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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 $\text{Aff}_+(\mathbb{R})$, is the semidirect product $(\mathbb{R},+) \rtimes (\mathbb{R}_*^+,\times)$, with the group law given by

$(u,\delta) \cdot (t,\Delta) = (u+\delta t, \delta\Delta)$.

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. $(0,1)$. If each gray mark represents the duration of a quarter note, then the second time-span corresponds to the group element $(1,1)$ and the third time-span to the element $(3,1/2)$. To see if you’ve assimilated the gymnastics of $\text{Aff}_+(\mathbb{R})$, 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 $g$ such that $g.(1,1)=(3,1/2)$. This gives $g=(5/2,1/2)$. In the right-action context, we want $(1,1).g=(3,1/2)$ instead. This gives $g=(2,1/2)$.

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 $\text{Aff}_+(\mathbb{R})$ 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 $(1,1/2)$, $(1,2)$, $(1,1/2)$, and so on. Part B begins with a time-span which corresponds to the group element $(0,1/2)$, so the successive time-spans are obtained by the right multiplication with $(1,2)$, $(1,1/2)$, $(1,2)$, 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 $\text{Aff}_+(\mathbb{R})$ so that it may act on multiple time-lines.

I got multi-dimensional rhythm…

Remember that our group of transformations is the semidirect product $(\mathbb{R},+) \rtimes (\mathbb{R}_*^+,\times)$, 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 $\mathbb{R}^n \rtimes \text{GL}(n,\mathbb{R})$, where $\mathbb{R}^n$ is the group of translations, and $\text{GL}(n,\mathbb{R})$ is the general linear group of degree n over $\mathbb{R}$. The elements of this group are of the form $(v,M)$, where $v \in \mathbb{R}^n$ is a vector, and $M \in \text{GL}(n,\mathbb{R})$ is an $n \times n$ matrix with real coefficients. The group composition is given by

$(v_1,M_1) \cdot (v_2,M_2) = (v_1+M_1 \cdot v_2, M_1 \cdot M_2)$

However tempting using this group may be, there is an immediate conceptual problem. In the 1-dimensional case, we use $M \in \text{GL}(1,\mathbb{R}_*^+)$ 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 $n \times n$ 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 $\text{GL}(n,\mathbb{R})$, 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 $\begin{pmatrix} \alpha & 0 \\ 0 & \beta \end{pmatrix}$ or $\begin{pmatrix} 0 & \alpha \\ \beta & 0 \end{pmatrix}$, with $\alpha,\beta \in \mathbb{R}_*^+$. In the first case, $\alpha$ and $\beta$ corresponds to the durations in time-lines 1 and 2 respectively. Notice in the second case that $\begin{pmatrix} 0 & \alpha \\ \beta & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} \alpha & 0 \\ 0 & \beta \end{pmatrix}$, wherein $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ 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 $\text{GL}(2,\mathbb{R}_*^+)$ we have just defined in the semidirect product with $\mathbb{R}^2$. The initial elements correspond to the group element $(\begin{pmatrix} 0 \\ 0 \end{pmatrix},\begin{pmatrix} 1 & 0 \\ 0 & 1/2 \end{pmatrix})$. The next two time-spans in both time-lines are obtained through the right multiplication with the unique group element $(\begin{pmatrix} 1 \\ 1 \end{pmatrix},\begin{pmatrix} 0 & 1/2 \\ 2 & 0 \end{pmatrix})$. 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 $\begin{pmatrix} 0 & 1/2 \\ 2 & 0 \end{pmatrix}$ 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 $\text{GL}(n,\mathbb{R})$ generated by $n \times n$ diagonal matrices with strictly positive real coefficients and by $n \times n$ 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 $\text{GL}(n,\mathbb{R})$. For example, I’m currently considering, in a 2-dimensional setting, the subgroup generated by matrices of the form $\begin{pmatrix} 1 & \alpha \\ 0 & 1+\alpha \end{pmatrix}$ or $\begin{pmatrix} 1 & 0 \\ \alpha & 1+\alpha \end{pmatrix}$, with $\alpha>0$. What do they mean musically ? Open questions…