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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 n_{\text{Maj}} a major chord whose root is the pitch-class [n], and identically, we notate n_{\text{Min}} a minor chord with the same root. We’ve seen that the T/I group is isomorphic to the D_{24} dihedral group, and is generated by two transformations, the first one being the transposition operator z, which transposes major and minor chords by one semitone :

z.n_{\text{Maj}} = (n+1)_{\text{Min}}

z.n_{\text{Min}} = (n+1)_{\text{Maj}}

The second generator is the inversion operator I_0, which switches from major to minor chords according to the following relations :

I_0.n_{\text{Maj}} = (5-n)_{\text{Min}}

I_0.n_{\text{Min}} = (5-n)_{\text{Maj}}

The group is given by the following presentation :

< z, I_0 | z^{12}=1, I_0^2=1, I_0.z.I_0 = z^{-1} > \cong D_{24}

The PLR group is also isomorphic to D_{24} and is generated by two transformations, L and R with the following action on chords :

L.n_{\text{Maj}} = (n+4)_{\text{Min}}

L.n_{\text{Min}} = (n+8)_{\text{Maj}}

and

R.n_{\text{Maj}} = (n+9)_{\text{Min}}

R.n_{\text{Min}} = (n+3)_{\text{Maj}}

The transformation P is not a generator, but is as useful as L or R in music analysis and has the following action :

P.n_{\text{Maj}} = n_{\text{Min}}

P.n_{\text{Min}} = n_{\text{Maj}}

with P = RLRLRLR = R(LR)^3. The group presentation is the following :

< L , R , P | P=R(LR)^3, (LR)^{12} = P^2 = (LRP)^2 = 1 > \cong D_{24}

It is a bit hard to find a relationship between these two groups apart from the fact that both are isomorphic to D_{24}.

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 \mathbb{Z}_{12}, 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 :

  1. As we said, we want our group to act simply transitively on the set of the 24 major/minor triads.
  2. 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 z in the T/I group.
  3. 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 I_0, then I_1 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 G of transformations. First, observe that the space of chord roots is naturally endowed with a group structure, since it is the pitch-class set [0], [1], ... with the natural action of the additive group \mathbb{Z}_{12}. Therefore, if there exists a group element z \in G which transposes major and minor triads, this means that the subgroup of G generated by z is isomorphic to \mathbb{Z}_{12}. In other words, there is an injective homomorphism i from \mathbb{Z}_{12} to G :

\mathbb{Z}_{12} \xrightarrow{i} G

Then, observe that there exists also a space of chord types, in this case a two-elements set \{\text{Maj},\text{Min}\}. We endow this set with a natural group action of \mathbb{Z}_2, wherein the elements of \mathbb{Z}_2 (which we will notate as 1_{\mathbb{Z}_2} for the identity element, and \curvearrowright for the other one) represent formal inversions. Now, as we have said in conditions 2 and 3, our group G contains either transpositions or inversions, so we have a mapping p from G to \mathbb{Z}_2. But p 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 G to be a formal inversion given by the composition of formal inversions of the individual group elements. We thus have

\mathbb{Z}_{12} \xrightarrow{i} G \xrightarrow{p} \mathbb{Z}_2

And finally, observe that transpositions, being operations which do not switch the chord type, should be mapped by p to the identity element of \mathbb{Z}_2. Hence we have Im(i)=Ker(p).

In other words, we have a short exact sequence of groups, which makes G a group extension of \mathbb{Z}_{12} by \mathbb{Z}_2 (Note: some authors refer to such a sequence as an extension of \mathbb{Z}_2 by \mathbb{Z}_{12}. I will not use this terminology here). Since \mathbb{Z}_{12} is the kernel of an homomorphism, it means that G contains a normal subgroup isomorphic to \mathbb{Z}_{12}.

The group extension structure restricts the number of possible groups: there are only 8 groups which are extensions of \mathbb{Z}_{12} by \mathbb{Z}_{2}. 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

< z, h | z^{12}=1, h^2=z^p, h^{-1}.z.h = z^q >

The 8 possible groups, with the (p,q) values are given below

  • \mathbb{Z}_{24}, (p,q)=(1,1),
  • \mathbb{Z}_{12} \times \mathbb{Z}_2, (p,q)=(0,1)
  • \mathbb{Z}_{12} \rtimes \mathbb{Z}_2=D_{24}, (p,q)=(0,11)=(0,-1)
  • \mathbb{Z}_4 \times S_3, (p,q)=(0,5)
  • \mathbb{Z}_3 \rtimes D_8, (p,q)=(0,7)
  • \mathbb{Z}_3 \times Q_8, (p,q)=(2,7)
  • \mathbb{Z}_3 \rtimes \mathbb{Z}_8, (p,q)=(3,5)
  • \mathbb{Z}_3 \rtimes Q_8 (\neq SL(2,3)), (p,q)=(6,11)=(6,-1)

We can see that we recover the D_{24} group, and the (p,q) values give the presentation we already saw for the T/I group. We also could have guessed the \mathbb{Z}_{12} \times \mathbb{Z}_2 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 P operation. The structure of the other groups is more intriguing and this is mainly due to the particular (p,q) 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 G can be written as pairs (z,h) with z \in \mathbb{Z}_{12} and h \in \mathbb{Z}_2. The composition of two elements is then given by :

(z_1,h_1) \cdot (z_2,h_2) = (z_1 \cdot \phi_{h_1}(z_2) \cdot \zeta(h_1,h_2),h_1 \cdot h_2)

where

  • \phi is an action of \mathbb{Z}_2 on \mathbb{Z}_{12}  by automorphism, i.e an homomorphism from \mathbb{Z}_2 to Aut(\mathbb{Z}_{12}).
  • \zeta is a 2-cocycle, i.e a function from \mathbb{Z}_2 \times \mathbb{Z}_2 to \mathbb{Z}_{12} satisfying
    g \cdot \zeta(h,k) + \zeta(g,hk) = \zeta(gh,k) + \zeta(g,h)

In our case, \phi is uniquely defined by the image of the non-identity element of \mathbb{Z}_2, which is mapped to one of the four automorphisms of \mathbb{Z}_{12} (since Aut(\mathbb{Z}_{12}) \cong \mathbb{Z}_2 \times \mathbb{Z}_2): z \to z, z \to 5z, z \to 7z, z \to 11z. Notice that this corresponds to the q value above. The 2-cocycle \zeta is uniquely defined by the image of the pair (\curvearrowright,\curvearrowright), which is either 1_{\mathbb{Z}_{12}}, z, z^2, z^3 or z^6 \in \mathbb{Z}_{12}. Again, notice that this corresponds to the p value above.

Let’s focus now on the D_{24} group, for which \phi maps the non-identity element of \mathbb{Z}_2 to the z \to 11z automorphism, and wherein the 2-cocycle \zeta maps to the identity of \mathbb{Z}_{12}. 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 G acts simply transitively on the set S of triads, there is a bijection between the elements of the set and the group elements of G, and this bijection is defined by the “choice of identity”, i.e which element of the set corresponds to the identity element of G. Let’s say that the C major chord, 0_{\text{Maj}} maps to 1_G = (1_{\mathbb{Z}_{12}},1_{\mathbb{Z}_2}). This means that major chords n_{\text{Maj}} are mapped to the group elements (z^n,1_{\mathbb{Z}_2}). The same can be checked for minor chords which are mapped to the group elements (z^n,\curvearrowright). We thus obtain the bijection \chi: G \to S.

To define the left action of a group element (z,h) on an element p \in S, we thus look for the group element corresponding to p, compose with (z,h) on the left, and determine the corresponding set element using \chi. In other words

(z,h).p = \chi( (z,h).\chi^{-1}(p))

As for the D_{24} group, it can readily be seen that (z,1_{\mathbb{Z}_2}) is the transposition operator, and observe that (z^5,\curvearrowright) has the following action:

(z^5,\curvearrowright).n_{\text{Maj}} = \chi( (z^5,\curvearrowright).(z^n,1_{\mathbb{Z}_2}))

= \chi( (z^5\phi_{\curvearrowright}(z^n),\curvearrowright)) = \chi((z^5z^{-n},\curvearrowright)) = (5-n)_{\text{Min}}

and

(z^5,\curvearrowright).n_{\text{Min}} = \chi( (z^5,\curvearrowright).(z^n,\curvearrowright))

= \chi( (z^5\phi_{\curvearrowright}(z^n),1_{\mathbb{Z}_2})) = \chi((z^5z^{-n},1_{\mathbb{Z}_2})) = (5-n)_{\text{Maj}}

therefore (z^5,\curvearrowright) is in fact our I_0 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 (z,h) on the right. Thus, the right group action is given by

p.(z,h) = \chi( \chi^{-1}(p).(z,h))

Still working in the D_{24} group, consider now the right action of (z^4,\curvearrowright). We have :

n_{\text{Maj}}.(z^4,\curvearrowright) = \chi((z^n,1_{\mathbb{Z}_2}).(z^4,\curvearrowright))

= \chi( (z^n\phi_{1_{\mathbb{Z}_2}}(z^4),\curvearrowright)) = \chi((z^nz^{4},\curvearrowright)) = (n+4)_{\text{Min}}

and

n_{\text{Min}}.(z^4,\curvearrowright) = \chi( (z^n,\curvearrowright).(z^4,\curvearrowright))

= \chi( (z^n\phi_{\curvearrowright}(z^4),1_{\mathbb{Z}_2})) = \chi((z^nz^{-4},1_{\mathbb{Z}_2})) = (n+8)_{\text{Maj}}

Lo and behold ! This is the action of the L operation. You can check that (z^9,\curvearrowright) acting on the right will give an action identical to that of the R operation.

We thus see now the relationship between the T/I group and the PLR group: both are isomorphic to D_{24} 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 \mathbb{Z}_{12} by \mathbb{Z}_2, 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 Z, and that similarly the chord types lie in a space endowed with a simply transitive group action of a group H. The elements of Z can be called “generalized transpositions” and the elements of H are formal inversions. If you are looking for groups G of transpositions and inversions acting simply transitively on your set of chords, then G will have the structure of a group extension of Z by H given by the short exact sequence

Z \to G \to H

For example, you could study groups of transformations in microtonal settings, with Z \cong \mathbb{Z}_{24} or \mathbb{Z}_{53} (if you like 53-equal temperament and can play the keyboard below for example).

MicrotonalKeyboard

You could also study more than two types of chords, with H \cong \mathbb{Z}_3 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 !

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