Topology of Musical Data

Transcription

1 Topoogy of Musica Data Wiiam A. Sethares Department of Eectrica and Computer Engineering, University of Wisconsin, Madison, USA, November 27, 2010 Abstract Techniques for discovering topoogica structures in arge data sets are now becoming practica. This paper argues why the musica ream is a particuary promising arena in which to expect to find nontrivia topoogica features. The anaysis is abe to recover three important topoogica features in music: the circe of notes, the circe of fifths, and the rhythmic repetition of timeines, often pictured in the neckace notation. Appications to fok music (in the form of standard MIDI fies) are presented and the bar codes show a variety of interesting features, some of which can be easiy interpreted. 1 Introduction Carsson and his coworkers 3] have recenty introduced a way of parsing arge data sets using barcodes that show the Betti numbers of an underying topoogica space. A parameter ɛ is sowy increased and a coud of points (which may be pictured in R n ) are identified whenever they are coser together than ɛ. For ɛ sma, the structure is trivia: a points are separated from a other points. For arge ɛ, the structure is again trivia, a points are identified together and the space is topoogicay equivaent to a singe point. In between sma and arge, cyces may appear in two dimensions, spheroids in three dimensions, and other higher dimensiona anaogs may appear and disappear as ɛ changes. Features which persist over a range of ɛ are caed persistent, and ikey refect some underying structure in the data. This technique is caed persistent homoogy, and a number of appications have begun to appear in areas such as image processing 6] and in the anaysis of bioogica data 14]. The search for such topoogica features in data is ony beginning, and it makes sense to ook in paces where one may reasonaby expect to find interesting structures. One such pace is in musica data. It is a commonaity that in musica scaes there is a circe of notes which may be pictured as in Figure 1. This figure encodes two important aspects of musica perception: notes that are near each other in frequency (such as C and C ) are perceptuay cose, and notes that are an octave apart (such as ow C and high C) are perceptuay cose, even though they differ by a factor of two in frequency. Obviousy, Figure 1 shows a circe, and it is reasonabe to ask if such circes can be recovered from an anaysis of musica data. 1

2 Figure 1: A surprising number of insights about musica structure are dispayed in this circe of notes, which is ike a cock face on which the hours of the day have been repaced by note names. A second we known topoogica structure in music theory is the circe of fifths, shown here in Figure 2, which is taken from the Wikipedia artice of the same name 16]. The circe of fifths is a standard way musicians and composers tak about the cose reationships between musica scaes and keys, and represents another way of interpreting the distance between musica chords and scaes. Again, Figure 2 shows a circe, and it is reasonabe to ask if this circe can be recovered from an anaysis of musica data. Representing tempora cyces as spatia circes is an od idea: Safî a-din a-urmawî, the 13th century theoretician from Baghdad, represents both musica and natura rhythms in a circuar notation in the Book of Cyces 1]. Time moves around the circe (usuay in a cockwise direction) and events are depicted aong the periphery. Since the end of the circe is aso the beginning, this emphasizes the repetition inherent in rhythmic patterns. Anku 2] argues that African music is perceived in a circuar (rather than inear) fashion that makes the neckace notation, shown here in Figure 3, particuary appropriate. Ceary, Figure 3 shows a circe, and it is reasonabe to ask if this circe can be recovered from an anaysis of musica data. This paper shows that a three of these circes can indeed be ocated in the barcodes drawn using the techniques of persistent homoogy. There are severa different types of musica data 12], incuding 1. spectra data that shows the interna structure of individua sounds, 2

3 Figure 2: The circe of fifths shows the reationships among the tones of the equa tempered chromatic scae, standard key signatures, and the major and minor keys. 2. audio data that provides a itera (numerica) representation of sound waves, 3. symboic note data (such as occurs in a musica score or in standard MIDI fies) that provides instructions for a performer to pay a specific piece of music, 4. anaytica data (such as Roman numera anaysis 10] or Schenkerian anaysis 11]) that can be derived from musica scores to provide theorists with toos to understand musica progressions, and 5. periodic data that reates to the tempora and rhythmic aspects of music 13]. Any kind of periodicity, if propery embedded, may ead to nontrivia topoogica structures, and periodicities may occur in rhythmic data at severa eves in a metric hierarchy. In principe, it may be possibe to ocate circes (of notes, of fifths, of periodicities in rhythmic or tona materia) or other topoogica structures in many of the above kinds of data. For this initia investigation, we use MIDI note data, since this is the eve at which the circes in Figures 1-3 are conceptuaized, and hence the eve at which success is most ikey. Section 2 defines a pitch-cass metric that aows a cear dispay of the circe of notes. Data is taken from a standard repertoire of traditiona fok meodies, and the barcodes are shown to dispay information about the musica scae used in the piece (using the Betti 0 barcode) and to refect the circe of notes itsef in the Betti 1 barcode. More generay, the note data can be gathered into higher dimensiona time-deay embeddings. When this is done for the same fok meodies, interesting new (and as yet unexpained) features arise in the barcodes. 3

4 Ewe (Ghana) Bemba (Centra Africa) time Yoruba (Nigeria) Figure 3: Traditiona rhythms of the Ewe (from Ghana), the Yoruba (from Nigeria) and the Bemba (from Centra Africa) are a variants of the standard rhythm pattern described by King 5]. These timeines are represented in the neckace notation as having different starting points. Section 3 ooks at the issue of measuring chord and scae data (i.e., when mutipe notes occur simutaneousy) using the techniques of persistent homoogy. A metric caed the chord-cass metric aows a cear dispay of the circe of fifths on speciay chosen synthetic data. Chord data is then taken from a traditiona fok meody and MIDI data is used to anayze a coection of four-voice Bach choraes. The barcodes are shown to dispay information about the musica chords used in the piece. More generay, the chord data can be gathered into higher dimensiona time-deay embeddings, and again new and unexpained patterns emerge in the barcodes. Section 4 anayzes some simpe rhythmic patterns (such as those of Figure 3) using the techniques of persistent homoogy. The appropriate metric is anaogous to those above, though it must be modified to refect the fact that tempora data is not perceived in a ogarithmic fashion. Once again, the circuar structures are immediatey evident from the barcodes. 2 Meodic Barcodes and the Circe of Notes The pitches of musica tones are generay perceived as a function of frequency in a ogarithmic fashion. Thus the 15.6 Hz distance from C to C is perceived to be the same size as the 24.7 Hz distance from G to A (refer to Figure 1 for the origin of these numerica vaues). Accordingy, it is reasonabe to consider a measure that operates on og frequency rather than on frequency itsef. A metric ike og 2 (f) og 2 (g) captures this aong with the notion that nearby tones (with fundamenta frequencies f and g) on the circe of notes shoud have a sma numerica distance. But this measure fais to capture the idea that the C at Hz and the high C at Hz are effectivey the same. This is what happens when a man sings aong with a woman (or when a woman sings aong with a chid): the same note is actuay a factor of two apart in 4

5 frequency. This second notion of coseness can be incorporated by using s = mod ( og 2 (f) og 2 (g), 1) (1) but this is not a metric since it fais the triange inequaity. It can, however, be made a metric by defining the distance between two notes f and g, expressed in terms of their fundamenta frequency in Hz, as d(f, g) = min(s, 1 s) (2) where s is from (1). This measure may be interpreted as a measure of distance between pitch casses 9], since it identifies a Cs, a C s, etc. into equivaence casses. We ca (1)-(2) the pitch-cass metric. To verify that this metric makes sense, consider a major scae consisting of the eight notes C, D, E, F, G, A, B, C with the frequencies as specified in Figure 1. The Pex software 8] is designed to cacuate the persistent homoogy of finite simpicia compexes... generated from point coud data. In this case, the point coud is defined by a matrix of distances between a pairs of the eight notes using the pitch-cass metric. The resuting barcodes are shown in Figure 4. Figure 4: Barcodes cacuated by the Pex software show the number of connected components in the Dimension 0 pot (top) and the number of circes in the Dimension 1 pot (bottom), as the size parameter ɛ varies. These two pots are straightforward to interpret. When the size parameter ɛ is sma, there are seven distinct notes. Though we input eight notes, the high C has exacty the same distances to a the other notes as the ow C under the pitch-cass metric (1)-(2), and thus the barcode merges these two tones even at ɛ = 0. When ɛ reaches 0.08, the two haf steps (the intervas between E-F and B-C) merge. When ɛ reaches 0.16, the five remaining connected components (a the major seconds) merge into one. Thus Betti 0 = 1 for a greater ɛ. At ɛ = 0.16, the Dimension 1 code shows a singe component, which persists unti ɛ = 0.4. This Betti 1 = 1 feature is exacty the circe of notes shown in Figure 1. The exampe of Figure 4 was buit specificay with the circe of notes in mind, so it is perhaps unsurprising that the circe appears. Wi such shapes appear in rea music? The website 7] contains a arge seection of traditiona meodies, with most tunes avaiabe in both sheet music and as standard MIDI fies. The musica score for 5

6 Figure 5: Barcodes for the traditiona fok tune Abbott s Bromey Horn Dance (see Figure 6) show many of the same features as the major scae barcodes of Figure 4. The distribution of whoe and haf steps are cear from the Betti 0 code for sma ɛ whie the circe of notes appears again in Betti 1 when 0.16 < ɛ < Abbott s Bromey Horn Dance is shown in Figure 6 and the corresponding barcodes are shown in Figure 5. The top barcode in Figure 5 shows eight ines, which correspond to the eight notes that appear in the score (observe again the insensitivity to octave). Four disappear at ɛ = 0.08, which correspond to the four haf steps (F -F, D -E, B-C, and D-D ). Three more disappear at ɛ = Aong with the constant bar, these correspond to the four whoe steps (E-F, G-A, A-B, and C-D). A of these join into one bar for a arger ɛ. The region 0.16 < ɛ < 0.33 is characterized by Betti 0 = 1 (one connected component) and Betti 1 = 1, one circe. This is again the circe of notes. In fact, a the meodies from the website 7] show this same structure, though the number of haf and whoe steps changes to refect the scae of the piece, and the exact extent of the Betti 0 =Betti 1 = 1 region is somewhat variabe. The anayses of Figures 4 and 5 may be somewhat naive because they suppresses tempora information in the meody. This can be addressed by using a time-deay embedding, which is a common procedure in time series anaysis. Suppose that a meody consists of a sequence of notes with fundamentas at f 1, f 2, f 3, f 4... These may be combined into pairs (a two-dimensiona time-deay embedding) by creating the sequence (f 0, f 1 ), (f 1, f 2 ), (f 2, f 3 ),... The distances between such pairs can be cacuated by adding the distances between the notes eement-wise using the pitch-cass metric. 1 Buiding a matrix of a such distances for Abbott s Bromey Horn Dance and cacuating the barcodes gives Figure 7. The Dimension 0 barcodes (the top pot in Figure 7) can be interpreted as showing the distances between pairs of notes as the meody progresses over time. Thus there are 11 pairs of notes that are at a distance of one-haf step, since 11 ines end at ɛ = There are 19 pairs that differ by a whoe step since 19 ines end at ɛ = Above this vaue, a pairs have merged into a singe connected component. This can be interpreted as saying that the meody progresses primariy by stepwise motion, and that no pairs of tones are isoated from any other pairs of tones (though of course there are many 1 Distances may aternativey be cacuated using the chord-cass metric d cc of (3). The differences in the resuting barcodes appear to be subte, at east for the traditiona meodies of 7]. 6

7 Figure 6: The traditiona meody Abbott s Bromey Horn Dance is taken from Chris Peterson s coection 7]. The standard MIDI version of this meody is anayzed using the ideas of persistent homoogy in Figures 5, 7, 8, 10, and 11, and a barcode anaysis of the rhythm is shown in Figure 16. 7

8 Figure 7: Barcodes for the two-dimensiona time-deay embedding of Abbott s Bromey Horn Dance (see Figure 6) are consideraby more interesting than those in Figure 5. 8

9 individua pairs with arger distances). The Dimension 1 barcodes (the second pot in Figure 7) shows the number of circes present at each vaue of ɛ, the Dimension 2 barcodes (the third pot) show the number of hoow spheres as a function of ɛ, and the Dimension 3 barcodes (the bottom pot) show the distribution of 4D hoes in 4-space (do these have a name?). It woud be interesting to try and interpret these higher dimensiona structures in terms of the underying piece of music. Other musica pieces (from the same ibrary) have quaitativey simiar structures, though the detais appear to differ in intriguing ways. Longer tempora information can be incorporated by using onger time-deay embeddings. If a meody consists of a sequence of notes with fundamentas at f 1, f 2, f 3, f 4..., these can be combined into tripets (a three-dimensiona time-deay embedding) by creating the sequence (f 0, f 1, f 2 ), (f 1, f 2, f 3 ), (f 2, f 3, f 4 ),... The distances between such tripets can be cacuated by adding the distances between the notes eementwise using the pitch-cass metric. Buiding a matrix of a such distances for Abbott s Bromey Horn Dance and cacuating the barcodes gives Figure 8. Again, it is straightforward to interpret the Dimension 0 barcodes as distances between tripets of notes in the meody. The higher dimensiona structures are again somewhat enigmatic, though presumaby they indicate something about the pieces being anayzed. 3 Harmonic Barcodes and the Circe of Fifths In order to ook for the second major topoogica feature that shoud exist in musica data (the circe of fifths of Figure 2) it is necessary to generaize the metric to consider scaar harmony, mutipe pitches considered simutaneousy. Perhaps the most straightforward generaization of the pitch-cass metric is to add the (pitch-cass) distances between a eements of the vectors, as was done for the time-deay embeddings of the previous section. This metric distinguishes chord inversions: for instance, a C major chord in root position (C-E-G) woud be distant from a C major chord in third position (G-C-E). Whie this is desirabe in some musica situations, it is undesirabe when ooking for structures that invove musica key, where (say) a C major chords are identified irrespective of inversion and a C major scaes are identified irrespective of the order in which the pitches are isted. For exampe, the ascending C major scae and the descending C major scae are both the same entity, and the metric shoud refect this ream of musica perception. In terms of the eves of musica data (items (1)-(5) on page 2), pitch-casses are appropriate for the symboic note eve data (item 3) whie the circe of fifths ies at the anaytica eve (item 4). Accordingy, et f = (f 1, f 2,...f n ) and g = (g 1, g 2,...g n ) be two n-tupes, and define the distance d cc (f, g) = min d(f, P g) (3) P where P ranges over a possibe permutation matrices and where d(, ) is the pitchcass metric of (1)-(2) appied in an eement-by-eement fashion. This chord-cass metric cacuates the (eementwise) pitch-cass distance between f and a the permutations 9

10 Figure 8: Barcodes for the three-dimensiona time-deay embedding of Abbott s Bromey Horn Dance (see Figure 6) show a remarkabe array of features that woud be great to understand. Whie the dimension 0 pot is straightforward (showing the number of meodic tripets at each distance), the Dimension 1 and 2 pots contain a fascinating coection of circes and higher dimensiona anaogs that persist over a nontrivia range of ɛ. 10

11 of g and hence is invariant with respect to chord and scae inversion; a reorderings of the eements of f and g are paced in the same equivaence cass. To verify that this metric makes sense, consider a progression that moves around the circe of fifths: C major to G major to D major etc, a the way back to F and finay C. Inputting these seven-note sets into Pex and cacuating the barcodes gives Figure 9. Under this metric, scaes that are a fifth apart (such as C major and G major) have a distance of 0.08 and this expains the tweve ines that merge down to a singe connected set at ɛ = 0.08 in the Dimension 0 (top) pot. For 0.08 < ɛ < 0.33, the Dimension 1 barcode shows a singe persistent bar; this is the circe of fifths! There are aso some higher dimensiona features for arger ɛ, but the exact meaning of these is not cear. Figure 9: Barcodes for a chord progression consisting of one cyce around the circe of fifths. The circe of fifths is the persistent ine from 0.08 < ɛ < 0.33 in the Dimension 1 pot. The same metric can, of course, be appied to chord progressions. The chords from the score of Abbott s Bromey Horn Dance in Figure 6 were entered manuay, and the barcodes cacuated in Figure 10. Since there are ony four chords (Em, B, Am, D), there is not much structure. The Dimension 0 barcode shows the distances between the four chords, and the dimension 1 barcode ony shows a circe for 0.33 < ɛ < 0.4. It is aso easy to add in tempora information using a time-deay embedding, and this is done for the same piece at the chorda eve in Figure 11. Here the tripets of chords have some structure, with one circe when 0.25 < ɛ < 0.32 and three circes when 0.32 < ɛ < 0.4. It woud be great to be abe to interpret this kind of thing! The fina exampes examine Bach s Chorae No. 19, with musica score shown in Figure 12. A MIDI fie of this piece, from 4], is parsed to extract the four voices. The dis- 11

12 Figure 10: Barcodes for the chord progression from the score of Abbott s Bromey Horn Dance in Figure 6. Figure 11: Barcodes for the dimension-three time-deay embedding of the chord progression from the score of Abbott s Bromey Horn Dance. 12

13 j Bach Chorae # 19 soprano ato tenor bass q = 76 & b 4 & b 4? b 4 L? b 4 Ó. œ œ œ b œ œ œ œ # œ œ k. œ # œ œ œ œ Ó. œ œ œ #_ œ œ œ œ b œ œ n œ œ œ _ œ œ œ _ œ Ó. œ œ œ œ œ _ œ _ œ œ œ œ œ œ œ œ œ Ó. œ œ œ b œ œ œ œ œ œ # œ œ œ b œ œ J œ œ j œ n œ œ b œ sop. ato ten. bass & b & b? b L? b # œ œ œ j œ k. œ œ œ œ œ œ œ œ J œ K. œ _ œ b œ œ œ œ œ œ œ œ b œ œ _ œ œ œ œ _ œ _ œ œ œ œ _ œ _ œ _ œ _ œ œ œ œ _ œ œ œ œ œ œ œ œ œ œ J b œ œ J b œ n œ œ # œ œ œ œ _ œ œ sop. ato ten. bass & b & b? b L? b œ œ # œ j œ k. œ # œ œ œ œ # œ œ œ Œ œ # œ œ _ œ œ _ œ œ b œ œ œ n œ œ œ œ Œ _ œ. _ œ J œ œ œ œ œ œ œ œ œ œ œ œ _ œ _ œ n œ Œ œ b œ œ œ œ œ œ œ n œ œ # œ œ _ œ œ Œ Figure 12: The standard MIDI version of Bach s Chorae No. 19 is anayzed using the ideas of persistent homoogy in Figures 13 and

14 tances between a four-part chords are cacuated according to the chord-cass metric (3), and the resuts are input to the Pex software in order to draw the barcodes. This is shown in Figure 13. The dimension 0 barcode shows a arge number of chords that are separated by ɛ = 0.08, a somewhat smaer number of chords that are separated by a distance of ɛ = 0.16, and two chords separated by ɛ = Above this vaue, a chords merge into one connected component. The dimension 1 barcode in Figure 13 shows Betti 0 = 3 connected components and one circe Betti 1 = 1 for 0.16 < ɛ = 0.23, and this structure then changes to Betti 0 = 1 and Betti 1 = 3 for 0.24 < ɛ = Features such as these appear to be unique identifiers of the particuar pieces, meaning that other Bach Choraes from the same data set have different Betti numbers that occur over different ranges of ɛ. Finding the origin of such variations is an interesting chaenge. Figure 13: Barcodes for Bach s Chorae No. 19. Finay, Figure 14 shows the two-dimensiona time-deay embedding of the Bach Chorae, where the zero dimensiona pot is interpretabe directy in terms of the distribution of chord pairs and how they custer under the chord-cass metric. Again, there is a coection of persistent Betti 1 circes. 4 Rhythmic Barcodes and the Neckace Notation Rhythmic notations represent time via a spatia metaphor. In standard musica notation, time is drawn ineary, though there are aternative notations such the neckace notation of Figure 3 that dispay the circuar nature of rhythmic patterns: each pass through the cyce is one repetition of the rhythmic motif. The pitch-cass metric (1)-(2) is not immediatey appicabe to the task of measuring such cyces because pitch is perceived 14

15 Figure 14: Barcodes for the two-dimensiona time-deay embedding of Bach s Chorae No

16 in a ogarithmic fashion whie time is not. Accordingy, the metric can be modified to measure the distance between two times f and g as d(f, g) = min(s, 1 s) where s = mod ( f g, 1), (4) and where one unit of time represents one period of the rhythm. Let s ca this the neckace metric. The Ewe rhythm of Figure 3 is transated into the vector of time points {0, 1 6, 2 6, 5 12, 7 12, 3 4, 11 }. (5) 12 The distance is cacuated between a of these time points under the neckace metric (4), and this set of distances is input into the Pex software. The resuting barcodes are shown in Figure 15. Figure 15: Barcodes for the standard rhythm 5] of Figure 3 show the distribution of time intervas in the rhythm in the top (dimension 0) pot and show the circuar structure with Betti 0 = 1 and Betti 1 = 1 in the bottom (dimension 1) pot. The dimension 0 barcode shows the custering of the points in time. In the sequence (5), the minimum distance is 1 12, and this occurs in two paces, between the third and fourth notes, and again between the 11th and the first notes. Accordingy, the barcode shows two ines that vanish when ɛ reaches Since the argest distance between any two adjacent time points is 0.16, a the points merge into one custer at ɛ = The dimension 1 barcode dispays a persistent Betti 1 bar from 0.16 < ɛ < This is the anticipated cyce around the neckace. As might be expected, more compex rhythmic patterns and higher dimensiona embeddings yied more compex barcodes. For instance, the rhythm of the first 4 measures (a 24 beat cyce) of Abbott s Bromey Horn Dance are shown in Figure 16. As usua, the distribution of short and ong intervas is shown in the dimension 0 barcode whie the circuar structure of the rhythm appears in the dimension 1 barcode for 0.08 < ɛ < There are aso interesting features to this rhythm in the two and three dimensiona barcodes. 16

17 Figure 16: Barcodes for the first four measure of the traditiona fok tune Abbott s Bromey Horn Dance of Figure 6 show the distribution of time intervas in the rhythm in the top (dimension 0) pot and show the circuar structure with Betti 0 = 1 and Betti 1 = 1 in the second (dimension 1) pot. Higher dimensiona features are aso readiy apparent. 17

18 5 Discussion This paper argues that an investigation of the topoogica structures inherent in musica data is feasibe using the ideas of persistent homoogy. Besides demonstrating that we known topoogica features can be derived from musica data sets, such anayses may be usefu in information retrieva, in anaysis of musica pieces, and in appications such as audio segmentation, meody recognition, and musica cassification. Musica data may be ideay suited as a vehice for exporation of the techniques of persistent homoogy because it is obvious (at east in retrospect) that there are significant topoogica structures present. Three exampes are given: the circe of notes shown in Figure 1, the circe of fifths shown in Figure 2, and the circuar form of the rhythmic neckace notation shown in Figure 3. These are readiy identifiabe in the barcodes derived from musica data dispayed in Figures 4, 9, and 15. There are a number of issues raised that may ead in fruitfu directions. Generaizing the homoogica anaysis to other musica domains such as spectra and audio rate data woud provide an important and nontrivia extension. For instance, harmonic musica instruments (such as those that make sounds using strings or air coumns) have waveforms that are approximatey periodic. Since periodic waves can be pictured as a function on the circe, they shoud exhibit nontrivia topoogica structure. A basic question is what metrics can be appied in these aternative domains since different metrics may be abe to provide different kinds of information. It is ikey that the requirement that the distance be given between a pairs of points in the data set is overrestrictive. If this can be reaxed, it woud aow the use of partia orderings, and might be appropriate for submajorization 15]. This coud be especiay usefu for perceptua data that does not conform to a metric or where the information may be incompete. Even with barcodes at the symboic (.mid) eve, questions remain. It is not yet cear what kind of geometric shapes correspond to the higher eve Betti numbers in the more compex barcodes (such as in Figures 7 and 8). From the musica perspective, it is important to ask how such topoogica structures can be interpreted in terms of the underying musica piece. A good approach might be to evauate a arger corpus of meodies, harmonies, and or rhythms with the goa of fuy deciphering such reationships. Even if higher dimensiona barcodes cannot be interpreted easiy, they might be usefu in cassification as a kind of signature or feature vector for subsequent processing. Simiary, they might be usefu in automatic segmentation to determine when something has changed (for instance in the underying scae or the underying meodic pattern). There are other ways of incorporating tempora information than using the timedeay embedding, and these might aso hep to provide a fuer topoogica anaysis that incudes both pitch and rhythmic anayses simutaneousy. Simiary, rhythmic patterns often occur in hierarchies and it woud be interesting to pursue the idea of ocating persistent homoogica structures from hierarchica musica data. 18

19 References 1] Safî a-din a-urmawî, Kitâb a-adwâr 1252, trans. R. Eranger in La Musique arabe, Pau Geuthner, Paris, ] W. Anku, Circes and time: a theory of structura organization of rhythm in African music, Music Theory Onine, Vo. 6, No. 1, Jan ] G. Carsson, Topoogy and Data, Buetin of the American Mathematica Society, Voume 46, Number 2, Apri 2009, Pages ] Cassica MIDI Archives 5] A. King, Empoyments of the standard pattern in Yoruba music. African Music, 2 (3) ] A.B. Lee, K.S. Pedersen, and D. Mumford, The noninear statistics of highcontrast patches in natura images, Internationa Journa of Computer Vision (54), No. 1-3, August 2003, pp ] C. Peterson, Traditiona Music, 8] 9] John Rahn, Basic Atona Theory, Prentice Ha Internationa, ] numera anaysis 11] H. Schenker; ed. O. Jonas, Harmony, trans. E. Mann-Borgese. Chicago: University of Chicago Press, ] W.A. Sethares, Tuning, Timbre, Spectrum, Scae, Springer-Verag, London, 2nd edn, ] W.A. Sethares, Rhythm and Transforms, Springer-Verag, London, ] G. Singh, F. Memoi, T. Ishkhanov, G. Carsson, G. Sapiro and D. Ringach, Topoogica structure of popuation activity in primary visua cortex, Journa of Vision, Vo. 8, No. 8, (11), pp. 1-18, ] D. Tymoczko, Geometry of Music, ] of fifths 19