D
o twentieth century, East London, acid jazz music and the ancient Aka Pygmy music of Central Africa have anything noteworthy in common? Yes, they do. There exist pieces of music in both domains that use rhythmic timelines that possess the rhyth-mic oddity property. But that is getting ahead of our story. First, we must backtrack half a
century to 1963, when a 33-year-old horn player with the symphony orchestra of an Israeli radio station received an invitation to work on a project spearheaded by the Israeli Ministry of Foreign Affairs: the setting up of a youth orchestra in the Central African Republic. The horn player’s name was Simha Arom, and although he was not overly enthusiastic about the project itself, he was excited by the possibility of discovering a world of music unknown to him. Besides, he was ready to break up the routine that had enveloped his life. When he first heard the music of the Aka Pygmies, he was instantly overwhelmed. He felt that their music not only had ancient roots, but that it also touched roots deep inside him.* The rest
is history. Arom went on to develop original methods of musicological research, and new tools with which to collect data. He made multiple recordings of African traditional music, and created a museum of the Arts and popular traditions. He studied the music of the Aka Pygmies for decades, becoming one of the foremost systematic ethnomusicologists in the world.
While studying the music of the Aka Pygmies of Central Africa, Arom noticed that their music contained rhythmic timelines that exhibited a property that he christened
rhythmic oddity. A rhythm with an even number of pulses in its cycle has this property if
no two of its onsets divide the rhythmic cycle into two half-cycles, that is, two segments of equal duration.† This property is not defined for rhythms with an odd number of pulses
since then it is impossible for two pulses to lie diametrically opposite each other on the rhythm circle (an odd number is not evenly divisible by two). It is quite easy in theory to construct examples of rhythms that have this property.‡ Figure 15.1 shows two such * Arom, S. (2009), p. 7.
† Chemillier, M. (2002), p. 176, Chemillier, M. and Truchet, C. (2003).
‡ It is more difficult to enumerate all rhythms that have the rhythmic oddity property. Chemillier, M. (2004), p. 615, shows
86 ◾ The Geometry of Musical Rhythm
examples: the rhythm on the left has onsets on the first four of its eight pulses, and the one on the right has onsets on the first six of its 12 pulses. It is obvious that in general, for any even number n of pulses, a rhythm with less than n/2 consecutively adjacent onsets has the rhythmic oddity property. However, rhythms constructed in this way are not particu- larly interesting musically, and are not used as timelines in world music except when the number of pulses in the cycle is a small number such as n = 4 or n = 6, in which case we may obtain, for example, the two-onset and three-onset rhythms with inter-onset intervals [1-3] and [1-1-4], respectively, which can be heard as ostinatos in several musical traditions. Its drawbacks notwithstanding, we shall identify this procedure as the Walk Algorithm, since we can think of starting a walk at pulse zero, taking k short steps of length one pulse each, where k is less than n/2.
In spite of their easy construction from the mathematical point of view, in practice, time- lines that possess the rhythmic oddity property are unusual in world music. For rhythms to be effective as timelines, they should in general not contain silent gaps longer than half of their cycle, and they should exhibit a certain degree of regularity or near-evenness. These two constraints are often enough to inadvertently prevent the rhythmic oddity property from being satisfied. Figure 15.2 illustrates four such examples of traditional rhythm time- lines that satisfy these two conditions but lack the rhythmic oddity property. In the rhythm on the left with inter-onset intervals [1-1-2], the first and last onsets violate the rhythmic oddity property. This simple pattern is used almost universally. For example, it is the baiaó
2 3 4 5 6 7 0 1 0 1 2 3 4 5 6 7 8 9 10 11
FIGURE 15.1 Two humdrum rhythm timelines that possess the rhythmic oddity property.
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 0 1 2 3
Rhythmic Oddity ◾ 87
rhythm of Brazil, as well as the polos rhythm of Bali. When it is started on the second onset, it turns into the catarete rhythm of the indigenous people of Brazil. Started on the third onset, it becomes an archetypal pattern of the Persian Gulf region,* the cumbia from
Colombia, and the calypso from Trinidad. It is also a thirteenth century Persian rhythm called khalif-e saghil, as well as the trochoid choreic rhythmic pattern of ancient Greece. Starting it on the silent pulse (anacrusis) yields a popular flamenco hand-clapping pattern (also compás) used in the flamenco styles called the taranto, the tiento, the tango, and the
tanguillo. It is also the rumba clapping pattern in flamenco, as well as another pattern used
in the baiaó rhythm of Brazil.† In the second rhythm with inter-onset intervals [3-3-3-3], the
rhythmic oddity property is violated twice, once with the first and third onsets, and again with the second and fourth onsets. This rhythm is the meter or compás of the fandango music of Spain. It is often accompanied by hand-clapping every pulse, but with loud claps at pulses zero, three, six, and nine. The third rhythm with inter-onset intervals [1-1-1-1-2] con- tains two violations of the rhythmic oddity property at pulses zero and three as well as one and four. It is the york-samai pattern, a popular Arabic rhythm, as well as a hand-clapping rhythm used in the al-medemi songs of Oman. Finally, the fourth rhythm with inter-onset intervals [2-1-1-1-1-1-1] contains three violations of the property at pulses zero and four, two and six, and three and seven. It is a typical rhythm played on the bendir (frame drum), and used for the accompaniment of women’s songs of the Tuareg people of Libya.‡
Let us now turn to rhythms that contain the rhythmic oddity property and that satisfy the above constraints. Two examples are the 24-pulse timeline rhythms used by the Aka Pygmies pictured in Figure 15.3. The rhythm on the left has nine onsets with inter-onset intervals [3-3-3-2-3-3-2-3-2] whereas the one on the right has eleven onsets with inter- onset intervals [3-2-2-2-2-3-2-2-2-2-2].
* Olsen, P. R. (1967), p. 31.
† The song Baião by Luiz Gonzaga uses the rhythms [x . . x x . . .] and [. . x . . . x]. See Murphy, J. P. (2006), p. 97. ‡ Standifer, J. A. (1988), p. 50. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
88 ◾ The Geometry of Musical Rhythm
The Aka Pygmies also use the five-onset, 12-pulse timeline shown in Figure 15.4 (left). It has inter-onset intervals [3-2-3-2-2] and possesses the desired properties. As an aside, it is interesting to note that if the five intervals are permuted to yield [2-2-3-3-2], we obtain the hand-clapping pattern and meter (compás) used in the seguiriya style of the flamenco music of southern Spain shown on the right.* With the two inter-onset intervals of length three
now adjacent to each other, the rhythmic oddity property is violated at pulses 4 and 10. Given that the music of the Aka Pygmies is characterized by having rhythmic timelines that possess the rhythmic oddity property, a natural ethnomusicological question arises: to what extent does this property manifest itself in other cultures such as, for example, West Africa, South Africa, or Cuba? To try to answer this question, consider an archetype timeline structure used extensively in these three geographical regions, that consists of seven onsets in a cycle of 12 pulses, with the constraint that all the inter-onset intervals must be of only two distinct durations: either one unit or two units. Three examples of such bell-pattern timelines are the bembé, tonada, and sorsonet pictured in Figure 15.5. Note that none of them possess the rhythmic oddity property. However, before we dismiss the usefulness of this property altogether, it is worth noting that the bembé contains one violation, the tonada† contains two, and the sorsonet three. This information may be used
to generalize the rhythmic oddity property as described in the following.
Arom defined the rhythmic oddity property in the form of a strict binary category, that is, a rhythm either has or does not have the rhythmic oddity property. This concept may be extended by means of a multivalued function that measures the amount of rhythmic oddity that a rhythm possesses. This function (rhythmic oddity) depends on the number of violations of the rhythmic oddity property present in a rhythm. Stated another way, a violation of the rhythmic oddity property yields a partition of the rhythmic cycle into two
* Fernández, L. (2004), p. 35.
† The tonada has one less onset than a popular traditional nineteenth century Cuban timeline called the clave campesina
given by [x . x x . x x . x x x . ] (see Mauleón, R. [1997], p. 10). With the additional onset in between the last two onsets of the tonada, the clave campesina has a third violation of rhythmic oddity at pulses three and nine.
0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11
FIGURE 15.4 A 12-pulse timeline used by the Aka Pygmies (left) and the seguiriya compás of the
Rhythmic Oddity ◾ 89
half-cycles by pairs of its antipodal onsets. Let us call such a partition of the cycle an equal
bipartition. Then, the fewer equal bipartitions a rhythm admits, the more rhythmic oddity
it possesses. The timeline in Figure 15.4 (left) used by the Aka Pygmies contains no equal bipartitions, whereas the seguiriya compás of the flamenco music (right) contains one.
Let us return to the three archetypal timeline necklaces from West Africa shown in Figure 15.6. Recall that if we disregard the rotations of a rhythm, so that all its rotations form an equivalence class, we call such an object a necklace. The relevance of necklaces here comes from the fact that the rhythmic oddity function is independent of the rotations of a rhythm; it is a property of the necklace. Figure 15.6 depicts three distinct necklaces, and each necklace determines seven different rhythms depending on which onset of the rhythm is taken as pulse zero (not counting the rotations that yield rhythms with anacrusis that start on a silent pulse). As it turns out, if the inter-onset intervals are restricted to the values one and two, these three necklaces are the only mathematical possibilities. The two short intervals of length one may be separated by two, one, or zero long intervals of length two, in the bembé, the tonada, or the sorsonet, necklaces, respectively. Our original ques- tion concerning the postulated preference of timelines then becomes: which of the three necklaces in Figure 15.6 are preferred in West African music? This question is not easy to
0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11
FIGURE 15.5 Three archetypal rhythmic timelines from West Africa.
0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11
90 ◾ The Geometry of Musical Rhythm
answer without first agreeing on the definition of “preference,” and spending time in the field performing listening experiments. In the absence of all these requirements, we may attempt to answer this question as an arm-chair musicologist by counting, for each neck- lace, how many of its rotations are used in musical practice. This definition of preference still needs elaboration since it is conceivable that one necklace appears frequently, but in only one of its rotations, whereas the other necklaces appear infrequently, but in all their rotations. However, if only one rotation of a necklace is used, then it is that rhythm that is preferred, and not the necklace that gives rise to the rhythm. What is intended by prefer- ence here is precisely the necklace. Which necklace is preferred by nature; which has the greatest fecundity?
It turns out that in West African music, the sorsonet necklace is one of the least pre- ferred of the three, yielding one timeline used in traditional music, the sorsonet rhythm of Figure 15.5 (right). However, the rotation with durational pattern [2-2-2-2-1-1-2] is the Persian rhythm kitāb al-Adwār,* and the rotation [2-1-1-2-2-2-2] is the rhythm of the Polish
polonaise.† Rotations of the tonada necklace are encountered more frequently, yielding two
West African rhythms, the tonada with intervals [2-1-2-1-2-2-2]‡ and the asaadua given by
[2-2-2-1-2-1-2], and one Persian rhythm, the al-ramal with intervals [2-2-2-2-1-2-1]. The bembé necklace is overwhelmingly preferred over the other two necklaces. Indeed, all seven rhythms obtained by starting the cycle at every one of its seven onsets are heavily used. It is evident, then, that among this family of rhythms there may have been an evolutionary pref- erence for those that admit as few as possible equal bipartitions, and thus a higher degree of rhythmic oddity. This is not to imply that there are no other mathematical properties that can produce the same preference ranking of these three necklaces. Perhaps, the most obvious one is the separation distance between the two short intervals in the cycle, which are separated either by a minimum of two, one, or zero long intervals. This implies that the same preference ranking may also be obtained by measuring how evenly the seven attacks are spaced out in the circle. Yet another method for obtaining the same preference ranking is by calculating the minimum number of elementary mutations required for each of the neck- laces to become a regular hexagon, which is another measure of evenness of these necklaces. An elementary mutation here either deletes an attack or inserts an attack. In the leftmost necklace of Figure 15.6, deleting the three attacks at pulses 10, 0, and 2, and inserting two attacks at pulses 11 and 1 does the job, yielding a total of five mutations. The necklace in the middle may be transformed into a regular hexagon by deleting the attacks at pulses 11 and 1, and inserting an attack at pulse zero, for a total of three mutations. Finally, the necklace on the right requires only the deletion of one attack at pulse zero. We will return to such muta- tion operations in more depth later in the book. All these mathematical methods are in effect theoretical explanations that fit the data. Whether any of these methods actually guided the evolutionary selection process is another matter altogether. It would be interesting to test experimentally which of these properties has the most perceptual reality. Under what
* Wright, O. (1995).
† Dahlig-Turek, E. (2009), p. 127.
‡ Kwabena Nketia, J. H. (1962), p. 85, lists this rhythm as a hand-clapping pattern of the Akan people of Ghana. However,
Rhythmic Oddity ◾ 91
circumstances, if any, is the degree of rhythmic oddity possessed by a rhythm, more easily perceived by humans than the amount of evenness? A dancing culture might have selected a timeline on the basis of rhythmic oddity, in as much as this property has a marked effect on the order of the upbeats and downbeats of the feet, thus rendering evenness as a by-product.
In the pitch domain, the three necklaces in Figure 15.6 are the three well-known scales called (from left to right) the diatonic scale, the ascending melodic minor scale, and the Neapolitan major scale.* Michael Keith proposes measuring the evenness of scales by
a suitable distance function between each note and the ideal note. The ideal notes are located at multiples of 12/7 on the circle, yielding the coordinate values along the circle: 0.0, 1.714, 3.428, 5.142, 6.856, 8.570, and 10.284. His measure called the scale-idealness also ranks the three necklaces of Figure 15.6 in decreasing order from left to right.† If we
compute the sum of the absolute values of the differences between these coordinates, and those of the attacks of the bembé, tonada, and sorsonet rhythms of Figure 15.5, we obtain the distances: bembé = 2.290, tonada = 2.566, and sorsonet = 4.994. Thus, the bembé is slightly more even than the tonada, and both are much more even than the sorsonet.
Let us return to the topic of generating rhythms that exhibit the rhythmic oddity prop- erty. At the beginning of this chapter, the Walk algorithm was presented that constructs rhythms that have the rhythmic oddity property but place all the onsets within a total dura- tion region that spanned less than one half-cycle, thus producing not the best of timelines. We close this chapter with a demonstration of a modification of the procedure that yields timelines that satisfy the rhythmic oddity property, and such that every half-cycle contains at least one onset. Furthermore, the timelines obtained in this way turn out to be better. This algorithm will be called the Hop-and-Jump algorithm. It falls in the general category of algorithms for obtaining the so-called generated rhythms, and later in the book we shall see its relation to other generative methods for producing deep rhythms. Thus, one application of the rhythmic oddity property is to the algorithmic generation of “good” rhythms.
Let us assume that we want to generate a rhythm with five onsets in a cycle of 12 pulses. The algorithm is illustrated with the five clock diagrams (left to right) in Figure 15.7. The first onset is placed at pulse zero. This implies that the diametrically opposite pulse six is now unavailable for placing an onset, since we want the rhythmic oddity property satisfied. To place the next onset, we hop to pulse two, making pulse eight unavailable. This process is continued, always advancing by hopping a distance of two units if this is possible. When this is not possible, as is the case when we want to hop to onset number four at pulse six (which is unavailable), we try the next pulse (here pulse seven). If it is available (as it is in this example), we take it. Otherwise, we continue skipping pulses until an available pulse is found. Since in this case we advanced by a distance of more than two pulses, we call this a
jump. Following a jump, we continue as before making hops of distance two if possible (or
jumps otherwise), here yielding the fifth onset at pulse nine.
The Hop-and-Jump algorithm is obviously guaranteed to yield rhythms with the rhythmic oddity property since it never places an onset on an unavailable pulse location.
* Rappaport, D. (2007), p. 322 and p. 323.
92 ◾ The Geometry of Musical Rhythm
Furthermore, by choosing the number of onsets and hop distance appropriately, we may guarantee that there are no silent gaps longer than a half-cycle. Finally, note that the result- ing five-onset rhythm obtained in Figure 15.7 is the fume-fume bell pattern (also the stan-
dard pattern) widely used in West Africa, and is the same as the Aka Pygmie rhythm of
Figure 15.4 either played backwards or rotated in a counterclockwise direction by four pulses.
Let us consider a few more examples with different numbers of onsets and pulses, and different sizes of hops, to substantiate our claim that the Hop-and-Jump algorithm is suc- cessful at generating good timelines.
For three onsets out of eight pulses, and hop-size two, the algorithm generates the rhythm with inter-onset intervals [2-3-3] as shown in Figure 15.8. Recall that this rhythm