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

easy to handle with any kind of filtering), and it ensures a melodic motion for each instrument.

2.1.8 Scales in quartertones

This initial idea of this problem has been suggested by a french musicologist, Jean-Michel Bardez. The goal is to find musical scales with quarter tones. Of course, the whole set of these is far too big for anyone to listen to. So we used constraint programming to express some properties on the scales, thus reducing the amount of generated scales. The variables are $V1...Vn$ , with domain at least ${1/2,1/4}$ (it is possible to add $3/4$ , $5/4$ ...). Here $n$ is not intrinsic to the problem and should not be fixed, since the scale-constraint simply states that $sum iVi = 6$ (the scale has to range over one octave).

We can add the following constraints, or any combination of them :

Symetry
$(Vn...V1) = (V1...Vn)$
Limited transposition
$E i < n$ such that $(Vi ...VnV1 ...Vi-1) = (V1...Vn)$
Pattern : the scale contains a fixed pattern $a ...a 1 k$
$(a1...ak) < (V1 ...Vn)$
Minimal number of semi and quarter tones, $m$ fixed.
$Card{i|Vi = 1/4$ ou $1/2} >= m$
Quartertone harmony : more interesting, this constraint is about the harmonies that can be constructed on the scale, in order to avoid a scale where most of the chord would by chance have intervals in semi tones. We fix a reference figuring $d1...dk$ , for instance $0$ , $4$ , $7$ by analogy with the major chord, and an integer $m$ . Then the chords with this figuring are built on all the degrees of the scale. The constraint states that among all those, at least $m$ have exactly one interval in quartertones (possibly, all theirs intervals in quartertone). Writing $Inti$ for the intervals of the $i$ -th chord, the constraint is $Card{i| E a (- (2* N+ 1)a/4 (- Inti}>= m$

Here the constraints are used in a very different manner as in the other problems. They allow to generate a database-like set with some properties. This is the only compositional case where we have to find all the solutions.

2.2 CSPs in computer assisted musical analysis

2.2.1 Nzakara canons

The three last problem are not about composition but about musical analysis.

This problem has been formulated by Marc Chemillier (Chemillier, 1995). The goal is to modelize a very precise structure found in the Nzakara harp repertoire,