Today, more about the math one may find behind rhythms and their transformations !

**I got rhythm….**

In the previous post, we’ve introduced a group of transformations which acts on time-spans, i.e. a duration with a precise position on a time-line. We have seen that this group, which we call , is the semidirect product , with the group law given by

.

This allows us, for example, to describe the transformations of successive time-spans on a time-line, like this one:

Here, we’ve identified the first time-span as the group identity element, i.e. . If each gray mark represents the duration of a quarter note, then the second time-span corresponds to the group element and the third time-span to the element . To see if you’ve assimilated the gymnastics of , can you determine the transformation from the second time-span to the third, both in the left- and right-action contexts ? The answer is below…

*Answer*: in the left-action context, we want to find such that . This gives . In the right-action context, we want instead. This gives .

So this is great, we can analyze the transformation between any time-spans on a time-line. But what about multiple time-lines ? Take this 2-part rhythm, for example:

Here, the gray marks correspond again to the duration of a quarter note, and I’ve highlighted the first quarter note in part A as I choose to identify it to the group identity element. For part B, I will choose the same time-span as the identity element. How can we analyze this short example ? The group only allows us to transform time-spans in a single time-line, so we have to determine the transformations separately for each part. I will choose here to work in a right-context action.

In part A, the successive time-spans are obtained by the right multiplication with , , , and so on. Part B begins with a time-span which corresponds to the group element , so the successive time-spans are obtained by the right multiplication with , , , etc.

This analysis is really clumsy as we have to treat each part independently, thus missing the obvious symmetry between the parts. What we need is a way to generalize the group so that it may act on multiple time-lines.

**I got multi-dimensional rhythm…**

Remember that our group of transformations is the semidirect product , which is in fact the general affine group for a 1-dimensional real vector space. If we want to consider the possible transformations of *n* time-lines, is tempting to consider the general affine group for an *n*-dimensional real vector space. Remember that in the general case, this group is given by the semidirect product , where is the group of translations, and is the general linear group of degree *n* over . The elements of this group are of the form , where is a vector, and is an matrix with real coefficients. The group composition is given by

However tempting using this group may be, there is an immediate conceptual problem. In the 1-dimensional case, we use is simply a strictly positive real value, which we identify with the duration of the time-span. But, for *n* time-lines, how can we think of an matrix as a duration ? And how can we avoid “negative durations” ?

In fact, this is an open question for me, and at the present time I don’t have an exact description of the suitable matrices one could use. Nevertheless, we can work with a reduced subgroup of , which will fit our needs. Consider the particular 2-dimensional case. We choose to work with a subgroup consisting of all matrices of the form or , with . In the first case, and corresponds to the durations in time-lines 1 and 2 respectively. Notice in the second case that , wherein is a permutation matrix. In other words, we have a way to switch time-lines and analyze the possible interactions between them.

Let’s consider again the short rhythmic example above. We use the subgroup of we have just defined in the semidirect product with . The initial elements correspond to the group element . The next two time-spans in both time-lines are obtained through the right multiplication with the unique group element . This is also the case for all successive time-spans: we are thus able to describe this example with the use of only one transformation ! Notice that the element reflects at the same time the alternative dilation and contraction of the note durations and the interchange between parts which is visible on the score.

**Generalizations**

You can of course generalize this example to any number of time-lines using the subgroup of generated by diagonal matrices with strictly positive real coefficients and by permutation matrices.

As we will see in a later post, the subgroup of the general *n*-dimensional affine group thus obtained has a nice categorical flavor, which will allow us for some further generalizations.

You could also experiment with other matrices in . For example, I’m currently considering, in a 2-dimensional setting, the subgroup generated by matrices of the form or , with . What do they mean musically ? Open questions…