Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

in Central Africa. We want to produce a sequence of $n$ bichords $B1 ...Bn$ , in a domain of size 5. Approximating in MIDI values, the domain is $(60,64)$ , $(60,67)$ , $(62,67)$ , $(62,70)$ , $(64,70)$ . The sequence is played repetitively, so the constraint must hold on all the sequences, though we shall not write the modulo it implies for legibility reasons.

The first constraint $C1$ states that the lower voice of the sequence reproduces the upper voice, with a time gap of $p$ (a fixed integer). Formally, we have a transposition-like function $t$ which maps $60$ to $64$ , $62$ to $67$ , and $64$ to $67$ . For every $k < n$ , the lower note of $Bk+p$ is the transposition of the upper note of the $Bk$ . The second constraint $C2$ is to avoid trivial sequences such as $(aaa)$ or $(ababab...)$ . It can be written as $Bk+p /= Bk$ and $Bk+p /= Bk$ (the Nzakara never repeat a bichord). An example is shown figure 5.

Figure 5:

A Nzakara canon of the limanza category.

Notice that the two constraints are contradictory. In the Nzakara sequences, the second constraint is always satisfied, so we have to allow some errors on the first one. A result from Chemillier (1995) shows that the number of errors in a Nzakara canon is at least $gcd(n,p)$ .

In the original Nzakara harp repertoire, one can find different values for integers $n$ and $p$ (respectively the total length of the repeated sequence and the distance of the canon). Sequences with $n = 10$ , $p = 4$ (as in figure 1) and $n = 20$ , $p = 4$ belong to the category called ngbakia. Sequences with $n = 30$ , $p = 6$ belong to the category called limanza (figure refnzakara-limanza). These sequences are played as ostinato, each piece of Nzakara poetry being sung with the accompagniment of such formula played on the harp. The categories ngbakia and limanza also refer to traditional dances, the harp formulas being adapted from rhythms and musical elements borrowed from the dance-repertoire played on the portable xylophone or the drum. Some of these sequences can heard on the two CDs mentionned in the references below.

2.2.2 Rhythmical imparity

This problem has been proposed by Marc Chemillier. Aka pygmies play rhythmical formulas on a regular beat, with irregularly distributed accents. These accents form groups of $2$ or $3$ beats, for instance $3$ $2$ $2$ $2$ $2$ $3$ $2$ $2$ $2$ $2$ $2$ . This formula has a property called rhythmical imparity by Simha Arom. Figure 6 shows the formula placed on a circle (the formulas are played in a continuous loop). It is impossible to divide the circle in two. This dissymetry is intrinsic to this formula, and to others of the same kind played in this Central African area.