# How to identify the two copies of $D_{24}$ in the homomorphisms of the 2 musical actions? [closed]

Let $$S$$ be the set of minor and major triads. Two sets of actions are defined on the set:

1) Musical transposition and inversion;

2) P, L, R actions $$P(C-major) = c-minor,$$ $$L(C-major) = e-minor,$$ $$R(C-major) = a-minor.$$

I already know that each action can be described as a homomorphism from our group into $$Sym(S)$$ ($$S_n$$). I just don't really know how to identify these 'distinguished copies'.

Apparently, each of these homomorphisms (of action 1 and 2) is an embedding so that we have two distinguished copies, $$H_1$$ and $$H_2$$, of the dihedral group of order 24 in $$Sym(S)$$. This is the duality in music described by David Lewin.

"The two group actions are dual in the sense that each of these subgroups $$H_1$$ and $$H_2$$ of $$Sym(S)$$ is the centralizer of the other!"

These notions are defined in the paper Musical actions of dihedral groups by Crans, Fiore, and Satyendra.

• You didn't fully describe P,L,R; e.g. even assuming that P takes "X-major" to "X-minor" (or "x-minor" if you prefer) for each of the 12 X's, you didn't tell us what P does to any "X-minor". I guess that each of P,L,R is an "involution", i.e. each of PP,LL,RR is the identity, so for instance P takes each "X-minor" back to "X-major", and likewise L(E-minor) = C-major and R(A-minor) = C-major. Is this what you intend? [P and R would then stand for "parallel" and "relative" in the music-theory sense; I don't remember what L would be.] Jun 6 '20 at 13:03

Understanding this action from what you said requires a basic understanding of the objects of music theory as members of a set with some structure. I’ll try and give the briefest of explanations possible.

The set of pitch classes in (Western chromatic) harmony is a twelve-element set, which we will identify with the set of integers modulo 12. The natural cyclic ordering on the set of pitch classes agrees with the cyclic ordering on $$C_{12} = \mathbb{Z}/12\mathbb{Z}$$, so we will use it.

An interval is a pair of pitch classes $$\{a,b\}$$, and its quality is the quantity $$q(a,b) = |a - b|$$ mod 12. A minor third is an interval of quality 3, and a major third is an interval of quality 4. A triad is an ordered triple of pitch classes $$(a,b,c)$$. We are concerned with major and minor triads. A triad is major if $$q(a,b) = 4$$ and $$q(b,c) = 3$$, and a triad is minor if $$q(a,b) = 3$$ and $$q(b,c) = 4$$. (In other words, we always have $$q(a,c) = 7$$, and we call the triad major or minor depending on the quality of the interval $$\{a,b\}$$.) In music theory there are other triads, but for this answer I will use triad as a shorthand for major or minor triad.

The set of triads is a 24-element set, for a triad $$(a,b,c)$$ is completely characterized by its root note $$a$$ and its quality, major or minor. Thus it is convenient to parametrize triads as elements of the set $$C_{12}\times C_2$$. For some reason I would like to think of $$C_2$$ as the group of units of the ring $$\mathbb{Z}$$, forgive me. If $$(a,b,c)$$ is a major triad, it corresponds to the element $$(a,+1) \in C_{12}\times C_2$$. Likewise if $$(a,b,c)$$ is a minor triad, it corresponds to $$(a,-1)$$.

The action of transposition and inversion is simple to describe: consider the group of bijection of $$C_{12}\times C_2$$ generated by

$$\begin{cases}\iota\colon (a,\pm 1) \mapsto (-c,\mp 1) = (-a-7,\mp 1) \\ \tau\colon (a,\pm 1) \mapsto (a+1,\pm 1). \end{cases}$$ (Here and throughout addition and subtraction are mod 12.)

Note that $$\iota$$ has order $$2$$, $$\tau$$ has order $$12$$, and that $$\iota\tau\iota (a,\pm 1) = \iota\tau(-a-7,\mp 1) = \iota(-a-6,\mp 1) = (a-1,\pm 1) = \tau^{-1}(a,\pm 1),$$ so this defines an action of the dihedral group of order 24—which I am fond of writing as $$D_{12}$$ because of its action on a $$12$$-element set, forgive me—on the set of triads.

The “P/L/R” action is maybe slightly more involved.

The parallel major/minor of a triad $$(a,\pm 1) \in C_{12}\times C_2$$ is the triad $$P(a,t) = (a,\mp 1)$$. Thus $$P^2 = 1$$, and $$P(0,3,7) = (0,4,7)$$.

The leading-tone exchange of a major triad $$(a,b,c)$$ is the triad $$L(a,b,c) = (b,c,a-1)$$. Thus if $$(a,b,c)$$ is a major triad, $$L(a,+1) = (b,-1)$$. The leading-tone exchange of a minor triad $$(a,b,c)$$ is the triad $$L(a,b,c) = (c+1,a,b)$$. So if $$(a,b,c)$$ is a minor triad, then $$L(a,-1) = (c+1,+1)$$. One checks that $$L^2 =1$$.

The relative major of a minor triad $$(a,b,c)$$ is the triad $$R(a,b,c) = (b,c,b+7)$$—that is, if $$(a,b,c)$$ is a minor triad, $$R(a,-1) = (b,+1)$$. The relative minor of a major triad is the inverse operation: $$R(a,b,c) = (b-7,a,b)$$, so if $$(a,b,c)$$ is a major triad, $$R(a,+1) = (b-7,-1)$$. Similarly one checks that $$R^2 = 1$$.

I claim that the action of $$RL$$ on major triads is given by $$RL = \tau^7$$. (We are acting on the left.) Indeed, suppose $$(a,b,c)$$ is a major triad. Then $$L(a,+1) = (b,-1)$$, which is the minor triad $$(b,c,b+7)$$, so $$RL(a,+1) = R(b,-1) = (c,+1)$$.

On the other hand, the action of $$LR$$ on minor triads is given by $$LR = \tau^7$$ (so on minor triads we have $$RL = (LR)^{-1} = \tau^{-7}$$). Indeed, suppose $$(a,b,c)$$ is a minor triad. Then $$R(a,-1) = (b,+1)$$, which is the major triad $$(b,c,b+7)$$, so $$LR(a,-1) = L(b,+1) = (c,-1)$$. Thus in both cases, the quality is preserved, and the root note has shifted up $$7$$ pitch classes.

Since $$7$$ is relatively prime to $$12$$, $$\langle R,L \rangle$$ is a quotient of a group with presentation $$\langle x,y \mid x^2 = y^2 = (xy)^{12} = 1\rangle$$, and the latter is a presentation of $$D_{12}$$. I suppose one has to show that the action is faithful; it is, but I’ll leave that to you.

I claim further that $$L$$ and $$R$$ are sufficient to recover $$P$$. I don’t immediately see a slick way of doing this, so I’ll leave it to you.

To show that each centralizes the other, it suffices to show that $$\tau R = R\tau$$, $$\tau L = L\tau$$, $$\iota R = R\iota$$, and finally $$\iota L = L\iota$$. Let me maybe just talk through a tiny piece.

Suppose first $$(a,b,c)$$ is a minor triad. Then

$$\tau R (a,-1) = \tau(b,+1) = (b+1,+1) = R(a+1,-1) = R\tau(a,-1).$$ The argument for a major triad is analogous.

On the other hand, note that $$L(a,-1) = (a-4,+1)$$ and $$L(a,+1) = (a+4,-1)$$. Thus we have $$\iota L(a,-1) = \iota(a-4,+1) = (-a-3,-1) = L(-a-7,+1) = \iota(a,-1).$$ The argument for a major triad is analogous.

Of course, even buying the gaps I’ve left, this doesn’t show that these copies of $$D_{12}$$ are the full centralizer of each other in $$S_{24}$$, but hopefully you have some ideas about how to prove it.

• One small nitpick: on my reading $q(a,c) = 7$ does not follow from the conditions: we could have $q(a,b) = 4$, $q(b,c) = 3$ but $q(a,c) = 1$. For example, take $(0,4,1)$, where $q(0,4) = |0-4| = 4$ (major), $q(4,1) = |4-1| = 3$, so this fits the definition of a 'major triad'. So I think we need to add $q(a,c) = 7$ to get triads in the required musical sense. Jun 6 '20 at 18:08
• @MarkWildon Sure, yes, that’s clearer. I meant for it to be implicit in “ordered”, but I imagine that wasn’t clear Jun 7 '20 at 23:19