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

cyclic shifts of one another. If $n 3$ is a power of $2$ , then $f(w)$ has exactly $|f(w)|= p$ cyclic shifts. Indeed, assume it has not, then $f(w)$ is a power of a shorter word, and since $n 3$ is a power of $2$ , $|f(w) | b$ is even, thus one can write $f(w) = gg$ . Then by proposition 5, one has $gg(e,e) = gg(u,v) = (u,v).(u,v) = (uu,vv)$ . It follows that $w = uuvv$ does not satisfy the rhythmic oddity property, since $h(uv) = h(vu)$ . This implies that $f(w)$ has exactly $|f(w)|= p$ different cyclic shifts $[]$

Corollary 2 If $n = 8 3$ , the number of solutions is the sum of the first squares.

Proof:. One has $j = 4$ , thus

X(p) = (p - 3)(p- 2)(p- 1)/4!

Since $n 2$ is odd, one can write $n = 2k - 1 2$ . Then $p = n2+ j = 2k - 1+ 4 = 2k+ 3$ . It follows $X(p) = k(k -1)(2k- 1)/6$ which proves that $X(p)$ is the sum of the $k$ first squares.

5 Results

The computation gives the following table.


$n3$	$n2$	Lyndon words	Rythmic patterns	Ethnic groups

2	1	ab	332	Zande

2	3	abbb	32322	Aka, Gbaya, Nzakara

2	5	abbbbb	3223222	Gbaya, Ngbaka

2	7	abbbbbbb	322232222

2	9	abbbbbbbbb	32222322222	Aka

4	1	aab	33332

4	3	aabbb	3323322
		ababb	3323232

4	5	aabbbbb	332233222
		ababbbb	332232322
		abbabbb	323232322

6	1	aaab	3333332

6	3	aaabbb	333233322
		aababb	333233232	Aka (mokongo)
		aabbab	333232332
		ababab	332332332	not primitive

Patterns actually used in Central African repertoires are indicated in the last column. There are reasons why those corresponding to the case $n3 = 2$ and $n2 = 7$ , and to the case $n3 = 4$ are not used (see Chemillier, 2002, for more details).