T
he rhythms considered in Chapters 5 through 9 were determined by cycles that had either 8 or 16 pulses. Such rhythms are here called binary rhythms, as are those with cycles of 2, 4, or 32 pulses. Note that all these numbers can be evenly divided by two, but not by three. Rhythms with cycles of 16 pulses are popular all over the world. In addition to 16, there is another number of pulses that also figures prominently in music of many parts of the world, most notably in sub-Saharan Africa and southern Spain, and this is the number 12. Such rhythms are here called ternary rhythms. Rhythms with 3, 6, and 24 pulses also belong to the family of ternary rhythms. The smallest binary and ternary rhythms with two and three pulses (also called duple and triple rhythms), and their com- binations, form the building blocks of most rhythms of the world, leading some scholars to label them as music universals. The reverend A. M. Jones writes: “When Europeans sing or play, their music will consist of rhythms which are essentially duple or triple or a combination of both. So it is with Africans. It is a fundamental natural law of rhythm and is therefore universal.”*The most illustrious ternary rhythm timelines in sub-Saharan Africa and the Caribbean have either five onsets or seven onsets, with durational patterns [2-2-3-2-3] and [2-2-1-2-2-2-1], respectively, in a cycle of 12 pulses. They are shown as polygons in Figure 10.1. Both rhythms have been called the standard pattern in the literature.† E. D. Novotney reviews the evolu-
tion of the terminology used for these rhythms and proposes his own: the five-stroke key pat-
tern and the seven-stroke-key pattern, respectively, on the basis that they play in sub-Saharan
African music, the same function that the “clave” rhythms play in Cuban music, and the word “clave” means “key.”‡ The rhythm on the left is sometimes called the fume-fume timeline§ and
* Jones, A. M. (1949), p. 293.
† Agawu, K. (2006), King, A. (1960), p. 51, Kubik, G. (1999), p. 53. ‡ Novotney, E. D. (1998), p. 165.
§ Klőwer, T. (1997), p. 176. This timeline pattern is also a clapping rhythm used by Ewe children in Ghana, who sometimes
dance four evenly spaced steps marking out the rhythm [x . . x . . x . . x . .]. See also Kubik, G. (2010b), p. 55, Kubik, G. (2010a), p. 45, Jones, A. M. (1959), p. 3, Collins, J. (2004), p. 29, Akpabot, S. (1972), p. 62, and Logan, W. (1984), p. 194. Bettermann, H., Amponsah, D., Cysarz, D., and Van Leeuwen, P. (1999), p. 1736, call this rhythm by the name of inyimbo.
46 ◾ The Geometry of Musical Rhythm
the one on the right the bembé pattern.* The fume-fume is often used as a clap pattern in West
African music, and A. M. Jones writes that: “This little clap-pattern is quite charming.”†
The reader has surely noticed that the ternary fume-fume clave has a geometric struc- ture very similar to that of the binary clave son. The two rhythms are shown together in Figure 10.2 for easy comparison.‡ In a subsequent chapter, both rhythms will be mapped
to a 48-pulse clock to quantify the absolute values of the differences between their cor- responding attack points. For now, it suffices to remark that visually the two pentagons are almost identical in shape and orientation. To be more precise, both rhythms contain an isosceles triangle rooted at the second onset, and both exhibit mirror symmetry about an axis that bisects the isosceles triangle at its apex. Furthermore, both contain obtuse isosceles triangles, which means that the triplets (onsets at pulses zero, two, and four in the
* King, A. (1960), p. 51, Malabe, F. and Weiner, B. (1990), p. 8, Logan, W. (1984), p. 194, Waterman, R. A. (1948), p. 28,
Temperley, D. (2000, 2004), p. 269, Kauffman, R. (1980), p. 397, Stone, R. M. (2005), p. 82, Kubik, G. (2010a), p. 44, Johnson, H. S. F. and Chernoff, J. M. (1991), p. 67, Chernoff, J. M. (1979), p. 145.
† Jones, A. M. (1954a), p. 33.
‡ Both of these rhythms are played in the Highlife popular dance rhythm of West Africa. See Chernoff, J. M. (1979), p. 145.
1 2 3 4 5 6 Fume-fume Son 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 7 8 9 10 11
FIGURE 10.2 The similar geometry of the fume-fume and son clave rhythms.
1 2 3 4 5 6 0 7 8 9 10 11 1 2 3 4 5 6 0 7 8 9 10 11
Binary and Ternary Rhythms ◾ 47
fume-fume and at pulses zero, three, and six in the son) are well separated from the twins (7 and 8 in the fume-fume, and 10 and 12 in the son) by the diagonals (11, 5) and (15, 7), respectively.
There are other musicological structural similarities between the two rhythms. For example, both numbers 12 and 16 may be evenly divided into four equal durations (quar- ter measures) without requiring additional pulses, by selecting the “north,” “south,” “east,” and “west” pulses numbered 0, 3, 6, and 9 in the ternary case and 0, 4, 8, and 12 in the binary case. These are the four most salient locations for regular metric beats in families of rhythms with 12 and 16 pulses. The fume-fume and son rhythms both have their first and last onsets on their “north” and “west” metric pulses, respectively. Since both regular meters, [3-3-3-3] and [4-4-4-4], can be easily aligned with each other, and the two rhythms are so similar, they can easily be interchanged during the performance of a piece, as is done in the Highlife music of West Africa. Jeff Pressing calls such timelines with unequal values of pulses in their cycles, but with similar inter-onset interval structures, transformational
analogues,* and Fernando Benadon explores their use as compositional and analytical
expressive transformations of each other.†
In addition to the fact that the two rhythms are quite similar to each other with respect to the exact locations of their attacks, they are in fact identical to each other if they are represented by their rhythmic contours. The rhythmic contour of a rhythm is obtained by coding the change in the durations of two adjacent inter-onset intervals using 0, +1, and −1 to stand for equal, greater, and smaller, respectively. The durational patterns of the fume-fume and son timelines are, respectively, [2-2-3-2-3] and [3-3-4-2-4]. Therefore, both rhythms have the same rhythmic contour: [0, +1, −1, +1, −1]. Rhythmic contours are relevant from the perceptual point of view because humans have an easier time perceiving qualitative relations such as “less than” or “greater than” or “equal to” than quantitative relations such as the second interval is four-thirds the duration of the first interval. It has also been found that often the reduced information contained in the contour is sufficient to effectively describe certain types of music.‡ On the other hand, two rhythms with the same
contour may also sound quite different, as is the case for the 16-pulse and 11-pulse rhythms with inter-onset intervals [4-3-2-3-4] and [3-2-1-2-3], respectively.§ Therefore, used in isola-
tion or in a context where the intervals can vary widely, the rhythmic contour suffers from
* Pressing, J. (1983), p. 43. † Benadon, F. (2010).
‡ Hutchinson, W. and Knopoff, L. (1987), p. 281, hypothesize that music style may be effectively described and discrimi-
nated on the basis of syntactic structures of three symbols used to code rhythmic contours, namely R (repetition), S (shortening), and L (lengthening), in effect, a three-letter alphabet for temporal groupings.
§ See Marvin, E. W. (1991) for the application of rhythmic contours to composition analysis. Contours have also been
explored in the pitch domain, where they are called pitch contours or melodic contours. See Schultz, R. (2008) for the application of melodic contours to the analysis of the nonretrogradable structure in the birdsong music of Olivier Messiaen. A structure is nonretrogradable if it is palindromic, that is, has the same structure when played forward or backward. Freedman, E. G. (1999), p. 365, writes that “musically experienced listeners can recognize both the contour and interval information, whereas musically inexperienced listeners rely predominantly on the contour information.” See also Callender, C., Quinn, I., and Tymoczko, D. (2008) for a more recent discussion on contour in the pitch domain.
48 ◾ The Geometry of Musical Rhythm
severe drawbacks as a representation from which to extract meaningful rhythmic similar- ity features.*
John Chernoff has suggested that for all practical purposes there is not much difference between the binary and ternary versions of the “standard” African bell pattern (fume- fume) when perceived relative to their underlying duple metric beats, [4-4-4-4] and [3-3- 3-3], and that these regular beats play a perceptually important role. However, while there is little doubt that these duple metric beats influence perception, it is not at all clear that this influence propels the listeners’ judgments of the two versions toward greater similar- ity. It may be argued to the contrary, that the duple underlying beats flesh out rather than camouflage their differences. Figure 10.3 shows the binary and ternary versions of the five- onset standard pattern superimposed on their duple metric structures, to more accurately examine their perceptual role. For either rhythm, imagine playing the metric beats (each highlighted with a ring) with the left hand on a bass drum, and the rhythm with the right hand on a woodblock. Let us denote with the letters R, L, and U the events consisting of striking the instruments with the right hand, left hand, and both hands in unison, respec- tively. While it is true that the sequence of onsets that describes the union of the metric beats and rhythm onsets yields the same alternating pattern for both the clave son and the fume-fume, namely [U-R-L-R-L-R-U], and although both rhythms start and end on the first and last beats of the cycle, these properties by themselves are not sufficient to engender greater perceived similarity. On the contrary, feeling the duple meter makes the listener more keenly aware of the differences in the placements of the third and fourth onsets of the rhythms, which in the clave son falls squarely in the middle of the interbeat intervals, whereas in the standard pattern fall closer to the beats, creating greater syncopation. In the fume-fume pattern, the third onset is twice as close to the second beat than to the third beat, and the fourth onset is twice as close to the third beat than to the fourth beat.
Furthermore, examples may be constructed of quite dissimilar rhythms that satisfy both these properties. To this end consider the two rhythms in Figure 10.4. The left rhythm
* Whether or not contours will play a significant role in music theory, they have already spawned interesting problems
in computer science. Demaine, E. D., Erickson, J., Krizanc, D., Meijer, H., Morin, P., Overmars, M., and Whitesides, S. (2008) consider the problem of reconstructing rhythms from the full and partial contour information.
1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 7 8 9 10 11
Binary and Ternary Rhythms ◾ 49
is a rotation of the bossa-nova clave, and the right is a mutation of the standard pattern, in which the second onset is moved from pulse two to pulse one, and the fourth onset is moved from pulse seven to pulse eight. Both rhythms have the same onset-to-meter place- ment pattern [U-R-L-R-L-R-U], and both have their first and last onsets on the first and last beats of the cycle, but they sound quite different from each other.
In closing this chapter, it should be noted that the terms binary and ternary are also used in music to describe the form of the compositions as a whole. In this context, form refers to the manner in which sections of the piece are structured. In the binary form, two main sections of the work are repeated to create patterns such as AABB, whereas in the ternary form, the sections are organized in patterns such as ABA.*
* Wright, D. (2000), p. 27. 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 11 0 10 11 12 13 14 15
FIGURE 10.4 A rotation of the bossa-nova rhythm (left) and a mutation of the fume-fume (right) embedded in a duple meter.
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C h a p t e r