S
yncopation is the spice of rhythm. It adds surprise to an otherwise bland rhythm. We can easily feel when a rhythm has syncopation, but translating that feeling to mathematical terms, my general goal with all musical properties explored in this book is easier said than done. A prerequisite for making progress in this direction is a precise constructive definition. Consider how some dictionaries explain what syncopation is. TheNew Oxford American Dictionary defines a syncopated rhythm as one in which the “beats
or accents” are displaced “so that strong beats become weak beats and vice versa.” From the mathematical point of view, this definition is not very satisfactory because the notions of “strong” and “weak” beats have not been defined. The Oxford Grove Music Online
Dictionary defines syncopation as: “The regular shifting of each beat in a measured pattern
by the same amount ahead of or behind its normal position in that pattern.” This definition also lacks mathematical rigor because the notion of “normal” has not been specified. The
Harvard Dictionary of Music defines syncopation as “A momentary contradiction of the
prevailing meter.”* This definition assumes we know what meter is, but more problemati-
cally, how do we interpret the words “momentary” and “prevailing?” As a final example, consider the lesser-known on-line Virginia Tech Multimedia Music Dictionary; it defines syncopation as the “deliberate upsetting of the meter or pulse of a composition by means of a temporary shifting of the accent to a weak beat or an off-beat.” Does this mean that if the shifting of the weak beat is not deliberate, there is no syncopation? Furthermore, what is the difference between a weak beat and an off-beat?
There must be more than 50 traditional definitions of syncopation adorning the pages of dictionaries, books, and Internet sites. Like the definitions offered here, most have their own particularities. However, to a mathematician, they all have one thing in common: vagueness. Of course this is not surprising considering that we are trying to define with precise mathematical tools a slippery human perceptual skill. We can all read text with- out effort, which implies that we recognize characters such as A, B, C, and so on, without difficulty. However, we do not know how to define what an “A” or a “B” is. This is a major
68 ◾ The Geometry of Musical Rhythm
problem in artificial intelligence.* Indeed, in the words of Michael Keith, “although syn-
copation in music is relatively easy to perceive, it is more than a little difficult to define precisely.Ӡ Some readers may believe that the desire for mathematical precision is com-
pletely inappropriate here, and that such demands lead inevitably to irrelevance regarding the psychological aspects of music. On the contrary, I concur with the philosopher Mario Bunge that we should mathematize everything we can, and the only way to know if a fuzzy concept can be successfully modeled mathematically is to try.‡ This is the sine qua
non and the hallmark of artificial intelligence, and it pushes the boundaries of the rele-
vant psychology of music. In spite of the difficulties that such a task may pose, it is possible to construct unambiguous mathematical definitions of notions that may be used as useful models that replace the traditional concept of syncopation. Furthermore, in due time, as the scientific and technological approaches of the study of music continue to expand, some of these mathematical versions of syncopation may supplant the traditional notion. Syncopation is very much a Western concept, and for some types of music, new math- ematical substitutes for syncopation, which are not culturally dependent, may be more appropriate and useful.§ Referring to sub-Saharan music, Simha Arom states the case
more bluntly: “terms such as . . . syncope . . . should be dispensed with as foreign to it.”¶
Jay Rahn reflects this evolving terminology by offering two definitions of syncopated, one descriptive of the Western culture, and another mathematically inspired.** His first defini-
tion of syncopated is “deviating from an oriental metrical organization in that one or both of the immediately adjacent presented moments to a given moment is not resolved.” Here, the term moment is used to mean an “irreducible portion of time.” His second definition of syncopated is “not commetric.” The term “commetric” here is synonymous with regu- lar, a well-defined mathematical notion. Thus, Rahn’s second definition of a syncopated rhythm is one that is irregular.
In 1996, Fred Lerdahl and Ray Jackendoff published a book titled A Generative Theory
of Tonal Music, in which they proposed a hierarchy of accents for musical rhythm inspired
by research work in linguistics.†† For a timeline of 16 pulses, their hierarchy of accents
or metrical weights may be expressed using the graph shown in Figure 13.1. One way to construct this graph is as follows. First, starting at pulse zero, and proceeding from left to right, assign a weight of one to every pulse (shown as shaded boxes). Second, in a similar manner, increment by one the weight of every second pulse. Third, increment by one every fourth pulse. Next, increment by one every eighth pulse, and finally every 16th pulse. The resulting height of the column at any pulse gives the weight or degree of accent given to an onset that occurs at that pulse. In other words, the pulse emphasized most strongly is pulse
* Longuet-Higgins, H. C., Webber, B., Cameron, W., Bundy, A., Hudson, R., Hudson, L., Ziman, J., Sloman, A., Sharples,
M., and Dennett, D. (1994).
† Keith, M. (1991), p. 133. ‡ Mahner, M., Ed. (2001).
§ Vurkaç, M. (2012), for example, finds the mathematical definition of off-beatness (to be explored later in the book) more
useful than syncopation, for the analysis of rhythms in traditional contexts.
¶ Arom, S. (1991), p. 183. ** Rahn, J. (1983), p. 248.
Syncopated Rhythms ◾ 69
zero with a weight of five. The next most salient pulse is number eight with a weight of four. Pulses 4 and 12 have a weight of three, pulses 2, 6, 10, and 14, have a weight of two, and all the remaining (odd-numbered) pulses have a weight of one.
This metrical hierarchy may be used to design a precise mathematical definition of syn- copation, which we shall call metrical complexity, as follows.* Consider the clave son time-
line shown in Figure 13.2 in box notation directly below the metrical hierarchy. The clave son consists of onsets at pulses 0, 3, 6, 10, and 12, with metrical weights equal to five, one, two, two, and three, respectively. These metrical weights express how normal or typical it is for a beat to occur at that pulse location according to the theory of Western music practice expressed by Lerdahl and Jackendoff. Therefore, the lower the weight for an onset, the more unexpected the onset, and thus the more syncopated it is as well. For the clave son, the onset with the lowest weight is the second onset occurring at pulse three that has a weight of one. Therefore, this onset is considered to be the most syncopated of the five onsets. Interestingly, in some Latin music such as salsa, a rhythm that accentuates this
* This measure of syncopation was first proposed in Toussaint, G. T. (2002). In a subsequent study by Thul, E. and
Toussaint, G. T. (2008a), it was compared with a large group of measures of rhythm complexity, irregularity, and syn- copation, against human judgments of performance and perceptual complexity, and gave superlative performance. Flanagan, P. (2008) proposes a mathematical measure of syncopation that computes an average with respect to many possible underlying meters.
5 4 3 2 1 0 0 1 2 3 4 5 6 7 Pulse number Me trical weigh t 8 9 10 11 12 13 14 15
FIGURE 13.1 The metrical hierarchy of Lerdahl and Jackendoff.
5 4 3 2 1 0 0 1 2 3 4 5 6 7 Pulse number Clave son Me trical weigh t 8 9 10 11 12 13 14 15
70 ◾ The Geometry of Musical Rhythm
second onset of the clave son is called bombó, also the name of a bass drum used in the Afro-Cuban comparsa music that is played in carnivals.*
To measure the total metrical expectedness (or simplicity) of the rhythm, we may add the metrical weights of all its onsets. Thus, for the clave son, the metrical expectedness is equal to 13. To convert this measure to a measure of metrical complexity or syncopation, it suffices to subtract the metrical expectedness value of a given rhythm with k onsets and n pulses from the maximum possible value that any rhythm with k onsets and n pulses may have. For a rhythm with five onsets and 16 pulses, the maximum expectedness value is 17, obtained by summing the column heights at pulses 0, 8, 4, 12, and any one of 2, 6, 10, and 14. This value is realized by several rhythms, including the popular classical music ostinato rhythm [4-4-2-2-4] with onsets at pulses 0, 4, 8, 10, and 12. Thus, the metrical complexity of the clave son is 17 − 13 = 4. For comparison, the more syncopated clave rumba that has its third onset at pulse number seven has a metrical complexity equal to 17 – 12 = 5.