzed according to a 13-pitch set, the development according to a 27-pitch set, and the r e c a p i t u l a t i o n according to a 14-pitch set. 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