The PLR group was the last thing we studied regarding the possible (simply transitive) actions on the set of the 24 major and minor triads. There is much to say about this group and its use in music analysis, but I will keep it for later. In this post, I’d like to go into details about the deep structure of the T/I group and the PLR group and the connection with a subject known to mathematicians as group extensions. The ideas I’m going to expose are taken from a paper I’ve published recently in Journal of Mathematics and Music :
- “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).
Summing-up known facts
First, let’s recapitulate what we’ve seen so far. We notate a major chord whose root is the pitch-class , and identically, we notate a minor chord with the same root. We’ve seen that the T/I group is isomorphic to the dihedral group, and is generated by two transformations, the first one being the transposition operator , which transposes major and minor chords by one semitone :
The second generator is the inversion operator , which switches from major to minor chords according to the following relations :
The group is given by the following presentation :
The PLR group is also isomorphic to and is generated by two transformations, and with the following action on chords :
The transformation is not a generator, but is as useful as or in music analysis and has the following action :
with . The group presentation is the following :
It is a bit hard to find a relationship between these two groups apart from the fact that both are isomorphic to .
Construction of a group of transformations
Now, let’s begin everything from scratch. We have a set which consists of 12 major triads, one for each pitch-class in , and 12 minor triads (idem), and we would like to know what are the groups which acts simply transitively on this 24-elements set. Obviously, the order of the group has to be 24 and any computational group theory software will tell you that there exists 15 non-isomorphic groups of that order.
However, we are going to put some conditions on the possible groups :
- As we said, we want our group to act simply transitively on the set of the 24 major/minor triads.
- Then, we would like to have a group element which fullfills the same role as the transposition operator in the T/I group. In other words, the action of this group element on the major and minor triads should be similar to that of in the T/I group.
- Finally, our group should also have group elements whose action on the triads switches their type, i.e from major to minor, or minor to major. Also, if we compose such elements, the “inversion type” (i.e whether it is an operation which inverts triads or not) of the resulting group element should correspond to the composition of the “inversion types” of the individual elements. For example, if we apply , then in the T/I group, we obtain a transposition, since an inversion of an inversion does not switch the chord type. Notice that, for those group elements which switch chords, we haven’t say anything about the root change.
These conditions seem reasonable, yet they lead to a very strong structure of the resulting group of transformations. First, observe that the space of chord roots is naturally endowed with a group structure, since it is the pitch-class set with the natural action of the additive group . Therefore, if there exists a group element which transposes major and minor triads, this means that the subgroup of generated by is isomorphic to . In other words, there is an injective homomorphism from to :
Then, observe that there exists also a space of chord types, in this case a two-elements set . We endow this set with a natural group action of , wherein the elements of (which we will notate as for the identity element, and for the other one) represent formal inversions. Now, as we have said in conditions 2 and 3, our group contains either transpositions or inversions, so we have a mapping from to . But is much more than a set-theoretic mapping ! It is in fact an homomorphism (a surjective one), since we want the composition of group elements from to be a formal inversion given by the composition of formal inversions of the individual group elements. We thus have
And finally, observe that transpositions, being operations which do not switch the chord type, should be mapped by to the identity element of . Hence we have .
In other words, we have a short exact sequence of groups, which makes a group extension of by (Note: some authors refer to such a sequence as an extension of by . I will not use this terminology here). Since is the kernel of an homomorphism, it means that contains a normal subgroup isomorphic to .
The group extension structure restricts the number of possible groups: there are only 8 groups which are extensions of by . We can even give their presentation, thanks to a result by Hempel
- “Metacyclic groups”, C. E. Hempel, Communications in Algebra 28/8 (2007), pp. 3865-3897
which states that their presentation is given by
The 8 possible groups, with the values are given below
We can see that we recover the group, and the values give the presentation we already saw for the T/I group. We also could have guessed the group, as it is the group generated by a transposition operator and a very simple inversion operator whose action is identical to that of the operation. The structure of the other groups is more intriguing and this is mainly due to the particular values.
Group operation and group actions
The group extension structure also gives a group operation which has tremendously useful consequences. Indeed, it is a known fact that group elements in can be written as pairs with and . The composition of two elements is then given by :
- is an action of on by automorphism, i.e an homomorphism from to .
- is a 2-cocycle, i.e a function from to satisfying
In our case, is uniquely defined by the image of the non-identity element of , which is mapped to one of the four automorphisms of (since ): , , , . Notice that this corresponds to the value above. The 2-cocycle is uniquely defined by the image of the pair , which is either , , , or . Again, notice that this corresponds to the value above.
Let’s focus now on the group, for which maps the non-identity element of to the automorphism, and wherein the 2-cocycle maps to the identity of . So now, we have a group of transformations and we now how to compose group elements. How can we define an action of this group on our set of triads ? Remember from the previous posts that we can either have a left or a right action, both being acceptable to define a Generalized Interval System (GIS). Let’s begin with the left action.
Left group actions
Remember that, since acts simply transitively on the set of triads, there is a bijection between the elements of the set and the group elements of , and this bijection is defined by the “choice of identity”, i.e which element of the set corresponds to the identity element of . Let’s say that the C major chord, maps to . This means that major chords are mapped to the group elements . The same can be checked for minor chords which are mapped to the group elements . We thus obtain the bijection .
To define the left action of a group element on an element , we thus look for the group element corresponding to , compose with on the left, and determine the corresponding set element using . In other words
As for the group, it can readily be seen that is the transposition operator, and observe that has the following action:
therefore is in fact our inversion operator. We thus recover our T/I group !
Right group actions
The procedure for right group actions is similar, but we will compose with on the right. Thus, the right group action is given by
Still working in the group, consider now the right action of . We have :
Lo and behold ! This is the action of the operation. You can check that acting on the right will give an action identical to that of the operation.
We thus see now the relationship between the T/I group and the PLR group: both are isomorphic to but they differ in their action, which is either a left action or a right action. Remember that the operations of the PLR group were called contextual. This stems directly from the difference between left and right actions, and I’ll explain it in another post, though you may already have a clue about it now.
Generalization and conclusions
I have mainly discussed about group extensions of by , but the construction is in fact very general. Assume you have a set of chords, each one having a unique root and type. Assume that the roots lie in a space endowed with a simply transitive group action of a group , and that similarly the chord types lie in a space endowed with a simply transitive group action of a group . The elements of can be called “generalized transpositions” and the elements of are formal inversions. If you are looking for groups of transpositions and inversions acting simply transitively on your set of chords, then G will have the structure of a group extension of by given by the short exact sequence
For example, you could study groups of transformations in microtonal settings, with or (if you like 53-equal temperament and can play the keyboard below for example).
You could also study more than two types of chords, with for example.
But what applies to chords can in fact apply to many other musical objects ! In the paper cited above, there is an application of group extensions to durations and rhythms, which I will explain in another post.
Oh, and for the first post of 2014, happy new year !