Music that not focused around melodic and harmonic structures

Analysis of Emergent Beat-Class Sets in Steve Reich's "Clapping Music" and the Yoruba Bell Timeline Author(s): Justin Colannino, Francisco Gómez and Godfried T. Toussaint Source: Perspectives of New Music, Vol. 47, No. 1 (WINTER 2009), pp. 111-134 Published by: Perspectives of New Music Stable URL: https://www.jstor.org/stable/25652402 Accessed: 11-05-2020 01:50 UTC

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Analysis of Emergent Beat-Class Sets in Steve Reich's

Clapping Music and the Yoruba Bell Timeline

Justin Colannino, Francisco Gomez, and godfried t. toussaint

1. Introduction

The history of music is often the history of humanity's reactions to it. A good example of this may be observed in Minimalism. Since the Second World War, mainstream classical music had been dominated by composers such as Boulez, Berio, Cage, Ligeti, and Stockhausen, among others. These composers represent postwar Modernism, either through postserialism, Boulez being its most prominent figure, or through

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I I 2 Perspectives of New Music

indeterminacy, where Cage is its most notable figure. Although the term Minimalism was originally used for the visual arts, it was later applied to a style of music characterized by an intentionally simplified rhythmic, melodic and harmonic vocabulary. Indeed, Timothy Johnson has argued that the term minimalism may be defined most fruitfully as a technique rather than an aesthetic or style. The music of LaMonte Young, Philip Glass, Terry Riley, and Steve Reich represent the major reaction to the Modernism epitomized by the aforementioned composers. Indeed, whereas Modernism is decisively atonal, Minimalism is clearly modal or tonal; whereas Modernism is aperiodic and fragmented, Minimalism is characterized by great rhythmic regularity; and whereas Modernism is structurally and texturally complex, Minimalism is simply transparent. Minimalism has different materializations depending upon the

particular composer, but minimalist works share concern for non functional tonality and reiteration of musical phrases, often small motifs or cells, which evolve gradually. For example, while Young uses sustained drones for long periods of time, Glass chooses recurrent chord arpeggios, and Riley and Reich incorporate repeated melodies and quick pulsating harmonies. No less significant is the fact that minimalist music possesses almost none of the main features of Western music (at least since the time of the Romantic period); that is, harmonic movement, key modulation, thematic development, complex textures, or musical forms with well-designed structures. On the contrary, this music deliberately skirts around any sense or awareness of climax or development, and seems to ignore the dialectic of tension and release, at least as it is usually posited in the classical music tradition. In the words of Roger Sutherland, "The listener is invited, not to follow a complex musical 'argument,' but to concentrate upon a slowly changing sound, and focus with microscopic awareness on different aspects of it." 1

It is probably Reich who most unhesitatingly repudiates the Western classical tradition. Reich objects to both European serialism and American indeterminacy because in these traditions the processes by which the music is constructed cannot be heard and discerned clearly by the listener. Before him, Pousseur and Xenakis had already pointed out that "where the most abstract constructions have been employed . . . one has the impression of finding oneself in the presence of the consequences of an aleatory free play."2

In his essay "Music as a Gradual Process," Reich states his principles as follows: "I am interested in perceptible processes. I want to be able to hear the process happening throughout the sounding music."3 For such processes to be accessible to the listener, they must flow in an extremely gradual manner. The process itself must be related to the idea of shifting

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Analysis of Emergent Beat-Class Sets I I 3

phases. First, a melody is played by two or more players, and after a while one of them gradually shifts phase. At the beginning of the phasing a kind or rippling broken chord is produced; later, as the process moves forward, the second melody is at a distance of an eighth note, and a new interlocking melody arises. The process continues until the two melodies are in phase again.

These ideas are fulfilled in many of Reich's works composed between 1965 and 1973. The experimentation started with It's Gonna Rain and Come Out (both composed in 1966), where he applied the phasing process to tape recordings; it continued with Piano Phase (1967), and Violin Phase (1967), where he experimented within an instrumental context (no electrical devices); and finally, Reich reached the highest development in Drumming (1970-71), Clapping Music (1972), and Music for Mallet Instruments, Voices and Organ (1973), where he incor porated gradual changes of timbre and rhythmic augmentation, among other musical resources. By the end of 1972, he abandoned the gradual phase shifting processes, because "it was time for something new."4

This paper is concerned with a mathematical comparative analysis of Clapping Music and the bell rhythmic timelines of West African Yoruba music. During the summer of 1970 Reich traveled to Ghana, where he studied African drumming. He learned Gahu, Agdabza, and other

musical styles, which influenced his music. Such an influence is most readily perceived in works like Drumming and Clapping Music, where the phasing is discrete, but still noticeable in his continuous phasing pieces such as Phase Patterns, Violin Phase, and New York Counterpoint, where he uses rhythmic ostinatos that are subsets of West African bell timelines.5 More specifically, we study Clapping Music with regard to syncopation, inasmuch as it forms an essential part of that piece. Could not Clapping Music be considered as a piece where the first performer keeps a fixed metrical context that the second performer contradicts with a rotation of the same pattern? In other words, we almost have here the definition of syncopation. The analysis of Reich's music from the point of view of the change in rhythmic properties as the piece progresses in time was pioneered by Epstein, and continued by Richard Cohn, who analyzed Phase Patterns and Violin Phase focusing on the cardinalities of the emergent beat-class sets (the number of attacks or onsets per metric cycle).6 Here we analyze Clapping Music and a Yoruba bell timeline focusing on the amount of syncopation of the emergent beat-class sets.

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I 14 Perspectives of New Music

2. Clapping Music

Clapping Music is a phase piece for two performers clapping the same pattern throughout the duration of the piece. The phasing is discrete, with one performer advancing an eighth note after several repetitions of the pattern, while the other imperturbably remains playing the pattern without shifting. See Example 1 for further details. In the following, variations produced by shifting are numbered in ascending order as {Vo, Vi, . . . Vn, V12}, where Vo = V12 indicates that the two performers

play the pattern in unison. This piece, in spite of its apparent simplicity, does not lack musical

interest. First of all, Clapping Music constitutes a synthesis and a refinement of Reich's ideas by means of a piece with few but well combined elements. Secondly, Clapping Music enjoys a profound metrical ambiguity (a common characteristic in Reich's music) as well as a great deal of interlocking rhythmic patterns. The analysis of these interlocking rhythmic patterns as the piece progresses (beat-class

modulation) elucidates the musical structure of the piece.7

3. Measuring Features of Clapping Music

When one listens to Clapping Music, a question that arises naturally is how Reich came to select that particular pattern. As the pattern shifts, a series of interlocking rhythms emerges, creating great variety. Further more, there is a sense of balance in the whole piece, between the resulting variations, as they create and release rhythmic tension. Once

V? VI V2

"**" [?9r r r' r r * s' r r ' |r r r' r r * b ' r r' |r r r' r r * s' r r'

Vlt vu vu

P1 |?r r r * r r * t1 r r' |r r r * r r * t' r r * |r r r * r r * t' r r *

example 1: the first and last few bars of clapping music

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Analysis of Emergent Beat-Class Sets I 15

the pattern is defined, however, the rulebook does not allow us to change it. The beat-class modulation throughout the piece is carved in stone. Therefore, the pattern must be carefully chosen to begin with. Phylogenetic graphs have been already used to analyze musical rhythms. In UA Mathematical Analysis of African, Brazilian, and Cuban Clave Rhythms," Toussaint offered a phylogenetic analysis of binary timelines. Timelines are rhythmic patterns repeated throughout a piece whose main functions include rhythmic stabilization as well as the organization of phrasing. Subsequently, such an analysis was extended to ternary timelines taken from the Sub-Saharan African tradition, and to the hand-clapping metric rhythms of flamenco music.8 In all cases, worthwhile conclusions were drawn from these phylogenetic analyses. In this paper we use phylogenetic graphs to both analyze the structure of Clapping Music itself and to compare it to the Yoruba rhythmic timeline.

One mystery in Clapping Music is how Reich was able to find a pattern that works so well within the constraints of process music. There is no doubt that his inspiration for the pattern came from his study of

African drumming in Ghana.9 In particular, we note an extraordinary resemblance between the pattern of Clapping Music and a bell pattern used by the Yoruba people of West Africa: only one additional, seemingly inconsequential, note has been added by Reich. The two patterns are shown in Example 2. We should mention that Reich composed Clapping Music in 1972,

two years after travelling to Ghana to study African drumming, but became interested in African music through the writings of A. M. Jones. In 1954 Jones published a comprehensive seminal paper on the topic of hand-clapping patterns used in African music.10 In this paper Jones describes a piece of music called Beer Dance performed by the Lala people of what was then central Rhodesia. This piece has three clapping patterns performed in unison, shown in Example 3. The resultant clapping rhythm is in fact the same necklace pattern (rotation) that

Clapping Music [] ^ ppp?pp^p^pp7 Yoruba Clave []ffi|?7p7p^7p7|?^7

EXAMPLE 2: THE PATTERN OF CLAPPING MUSIC AND THE YORUBA CLAVE

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i i 6 Perspectives of New Music

Reich used in Clapping Music. It appears that Reich considered the pattern quite successful, motivating him to use it again in 1984 in his composition Desert Music, where it is played on marimbas (see Example 4).

The Yoruba people live on the west coast of Africa, mainly in Nigeria, although they can be found also in the eastern Republic of Benin and Togo. Because most of the slaves were taken from West Africa, a

Clap 1-j-j-j Clap 2 - jjj. j j.

cpsITTjOTX: Resultant

EXAMPLE 3: MUSIC FROM THE LALA PEOPLE

EXAMPLE 4: RHYTHMIC PATTERNS FROM DESERT MUSIC

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Analysis of Emergent Beat-Class Sets I 17

diaspora evolved and the descendants of the Yoruba people can also be found in Brazil, Cuba, the Caribbean, the United States and the United Kingdom. They are one of the largest cultural groups in Africa, and musically speaking are of great relevance. Yoruba music has exerted much influence on the music of the surrounding countries.

The clave considered here is widely employed as a timeline in the sacred music of the Yoruba people.11 Bettermann calls this rhythm the Omele. It is also found in Cuba, where it is used in several styles like the rumba Columbia.12

It might be argued that there may exist other clave patterns very similar to the Clapping Music pattern. See for example the collection of candidate timelines assembled and discussed by Jay Rahn.13 That this is not the case is made clear in the following section. Once we formally introduce the distance function used to measure dissimilarity between a pair of rhythms, we will verify that the Yoruba clave is the rhythm closest to the Clapping Music pattern among a great number of African ternary timeline patterns. These reasons constitute our primary

motivation for comparing these two patterns. In fact, an intriguing natural question is whether the Yoruba clave itself works just as well as the pattern employed in Clapping Music.

The musical effectiveness of Clapping Music is due in part to the way in which syncopation is dealt with, thus providing motivation to mea sure the amount of syncopation of the emergent beat-class sets as a function of time. The problem of defining a mathematical measure of syncopation with formal precision has not been addressed until re cently.14 In their 2005 article, Gomez, et al. reviewed several mathemat ical measures of syncopation, and proposed a new measure, the weighted note-to-beat distance measure (WNBD measure from here on). This measure will be used here to compute the syncopation of Clapping Music and the Yoruba clave pattern, and thus flesh out their underlying beat-class modulation structures.

4. Phylogenetic Analysis

Phylogenetic graphs were originally used in biology to determine the proximity and evolution of species. Biologists measure the degree of proximity between two species by comparing their genes. In our con text, rhythmic patterns take the place of genes, and as a consequence,

we define a new measure of proximity (similarity) between rhythmic patterns (measures used in biology are not appropriate in the context of music). The question of how to define similarity measures for rhythms

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I I 8 Perspectives of New Music

and other interval sets such as pitch-class sets has received a great deal of attention in the past.15 More recently, in the area of voice-leading, dis similarity or distance measures similar to the edit distances used here have entered the stage.16 A variety of dissimilarity measures have recently been compared in the context of phylogenetic tree analyses.17 Among the similarity measures compared (Euclidean interval vector distance, interval-difference vector distance, swap distance, etc.), the most satisfactory one in these studies has been the directed-swap distance, which is a generalization of the simpler swap-distance, first introduced by Jose-Miguel Diaz-Banez. This was the motivation for using this distance measure in this study.

Consider first the simpler swap-distance inspired by the most elementary mutation model used in bioinformatics. Let Pand Q be two rhythms with the same number of onsets. Positions of the rhythm that contain an onset will be referred to as occupied positions. The swap distance is the minimum number of swaps required to convert P to Q. In this case the solution is simple: the first onset of P must move the position occupied by the first onset of Q the second to the second, and so on. This is equivalent to computing the Li norm between the index vectors of the onsets' time coordinates of the two rhythms, and corresponds to the distance measure preferred in voice-leading.18 The directed-swap distance is a generalization of the swap distance designed to handle comparison of rhythms that do not have the same number of onsets. Let P have more onsets than Q. The directed-swap distance is the minimum number of swaps required to convert P to Q with the constraints that each onset in P must move to some occupied position of Q, and all occupied positions of Q must receive at least one onset from P. For example, the directed-swap distance between Player-1 and Player-2 in variation Vi is 4, since we have to perform four swaps in Player-1 at positions 3, 6, 8, and 11 to convert Player-1 to Player-2 (see Example 1). In our case, since all rhythms have the same number of onsets, computing the directed-swap distance is easy because it reduces to the simpler swap-distance and thus the computation of a sum of a linear number of terms.19

As mentioned before, one of the reasons for comparing the pattern of Clapping Music to the Yoruba timeline is that this timeline is the rhythm closest to it. The distance between them was measured with the directed-swap distance, and the set of timelines used in the comparison was taken mainly from well-established African musical traditions.20 The distance matrix corresponding to the directed-swap distance is shown in Example 5. Box notation is used for the variations of Clapping

Music. For each rhythm the bottom of each column of Example 5 indicates the sum of the swap distances to all other rhythms.

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Analysis of Emergent Beat-Class Sets I 19

Surprisingly, the sums take on only two values, 48 and 74, where Vo, V3, V6 and V9 are the variations that obtain the highest score. Also noteworthy is the diagonal below the zeros in Example 5, that is, (4, 4, 4, 8, 4, 4, 8, 4, 8, 4, 4). This diagonal gives the directed-swap distances between consecutive variations. It takes on only two values, 4 and 8, but these values differ considerably, and change six times over a total of twelve variations.

Whereas traditionally, phylogenetic graphs have been used in the analysis of sequence data such as molecular sequences in biology in order to construct a phylogeny of a group of species, in this study they are used for analyzing musical rhythms, with the goals of visualization, cluster analysis, classification, and the reconstruction of "ancestral" rhythms to determine how useful they are as a new tool for musico logists. Example 6 shows the phylogenetic graph computed for the distance matrix. This graph was constructed with the program SplitsTree, which draws a graph in the plane such that the distances travelled along the edges of the graph between any pair of rhythm nodes (variations

marked with black dots) approximates as closely as possible the distances between the corresponding rhythms in the distance matrix.21 The program also computes a fitness value, listed as a percentage value, which shows how well the distances in the graph match those in the matrix. In our case we obtain a fitness value of 100%, meaning that the distances in the graph are all exactly the same as in the distance matrix.

Variations I V0 I V1 I V2 I V3 I V4 I V5 I V6 I V7 I V8 I Vg I V10 I Vn Vb=xxx.xx.x.xx. 0 Vl=xx.xx.x.xx.x 4 0 V2=x.xx.x.xx.xx 8 4 0 V3=.xx.x.xx.xxx 12 8 4 0 l/4=xx.x.xx.xxx. 4 2 4 8 0 l75=x.x.xx.xxx.x 8 4 2 4 4 0 V6=.x.xx.xxx.xx ~12 8 4 2 8 4 0 Vr7=x.xx.xxx.xx. 44482480 F8=.xx.xxx.xx.x 8 4 4 4 4 2 4 4 0 V9=xx.xxx.xx.x. ~2 4 8 12 4 8 12 4 8 0 Vio=x.xxx.xx.x.x ~T~ 2 4 8 ~~4 4 8 2 4 4 0 Vn=.xxx.xx.x.xx ~8~ 4 2 ~4~ 4 ~4 4 4 2 8 4 0

Y, I 74 j 48 j 48 j 74 1 48 j 48 j 74 1 48 j 48 j 74 j 48 j 48

EXAMPLE 5: THE DIRECTED SWAP DISTANCE MATRIX OF THE CLAPPING MUSIC PATTERN

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I 20 Perspectives of New Music

Fit= 100% I_I >v_^^rr^^^^^^ V3 ? C2 ^^^^ EXAMPLE 6: THE PHYLOGENETIC GRAPH OF THE CLAPPING MUSIC PATTERN

In addition, the graph also contains nodes (without black dots) that imply "ancestral" rhythms. The ancestral node of the entire collection of variations is the center of the graph, in this case the node labelled A. This phylogenetic graph provides valuable information about the

beat-class modulation structure of Clapping Music. For the moment, we consider the central node A. Such a node corresponds to an "ancestral" rhythm, and is also the center of the graph (i.e., it is the vertex that minimizes the maximum distance to any other vertex in the graph). Note also that as the variations Vo, Vi, . . . Vu progress throughout the piece, they exhibit considerable switching from one side to the other with respect to node A. Therefore, it seems that this central node plays a key role in the piece. There is as yet no known efficient algorithm to compute the "ancestral" nodes for rhythms in phylogenetic graphs constructed with our distance measure. However, in this case, given the small number of rather short rhythms involved, the "ancestral" rhythm can be reconstructed by hand without much difficulty. It turns out to be the rhythm shown in Example 7. This fundamental rhythmic pattern is none other than a group of

trochees. A trochee is a rhythmic grouping consisting of a long note followed by a short note. This "ancestral" rhythm has a strong metric time-keeping character. The trochee, expressed in box notation as [x.x],

gj j>j j>j j>j t>\

EXAMPLE 7: THE "ANCESTRAL" RHYTHM OF CLAPPING MUSIC

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Analysis of Emergent Beat-Class Sets 121

is a common Afro-Cuban drum pattern, also found in disparate areas of the globe. For some examples in Latin American music, it is found in the Chilean cueca, and the Cuban coros de clave.22 It is common in Arab music, as for example in the Al Tder rhythm of Nubia.23 It is also a rhythmic pattern of the Drum Dance of the Slavey Indians of Northern Canada.24 Furthermore, the entire pattern of eight onsets in a time span of twelve pulses shown in Example 7 is also the Euclidean rhythm E(S; 12), which distributes the onsets as evenly as possible.25 The phylogenetic graph has four distinguishable clusters CI, C2, C3

and C4, that can be easily seen in Example 6. When Clapping Music is performed these clusters appear in the order given in Example 8. From this sequence of clusters we may observe the evolution of the variations through time. There is a first section formed by variations Vo to V3; in it, each variation moves further away from Vo. In a second section,

which goes from V? to V6, variations are still kept away from Vo. In the third section, variations V7 and Vs remain around the center of the graph, and represent a turning-point after which the subsequent varia tions move towards Vo. The variations in section four, consisting of V9,

V10 and V11, tend towards Vo. Lastly, Clapping Music closes by coming back to the main pattern (Vo = V12) played in unison. This evolution may be detected, although less visually, on the diagonal below the zeros in the directed-swap distance. There is another interesting property deducible from the phylogenetic

graph. All rhythms in clusters CI and C4 are at a distance of 6 from the central node A; those in clusters C2 and C3 are at a distance of 2 from A. However, rhythms to the left of A are converted into A by "pushing" their onsets to the right, whereas rhythms to the right of A are converted into A by "pushing" their onsets to the left. For example, to turn Vo = xxx.xx.x.xx. G CI into A, six swaps are needed, but all of them are forward swaps. On the other hand, V3 = .xx.x.xx.xxx E C4 is converted into A by performing six backward swaps (see Example 9).

Clusters || CI | C2 | C3 | C4 11 Vo I Vi 1 V2 1 VT

_"____t__V5_Ve_ _'__J___Vs_ _"_V^_^o__Vii__

11 Vn I I I

EXAMPLE 8: CLUSTERING IN CLAPPING MUSIC

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I 22 Perspectives of New Music

X.XX.XX.XX.X

x x x, x x . x . x x. ^ Tx^TTxTx X . X X X Vo Vs

EXAMPLE 9: CONVERTING Vo AND F3 INTO THE CENTRAL PATTERN OF NODE A

Let us now compare the Clapping Music pattern to the Yoruba timeline. To start with, we present the swap distance matrix For the Yoruba bell pattern in Example 10, and its corresponding phylogenetic graph in Example 11. By looking at the bottom row of the matrix we see that the sums of the distances take many different values. The most similar rhythm is Vs and the most different one is V*. That is not surprising: Vs is the "standard pattern" used in Bembe, a very popular ternary rhythm. Toussaint already proved that it is one of the most similar of an important family of ternary timelines.26 Here we note that the standard pattern is the most similar rhythm in its own wheel (the wheel of a rhythm consists of those rhythms obtained by its rotations that begin on an onset). The diagonal below the zeros is (5, 7, 5, 7, 5, 5, 7, 5, 7, 5, 5). The difference between two consecutive variations is smaller than in the case of Clapping Music. Changes are more frequent in the case of the Yoruba clave.

The graph is a chain with a rather disappointing fit of 89%, with no ancestral nodes. Take into account that if the fit is not 100%, reasoning on the graph does not accurately reflect reasoning on the distance matrix, and accordingly, neither on the rhythms. For example, on the graph the distance from Vo to Vi is 2.5, but in the matrix it is actually 2. The role of A could be played by variation Vs in this phylogenetic graph. It is the center of the graph, and as before, variations alternately go from left to right and from right to left around Vs. Nevertheless, this does not seem to yield any particular insight into the structure of the Yoruba clave. Hence, there is no remarkable clustering analysis to be discussed. In addition, the graph does not exhibit special symmetries or regularities of musical significance. In conclusion, when the musical process of Clapping Music is carried out on the Yoruba timeline, a rather awkward result is obtained.

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Analysis of Emergent Beat-Class Sets 123

Variations I V0 I Vj I V2 1 V3 I V\ \ Vs \ Vg \ V7 \ Vs I V9 I Vjo I Vii Vq=x.x.xx.x.xx. 0 Vi=.x.xx.x.xx.x 5 0 V2=x.xx.x.xx.x. 2 7 0 V3=.xx.x.xx.x.x 3 2 5 0 Vi=xx.x.xx.x.x. 4 9 2 7 0 V5=x.x.xx.x.x.x 1 4 3 ~2~~ 5 ~0~ I ()=.x.xx.x.x.xx ~~6 1 8 3 10 5 0 V7=x.xx.x.x.xx. 1 6~ 1 4~ 3 2 7 ~0~ V8=.xx.x.x.xx.x 4 I 6 1 8 3 2 ~5 0 V9 =xx.x.x.xx.x. 3 8 1 ~6 1 1 9 2 7 0 Vjo=x.x.x.xx.x.x 2 3 4 1 6 1 4 3 2 5 0 Vji=.x.x.xx.x.xx j 7 2 9 4 11 6 1 ~8 4 9 5 CT

E I 38 I 48 j 48 j 38 I 66 j 36 1 56 1 42 j 43 j 55 | 41 | 66

example 10: the directed-swap distance matrix of the yoruba timeline

Fit=89% a v10 vi v6

- v8 v3 -

example 11: the phylogenetic graph of the yoruba timeline

5. Syncopation Analysis of Clapping Music

The characterizing feature of any definition of syncopation includes a momentary contradiction of the prevailing meter. Accordingly, the WNBD measure is based on the durations of notes, and how they cross over the strong beats of the meter. This measure differs from the others

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124 Perspectives of New Music

found in the literature, such as the measure proposed by Longuet Higgins and Lee based on the height of a tree generated by a metrical grammar, Keith's measure based on the combinatorial structure of metrical levels, Toussaint's measure of metrical-complexity based on the metrical structure of Lerdahl and Jackendoff, or Toussaint's off-beatness measure based on the underlying polyrhythmic group-theoretic struc ture of the meter.27 Gomez et al. proved the WNBD measure to have

wider applicability and validity than the others.28 The WNBD measure, which will be used here to analyze the synco

pation of Clapping Music, may be described as follows. We assume that a note ends where the subsequent note begins. Let a, a+i be two consecutive strong beats in the meter. Also, let x denote a note that starts after or on the strong beat a but before the strong beat a+\. We define J[x) = mm{d(x-, a), d(x; ei+i)), where d denotes the distance between notes in terms of durations. Here the distance between two adjacent strong beats is taken as the unit, and therefore, the distance d is always a fraction (see Example 12(a)).

The WNBD measure D(x) of a note x is then defined according to the following cases:

(1) D(x) = ^ , if note x &a ends before or at a+i. 2

(2) D(x) = ^ , if note x ends after a+\ but before or at a+i.

(3) D(x) = ^ , if note x ends after a+i.

(4) D(x) = 0, if a; = a.

See Example 12(b) for an illustration of this definition. Now, let n denote the number of notes of a rhythm. Then, the WNBD measure of a rhythm is the sum of all D(x), for all notes x in the rhythm, divided by n.

" 1 -4 .-. ,i *_ i+1 ei+2 1 (3) ?*???_Bl I I (a) (b)

EXAMPLE 12: DEFINITION OF THE WNBD MEASURE

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Analysis of Emergent Beat-Class Sets I 25

Example 13 plots the WNBD measure of the variations in Clapping Music with respect to a 12/8 meter. The measure produces only three different values, namely, 24/8; 21/8 and 12/8, but the graph is quite revealing about how syncopation works in Clapping Music. Variation Vo by itself has a high value of syncopation. Two identical ascending descending cycles, V\?Vi?Vz?Va and Vi?Vs?V6?Vz, follow after Vo. From Vi, we find a symmetric cycle with respect to the previous cycle, namely, Vz?Vs?V9?Vio. Finally, we discover an ascending path to Vu (which is a half of the previous cycle). If variation Vi were moved after F12, then the resulting graph would have a perfect mirror symmetry about Vi. Therefore, a strong symmetry in musical form is evident in Clapping Music.

On the other hand, a look at the graph of the WNBD measure for the Yoruba clave, depicted in Example 14, reveals several differences. As in Clapping Music, the measure only takes three values, namely, 21/7, 18/7 and 15/7. The range of the syncopation values is much smaller than in the case of Clapping Music, and consequently, so is its rhythmic variety. The smaller the range of the measure is, the less interesting the rhythms are from the syncopation standpoint. The graph of the Yoruba timeline exhibits the same quasi-symmetry about Vi.

6. Concluding Remarks

Although phylogenetic graphs have already been used for analyzing inter-relationships within families of rhythms, in this paper we have used them for studying the modulation of the dissimilarity of the emergent beat-class sets of process music, in particular Reich's Clapping Music. The resulting phylogenetic graph allows us to explore a variety of musical properties of the piece, such as the classification, evolution, and

example 13: the graph of the syncopation measure of CLAPPING MUSIC

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I 26 Perspectives of New Music

VO VI V2 V3 V4 V5 V6 V7 V8 V9 VIO Vll V12

EXAMPLE 14: THE GRAPH OF THE SYNCOPATION MEASURE OF THE YORUBA CLAVE

transformation of variations, or the structure of the musical form. Our positive results lead us to advocate the use of phylogenetic graphs as a useful tool for musical analysis in general, especially for musical styles as characteristic as Minimalism. Reich's other pieces in this style, such as Music for Pieces of Wood, would likely benefit from this kind of analysis. We have compared Clapping Music to the Yoruba bell timeline. One

would expect that changing only one note could not make such a significant difference in musical terms. Quite to the contrary, as we have seen, the phylogenetic graph of Clapping Music has a richer structure than that of the Yoruba clave. This is a consequence of its inherent musical structure.

The WNBD measure of syncopation produces interesting conclusions as well, mainly that certain features of beat-class modulation such as rhythmic variety, at least from a syncopation standpoint, can be mea sured in terms of the range and distribution of the syncopation values. Previously, Haack proved that Clapping Music is unique from a

combinatorial point of view. In this paper we have added arguments of a geometrical nature to support this conclusion. The properties exhibited by the phylogenetic graphs and the WNBD

measure pose several open problems. For example, one may ask which rhythms yield nice graphs, that is, graphs with good properties (symmetry, clustering, 100% fit, etc.) The WNBD measure gives poor results when it is used to measure the interlocking melodies (for that we just use the Clapping Music pattern as the meter, and recompute the measure). Therefore, finding a function that measures the overall complexity of the union of two rhythms is an interesting open problem. Finally, it is not known which other rhythmic patterns would work just

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Analysis of Emergent Beat-Class Sets I 27

as well as the pattern used in Clapping Music, or whether one may characterize those rhythms that guarantee musically interesting results when used in phasing music.

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I 28 Perspectives of New Music

Notes

1. Sutherland (1994), 1.

2. Pousseur (1966), 93. For further discussion on these topics, see Mertens (1983;, Sutherland (1994), and Reich (2002).

3. Reich (2002), 34.

4. Reich (2002), 68.

5. See Cohn (1992), and Roeder (2003).

6. For a discussion of Cohn's formalist analysis of Reich's music, and the redeeming factors of formalist methods applied to process music in general, see Quinn (2006).

7. While Clapping Music has received litde musical analysis despite this rich structure, Cohn (1992), Haack (1991; 1998), and Toussaint (2002) have all offered analyses from mathematical perspectives.

8. Sec Toussaint (2003), and Diaz-Banez, et. al. (2004).

9. This interpretation is reflected in Michael Nyman's article in The Musical Times.

10. See Epstein (1986). 11. For discussion of the Yoruba timeline as used in sacred music see

Men Rodriguez (1998), Euba (1991), and Pressing (1983).

12. See Men Rodriguez (1998) and Ortiz (1998).

13. Jay Rahn (1987; 1996).

14. See Smith and Honing (2006).

15. The bibliography offered by Kramer (1985) lists several such sources. Additionally, the references included in the current work provide more recent sources, including the research of Buchler, Cohn, Morris, Quinn, and others.

16. See Callender, et. al. (2007), Straus (2003), and Tymoczko (2005).

17. See Diaz-Banez, et. al. (2004), and Toussaint (2002, 2003, 2004a).

18. See Callender, et. al. (2007), and Tymoczko (2005).

19. See Diaz-Banez (2004), and Toussaint (2004a).

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Analysis of Emergent Beat-Class Sets 129

20. To obtain precise details about these timelines consult Toussaint (2003) and the references therein.

21. For a discussion of the SplitsTree program, see Hudson (1998).

22. See Klower (1997), van der Lee (1995), and Rodriguez (1997), respectively.

23. See Hagoel (2003).

24. See Asch (1975).

25. See Toussaint (2005a).

26. Toussaint (2003).

27. See Johnson-Laird (1991), Longuet-Higgins (1984), Smith (2006); Keith (1991); Toussaint (2002); Lerdahl and Jackendoff (1983); Toussaint (2004b, 2005b), respectively.

28. See Gomez et al. (2005).

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I 30 Perspectives of New Music

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