input can be reduced to a set of integer durations . A priori, they have no particular structure, though we know that they shall correspond to a rhythm, ie they are approximations of a pattern with rational and quite simple ratios. We could have for instance the input , , , that anybody would recognize as »two eigth notes - one quarter note«, ie an approximation of , , .

The goal is to find the tempo that fits best the real durations, in the sense that the input is close to the rhythmical patterns played on this tempo. We fix a set of allowed time ratios , for instance for a half note, for triplets, etc. The variables are the durations of the symbolic pattern to find. From the there are several tempi possible, we can take for instance .

The first constraint states that the pattern is close to the real rhythm : , The second constraint restricts the kind of symbolic patterns authorized. Without it, any serie of integer durations could be written in a symbolic rhythm, but far too complicated to be musically considered. For all , we impose that belongs to . An option is to refine this constraint by asking only the denominator of is in , to allow rhythms such as »quarter note - eigth note in a triplet« in the symbolic rhythm.

2.1.6 Accelerando

This problem has been given by the canadian composer Pierre Klanac. He wanted to find local tempi , forming a smooth accelerando, so the ratio of two successive tempi should be close to , say . As a constraint, (). But on a score, a tempo change is usually written as a musical equivalence, such as »dotted quarternote = quarternote«, with the underlying statement that the tempi ratio is rational, and >simple<, (for instance in the previous case). With a ratio of between and for instance, this is not possible. Thus we add sort of a second order constraint, on the ratios between and , stating that (), and even two other constraints (), and ().

Here, a mathematician would argue that fixing the first tempo , all the other ones are fixed by as a geometric series and thus , and thus should be replaced by a ratio of , by a ratio of , and by a ratio of . Of course, we cannot apply this because the tempi have to be integers. So we have to consider this as an optimization problem : find the tempi with the closest ratios as possible from the given values.

Searching for around tempi in a range from to , Pierre Klanac decided to order the constraints this way : most important, optimize the number of exact ratios for and , with at least five exact ratios. Then optimize , but not by counting the exact ratios, just trying to minimize . Although all the constraints are stated in the same way, the difference of the notion of optimization between and the others is musically logical, since ensures the smoothness of the accelerando, while the others constraints guarantee a possible musical notation.