ABSTRACT
Consider the bembé timeline played in the usual sub-Saharan African context of an underlying meter that places an accent at every third pulse starting with pulse zero. In Figure 14.1, these four metrically strong pulses {0, 3, 6, 9} are indicated by the vertical and horizontal lines (a four-beat measure). Note that the bembé (le ) has attacks on the rst and last metric accents at positions zero and nine. Examine what happens when this rhythm is rotated clockwise by one pulse so that the new rhythm starts on the last onset of the bembé, as shown in Figure 14.1 (right). e new rhythm contains attacks on the rst, second, and third metrically strong pulses at positions zero, three, and six. is is a considerable change, and not surprisingly, if I play this rhythm on a bell with my hands, while playing a bass drum with my foot on pulses {0, 3, 6, 9}, the new rhythm sounds and feels quite dierent from the bembé. Indeed, it still feels considerably dierent even if I do not play the bass drum, and just mentally partition the cycle into a [3-3-3-3] metric subdivision. erefore, from the point of view of music-making, we may consider that these two rhythms are dierent. However, it is obvious that the interval contents of these two durational patterns and their resulting histograms are identical, since the interval content of a rhythm is invariant to its rotations. erefore, from certain analytical perspectives,
the two rhythms may be considered to be the same. In the mathematical eld of combinatorics, the two rhythms in Figure 14.1 are said to be instances of the same necklace.* In the pitch domain in music theory, a necklace corresponds to a chord type.†
A necklace is a closed string of beads (or pearls) of dierent colors, such as one might wear around one’s neck. We are interested here in binary necklaces, that is, necklaces with pearls of two colors: black and white. Two necklaces are considered to be the same if one can be rotated so that the colors of its beads correspond, one-to-one, with the colors of the beads of the other necklace. Figure 14.2 shows two more instances of identical necklaces. e rhythm on the right is obtained by rotating the one on the le clockwise by three pulses.
