ABSTRACT

In Chapter 19, it was demonstrated with numerous examples taken from music from around the world that for a multitude of numerical values of n and k, the number of pulses and onsets, respectively, there exist many rhythm timelines in cultures all over the world that have the property that they are Euclidean or maximally even. However, for every xed pair of values of n and k, the Euclidean algorithm yields only one rhythm necklace. Consider for instance the Euclidean rhythm obtained when n = 8 and k = 2. Since eight is divisible by two, the rhythm obtained is [x . . . x . . .] = [4-4]. Repeating this pattern yields only a steady pulsation, not a very interesting rhythm. However, if we displace the second attack by one pulse, say to the le , we obtain the very interesting rhythm [x . . x . . . .] = [3-5]. is rhythm is quite common in Afro-Cuban music, where it is called the conga.* It has also been incorporated into rock-n-roll music, perhaps most notably as the mid-song electric guitar solo in the Beach Boy’s 1964 best-selling ballad Don’t Worry Baby, ranked by the Rolling Stone Music magazine as the 178th greatest song of all time.†

Adding one more attack to the conga, so that n = 8 and k = 3, yields again only one Euclidean rhythm E(3,8) = [x . . x . . x .] = [3-3-2], which when rotated yields the additional rhythms [3-2-3] and [2-3-3]. On the other hand, several other rhythms used in practice also consist of three onsets among eight pulses, but they are neither Euclidean rhythms nor rotations thereof. As the two examples above illustrate, to generate a  larger,  more inclusive class of “good” rhythms, the concept of maximally even has to be relaxed. One approach is to modify slightly, or mutate, a maximally even rhythm to generate other rhythms that are almost maximally even. In a brute-force approach, for a given pair of values of n and k, we could rst generate all possible rhythms, then calculate according to some chosen distance function, the distance between all these rhythms and the maximally even rhythm, and nally select those rhythms that are close enough to the maximally even rhythm according to a preselected threshold. However, such an approach requires much computation. ere are other more direct

ways of generating rhythms that are close to maximally even rhythms. As an example, consider again the ubiquitous tresillo rhythm at the top of Figure 21.1, and dene a mutation operation as the displacement of one of its onsets (other than the rst) by a duration of one pulse toward the le (anticipation). We can generate three new rhythms this way by moving either the second onset from pulse three to pulse two, the third onset from pulse six to pulse ve, or both of these. e three rhythms produced in this way are used in music in many parts of the world. e second rhythm [3-2-3] is used in Beijing Opera, the third [2-4-2] is the catarete from Brazil, and the fourth [2-3-3] is a bossa-nova rhythm from Brazil, as well as the nandon bawaa from Ghana. Two of these are rotations of the tresillo, and therefore, they are also obviously maximally even, but the catarete is not.