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In our last post, we have seen that mathematical groups arise quite naturally in music. Today, I’d like to go deeper that way and introduce new groups for some various musical structures. I haven’t yet explained why groups are useful in the first place, apart from saying that they are related to the work of David Lewin, and I will come back to this important point in a later post. From now on, I’ll assume that the reader is familiar with basic group theory.

So, we have seen that an infinite piano keyboard (or an infinite fretted guitar neck… ok I’ll stop with that goofy example) comes with a natural group structure, in which the group is isomorphic to $\mathbb{Z} = < T >$, the additive group of the integers generated by $T$, the operation of “passing one key”. In our previous example, the elements of this group are the operations $T^n$: “passing $n$ keys, $n \in \mathbb{Z}$” (Note: in my previous post, I used the notation $T_n$ for group elements, but I’m going to use the more traditional multiplicative notation for group elements). Musicians call these operations transpositions. On a usual piano keyboard, we transpose the notes by half-tones, but you could also consider microtonal transpositions if you have the appropriate keyboard.

Now before we continue, one word about tunings. I’m assuming here that your piano keyboard is tuned to a twelve-tone equal temperament, meaning that we consider that the octave is divided equally in 12 half-tones. This is not the only tuning for musical instruments. In fact, baroque music often used very different tunings, such as the Pythagorean tuning, or just intonation tunings. In these tunings, the octave is not divided equally, and therefore group operations for transpositions of notes cannot be defined as easily. Tunings have a lots of interesting mathematical aspects, though it is not my goal to discuss them in this post. If you want to know more, you can check the following books :

• “Tuning and Temperament: A Historical Survey”, by J.M. Barbour, readily available from quite any bookstore.
• “Mathématiques des Systèmes Acoustiques”, by F. Jedrzejewski (in French), which describes a huge collection of different tuning systems.

So from now on and in the next posts as well, I’ll assume we are working in an equal temperament, so that transpositions can be defined easily, and more particularly in 12-TET (twelve tone equal temperament). You could work in other $n$-TETs if you want. By the way, have you ever heard 19-TET in action ? Check out the audio reconstruction by Roger Wibberley (and his associated work) of Seigneur Dieu ta pitié, written originally in 19-TET by Guillaume Costeley in 1570 (yes, XVIth century !).

So far, we spoke mainly about notes and half-tones, but other musical objects can be considered as well. Take for example note durations. We have whole notes, half notes, quarter notes, eigth notes, sixteenth notes, etc. The obvious pattern behind these durations is multiplication (or division) by 2.

Here too, we have a group, which is the subgroup of the multiplicative group of the integers generated by multiplication by 2, which is therefore isomorphic to the additive group $\mathbb{Z}$. But what if we want to include dotted notes or triplets ? Then we would have to be able to multiply a note duration by any possible rational value, which would give us $\mathbb{Q}$.

Or what if we forget about the durations and we consider instead the position of a quarter note in time ? Let’s pick a duration as our reference unit duration. Below is a picture in which the arrow is the arrow of time, and wherein beats have been marked (1 beat = 1 unit duration).

A quarter note sits on one beat. How do we move it to the next beat ? We introduce an operation which is “moving a note by one beat”, and similarly to notes and half-tones, we’ll call that operation a (temporal) transposition. The group generated by this operation is obviously (again) $\mathbb{Z}$.

Let’s return now to notes. As you probably know, if you play all keys in order on your keyboard, you’ll notice that notes repeat themselves every 12 keys, albeit one or more octave higher or lower. We can assume that two notes an octave or more apart are equivalent (mathematically, this is an equivalence relation), and we will thus “fold” our keyboard into a circle, the circle of 12 tones which is represented below, assuming enharmonic equivalence (since we work in 12-TET).

This circle of 12 tones also has an underlying group structure, namely $\mathbb{Z} / {12}\mathbb{Z}$, the cyclic group with 12 elements. The presentation of this group is :

$\mathbb{Z} / {12}\mathbb{Z} = < T | T^{12}=1>$

wherein $T$, the generator of this group, is the operation “transposing by one half-tone”. The added relation $T^{12}=1$ expresses the fact that applying $T$ 12 times, we end up exactly on the same note. What if you want to apply it 13 times ? Since you are working in $\mathbb{Z} / {12}\mathbb{Z}$, you have to use modular arithmetic. In this case, $13 \equiv 1 \pmod {12}$, so $T^{13} = T$. If you chose to work in 19-TET instead, then you would have to consider $\mathbb{Z} / {19}\mathbb{Z}$ in which $T^{19}=1$, and operations are performed modulo 19.

The cyclic group $\mathbb{Z} / {12}\mathbb{Z}$ is one of the most basic group structures in mathematical music theory, and it pops up everywhere. In our next posts, we’ll see what other groups can be considered, especially for tones.

Now for those who want more, I’m going to give you a little bonus. At the same time, I’m going to raise the math level just a little bit. We have seen above that the temporal transposition by one beat (one unit duration) generates a group isomorphic to $\mathbb{Z}$. For reasons which will become clear soon, let’s call this transposition $T_0$ (not to be confused with $T^0=1$; the $0$ is now an index). In our graphic example, the quarter note sits exactly on a beat, but what if we want to move by half a beat ? Or a quarter of a beat ? Like the examples below :

How can we do that with the operation $T_0$ ? The short answer is: we can’t. The operation $T_0$ only moves notes by one whole beat. So we need to introduce another operation $T_1$, which moves notes by half a beat. But we won’t be able to move the note by a quarter beat, then, so we need another operation $T_2$ for that, and another one $T_3$ for an eigth of a beat, and so on… In fact we have to introduce an infinity of operations, $T_i, \forall i \in \mathbb{N}$, to cover all possible transpositions by a fraction $\dfrac{1}{2^i}, i \in \mathbb{N}$ of a beat. However, we are not finished yet, for if we consider the group $G = < T_0, T_1, T_2, ... >$, generated by all $T_i$, we see that this is an infinitely generated free group, which is missing some crucial relations. Indeed, what happens if we move twice a note by half a beat ? Then we will actually have moved the note by a beat, so we have $T_1^{2}=T_0$. In fact, we have $T_{i+1}^{2}=T_i$ for any $i \in \mathbb{N}$, so we have to consider the group

$G = < T_0, T_1, T_2, ... | T_{i+1}^{2}=T_i, \forall i \in \mathbb{N}>$

This group is actually well-known: it is in fact isomorphic to $\mathbb{Z}[1/2]$, the additive group of the dyadic rationals, i.e numbers of the form $\dfrac{a}{2^b}, a \in \mathbb{Z}, b \in \mathbb{N}$. The dyadic rationals are exactly the position a note can take when transposed temporally by the $T_i$. Algebraically, $\mathbb{Z}[1/2]$ is obtained as the direct limit of a system of groups, in this case the groups $\mathbb{Z}$ corresponding to individual transpositions by the $T_i$, where in the generators are amalgamated according to the above relations. Of course, you can define similar groups if you want to work with transpositions by a different fraction of the unit duration. We’ll hear more about this group when we’ll investigate the group structure associated with time-spans.