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Back to music and mathematics, but with a change as today we will not talk about chords. Instead, I propose we focus on the transformational theory of time and note durations.

Again, we will see plenty of applications for groups and group actions so let’s start with something basic (you may have seen it in earlier posts but it’s worth recalling it here).

The group of onset transformations

Here is a note on a time-line :

As you can see, this note has a precise duration (it’s a quarter note, and you can get its duration in seconds if you have set a metronome reference). It also has a precise location on the time-line, which we will call its onset, as soon as you fix one point of the time-line as the “zero” time. This onset can be given a value $t \in \mathbb{R}$.

You can move the onset of the note by any amount of time, like this

It is quite easy to see that there exists a group of onset transformations, which is nothing else but the additive group of the reals $(\mathbb{R},+)$.

This group has an interesting subgroup, which we will see later again. We can actually build it from scratch, without any reference to $(\mathbb{R},+)$. Suppose you’re interested in the onset shifts by an amount equal to the quarter duration (or as we can call it, unit). It is easy to see that the transformation $z_0$ which shifts the quarter by one unit (see below) generates a group isomorphic to $(\mathbb{Z},+)$.

Suppose now you also want to shift this quarter note by half-units. The transformation $z_0$ alone cannot do it, so we need to introduce a new transformation $z_1$ which does, as pictured below

Notice that we have the following relation: $z_1^2=z_0$. What about shifting the note by one quarter of a unit ? Again, we need to introduce a new transformation $z_2$, whose action is pictured below

And we have $z_2^2=z_1$. By repeating this procedure indefinitely, we obtain a group whose presentation is

$< z_0, z_1, z_2, \cdots | z_{n+1}^2=z_n, \forall n>0>$

It is well known that this group is isomorphic to $\mathbb{Z}[\frac{1}{2}]$, the additive group of the dyadic rationals, i.e. numbers of the form $\frac{a}{2^b}$ where $a$ is an integer and $b$ is a natural number. Hence we have built a group of transformations whose elements shift the onset a note by any dyadic amount on its time-line.

The group of duration transformations

The case of durations is a bit different. The note durations can be given a value $\Delta \in \mathbb{R}_*^+$, the set of strictly positive real numbers. We can’t use an additive group structure on this set, since negative durations are not allowed. Instead we use the multiplicative group $(\mathbb{R}_*^+,\times)$, whose elements transform the note durations by multiplication.

This is in fact quite natural. For example, most musicians have been told “two eight notes make a quaver, two quavers make a quarter, etc.”. So in that discrete case, we have a unique generator which corresponds to multiplication by 2 :

The group we obtain is in this case isomorphic to the additive group of the integers $(\mathbb{Z},+)$.

Transformations of time-spans

As said before, a note on a time-line is uniquely identified by its onset $t$ and its duration $\Delta$. The pair $(t,\Delta)$ is also called a time-span. As with many subjects in transformational theory, David Lewin was the first to define a Generalized Interval System (GIS) for time-spans, which you can find in his book

In his GIS (see pp. 60-87 in the above mentionned book, and in particular p. 75), the interval between two time-spans $(t_1,\Delta_1)$ and $(t_2,\Delta_2)$ is $(\frac{t_2-t_1}{\Delta_1},\frac{\Delta_2}{\Delta_1})$, an element of the affine group of $\mathbb{R}$ as a unidimensional vector space.

I will now show how a group of transformations for time-spans can be built in the framework of group extensions that we have seen before. We will thus recover Lewin’s example as well as a new one. You can check for more information from a paper I published recently:

• “Building Generalized Neo-Riemannian Groups of Musical Transformations as Extensions”, A. Popoff, Journal of Mathematics and Music, 7 (1), 2013, pp. 55-72. (There is an older version of this article available on Arxiv).

The fact that a time-span is uniquely defined by a pair $(t,\Delta)$, $t \in \mathbb{R}$, $\Delta \in \mathbb{R}_*^+$ should remind you of the previous posts in this blog, wherein major and minor chords where uniquely identified by their root $n$ and their type $M$ or $m$ (i.e. pitch-class sets [0,4,7] or [0,3,7]). So, if we want to build a group of transformations in a similar manner, we will look at extensions of the group of onset transformations by the group of duration transformations. In the most general case, we will thus look for groups $G$ such that we have a short exact sequence

$1 \to (\mathbb{R},+) \to G \to (\mathbb{R}_*^+,\times) \to 1$

There is of course the direct product $G=(\mathbb{R},+) \times (\mathbb{R}_*^+,\times)$. It is an abelian group and thus has no contextual transformations. I won’t speak much about it, but you can check Lewin’s arguments against using this group for music analysis.

More interestingly, there is also the semi-direct product $G=(\mathbb{R},+) \rtimes (\mathbb{R}_*^+,\times)$, wherein $(\mathbb{R}_*^+,\times)$ acts on $G=(\mathbb{R},+)$ by multiplication (i.e $m \to (x \to mx)$). We will call $\text{Aff}_+(\mathbb{R})$ this subgroup of the affine group of $\mathbb{R}$. Since it is a semi-direct product, we can expect different actions of a group element on a time-span, depending on whether it acts on the left or on the right. Can you already guess what would be the results ?

So, if you haven’t figured it out, here it is. We can associate to a given time-span the group element $(t,\Delta)$ of $\text{Aff}_+(\mathbb{R})$. Consider the left action of the group element $(u,\delta)$ on this time-span (we drop the distinction between set elements and group elements for clarity. See the previous posts for an explanation of this correspondence). We have

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

This action corresponds to a global dilation of the time-line around the zero-time point by a factor of $\delta$, followed by a shift of the transformed time-span by $u$ units of time.

Consider now the right action of that same element. We have

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

This action corresponds to a shift of the time-span by $u$ times its duration, followed by a dilation of its duration by a factor of $\delta$.

The right action is clearly contextual, as the amount of time shifting directly depends on the time-span (more particularly, its duration). In the case of the left action, the time-spans are all transformed in the same way. The GIS of Lewin corresponds to the right action (check it !), and he never actually talks about the left action. I don’t know if he ever considered it, but his book makes it clear that he was reluctant to the idea of having a precise reference time-span (the time-span we would identify with the identity of the group). Where would we fix, for example, the zero-time point (the onset of the initial time-span) ? The beginning of the score, or the beginning of the concert, or maybe the beginning of the universe ? He was more attracted to the right action, as in this case the determination of the interval between two time-spans always suppose that we take the first time-span as a reference (this is in fact the “change of basis” which we have seen in this post).

For example, consider the two time-spans in black below. I have chosen a particular time-span on this time-line (in orange) as the reference time-span.

The first time-span can be associated with the group element $(1,1)$, whereas the second is associated with the group element $(3,1/2)$. The interval between the second time-span and the first, in a left-action context, is the unique group element $g$ such that $g.(1,1) = (3,1/2)$. We can check that we have $g=(5/2,1/2)$.

But suppose that I’ve take another time-span as the reference, like this for example

The two time-spans are now associated with $(4,2)$ and $(8,1)$ respectively and the interval (still in a left-action context) is now $g=(6,1/2)$.

On the other hand, if we consider right actions, then the unique $g$ such that $(1,1).g = (3,1/2)$ is $g=(2,1/2)$, and we can check that $(4,2).(2,1/2) = (8,1)$. The interval is thus independent of the reference time-span we use !

The Baumslag-Solitar group BS(1,2)

So far, we have used $\text{Aff}_+(\mathbb{R})$ as our general group of transformations for time-spans. This group contains some interesting subgroups, among which the Baumslag-Solitar groups $BS(1,n)$.

We have seen before a specific group for time-spans having onsets in the dyadic rationals, i.e. the group of onset transformations is $\mathbb{Z}[\frac{1}{2}]$. Suppose now these time-spans have the durations of whole notes, half-notes, quarters, quavers, eighth note, etc. In other words, their group of durations is $(\mathbb{Z},+)$ generated by multiplication by 2. We can form a group of transformation of such time-spans by considering a semidirect product of $\mathbb{Z}[\frac{1}{2}]$ by $\mathbb{Z}$ wherein the action by automorphism is given by multiplication.

It is known that such a group is isomorphic to a HNN extension of $\mathbb{Z}$ by $\mathbb{Z}$ which has the following presentation :

$\mathbb{Z}[\frac{1}{2}] \rtimes \mathbb{Z} \cong < t,s | t^2s = st >$

This group is called the Baumslag-Solitar group $BS(1,2)$ and belongs to the family of Baumslag-Solitar groups $BS(p,q)$ which have the general presentation

$BS(p,q) = < a,b | ba^pb^{-1} = a^q >$

These groups are actively studied in geometric group theory, and it is interesting to note they appear in music theory as well. In $BS(1,2)$, the generator $t$ can be considered as the time-shift transformation, whereas the generator $s$ is responsible for the dilation of durations. If you want, you can try to determine its left and right actions on time-spans.

Remarks

You may wonder why I didn’t talk about rhythms in this post. There is actually a lot of work being done on the mathematics of rhythms, though not always from a transformational point of view, so I will probably talk about that in later posts.

In addition, the group of time-span transformations I have introduced only works for the transformation of a single time-span. A rhythm is typically made of multiple time-spans, and you could wonder if we can define transformations of rhythms as well. In fact, you can actually define global transformations of rhythms using the left action of an element of $\text{Aff}_+(\mathbb{R})$. However, things will get tricky if you try using the right action, as you may end up with overlapping time-spans.