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I have introduced before the notion of Generalized Interval System (GIS) (and its mathematical aspect), an I’ve promised I would introduce a GIS for chords, in particular major and minor chords. For that, we need to introduce a few more musical notions.

By now, you should be familiar with the circle of semitones (see picture below), which represents pitch classes assuming octave equivalence. Here, the pitch classes are represented by their name, but I will also number them using the notation [n] and assuming that C = [0]. Therefore C\# = [1], D=[2] and so on.

CircleTones

We have seen that there is a natural GIS associated with the set of pitch-classes: if we take the cyclic group of order 12 \mathbb{Z}_{12} = < z | z^{12}=1 >, there is a natural action of the generator z on pitch classes given by z.[n] = [n+1] \pmod {12}. Elements of the group are called transpositions: z is the operation which transposes pitch classes by one semitone.

Besides transpositions, musicians are also interested in another class of operations which are called inversions. As their name indicates, these operations invert a pitch class (pc) around another reference pc (you can also think of inversions as “turning upside-down”). For example, let’s consider the inversion operator I_0 (the reason for this notation will become clear soon) which inverts pcs around C. Naturally, the action of I_0 on C results in C. If I_0 acts on C\#, we will obtain B, and if it acts on D, we will obtain A\#, and so on. Mathematically, we have I_0.[n] = [-n] \pmod {12}. This can be represented graphically as pictured below :

InversionI0What happens if apply I_0 twice ? You can easily check that you will obtain the same pc you began with. In other terms, we have (I_0)^2=1, i.e the group generated by I_0 is cyclic of order 2. When such operations are their own inverses, we call them involutions.

So what if we want a group with both transpositions and inversions ? We could certainly add the element I_0 to \mathbb{Z}_{12}, but to respect the definition of a mathematical group, we need to add more elements, for example (z.I_0)(z^2.I_0), …, (I_0.z.I_0), etc.

Let’s check what those elements correspond to. Take the group element (z.I_0) for example. It takes a pitch class [n], invert it around [0] to give [-n] (all operations are now understood modulo 12) and then transposes it by one semitone, thus giving [-n+1]. So the action of (z.I_0) on pc [n] gives pc [-n+1], which can be represented graphically as

InversionI1This is another inversion ! We’ll call this one I_1. You can check that all elements of the form (z^p.I_0) are in fact inversions, and we will notate them as (z^p.I_0) = I_p. Since there are 12 transpositions, we thus obtain 12 inversion operators.

But wait, there is more. What happens if we consider the element (I_0.z.I_0) ? Since (z.I_0) = I_1 is an inversion, this is in fact I_0.I_1. Intuitively, we may think that inversions of inversions cancel out. Let’s check: we have (z.I_0).[n] = [-n+1], thus (I_0.z.I_0) = [n-1]. Lo and behold ! This is a transposition ! And this is z^{11}=z^{-1} to be exact.

This means that we have the relation I_0.z.I_0 = z^{-1} and that the 12 inversion operators are all that we need to add to form a group. Indeed, we the above relation, you can reduce any operation to either a transposition or an inversion. To make it clearer, we have

  • z^p.z^q = z^{p+q}
  • z^p.I_q = z^p.z^q.I_0 = I_{p+q}
  • I_p.z^q = z^p.I_0.z^q = z^p.z^{-q}.I_0 = I_{p-q}
  • I_p.I_q = z^p.I_0.z^q.I_0 = z^{p-q}

The new group G we have formed is therefore generated by z and I_0 with the relations

G = < z, I_0 | z^{12}=1, I_0^2=1, I_0.z.I_0 = z^{-1} >

Mathematicians are familiar with this type of group structure. Indeed, this is definition for the dihedral group D_{24} of order 24 (it has 24 elements, 12 transpositions and 12 inversions). In fact, you could have worked with any possible division of the octave, starting with a group of n transpositions, and adding inversions to obtain the dihedral group D_{2n} of order 2n.

So, we had before a group of transpositions acting on the set of pitch classes, and we now have a new group D_{24} which also has an action on this set. Can we form a GIS with D_{24} ?

Hint: NO. This should be pretty obvious: to pass from pitch-class to another, there may exist more than one possible group element. For example, to go from C to D, you can use z^2 or I_2. What good is this group for then ?

It gets really interesting when one considers chords instead of just pitch classes. Chords are, at first, a collection of pitch classes. Take for example, a C-major chord: it is comprised of pitch classes C, E and G. The C-minor chord includes C, Eb=D# and G. In our circle of semitones, we can picture chords as such

C-Maj

A C-major chord

Cs-Maj

A C#-major chord

C-Min

A C-minor chord

Mathematically, we will notate a major chord as a set of pitch classes of the form n_{\text{Maj}} = [n, n+4, n+7]. Similarly, a minor chord will be notated as n_{\text{Min}} = [n, n+3, n+7]. In the case of major and minor chords, we always make a reference to the root of the chord: a C-major chord, an A-minor chord, etc. In our notation, this root is n. If we encounter an unordered set, for example [0, 9, 5], we will always try to find the correct form for the chord: here it would be [5, 9, 0], an F-major chord. Some chords don’t have a clearly identifiable root, for example triads of the form [n, n+4, n+8], but we will not consider these chords for now. Let’s concentrate on major and minor chords.

The first thing to notice is that we can transpose chords the same way we do with pitch classes. Indeed, we have an action of the transposition operator z on major minor chords as

z.n_{\text{Maj}} = z.[n, n+4, n+7] \\ = [(n+1), (n+1)+4, (n+1)+7] = (n+1)_{\text{Maj}} \\

and

z.n_{\text{Min}} = z.[n, n+3, n+7] \\ = [(n+1), (n+1)+3, (n+1)+7] = (n+1)_{\text{Min}} \\

Hence, we have 12 major chords and 12 minor chords.

And what about inversions ? In a similar manner, we will apply inversion operators to each pitch-class of the chord. This is more easily understood on the picture below, where I_0 acts on a C-major chordChordInversionWe can observe that I_0 inverts the type of the chord: applied on a C-major chord, we obtain an F-minor chord. Since I_0 is an involution, if we apply it on an F-minor chord, we will obtain a C-major chord again. What is the general action on a chord n_{\text{Maj}} or n_{\text{Min}} ? You can check that we have I_0.n_{\text{Maj}} = (5-n)_{\text{Min}} and that similarly, I_0.n_{\text{Min}} = (5-n)_{\text{Maj}}.

So we have an action of D_{24} on the set of the 24 major and minor chords. And now for the good part: this action is simply transitive ! (Prove it !). Hence, the group D_{24} and the set of major and minor chords form a GIS ! This means we can analyze any progression of chords using unique group elements, transpositions and inversions, from D_{24}.

To conclude, I’d like to give such an example, which I take from Lewin’s book:

In it, Lewin analyzes Stockhausen’s Klavierstück III with the tools we have introduced. Here is the beginning of Klavierstück III, which shows only the pitch classes, without the rhythmStockhausen

Lewin considers a particular pentachord, i.e a chord containing five pitch classes, of the form pictured below

PentachordAlphaand its inversion

PentachordBetaand segments the beginning of Klavierstück III so that the progression of these pentachords clearly appear :LewinStockhausenAnalys

Observe the use of inversion and transpositions to pass from one chord to another.

We have seen in this post the action of D_{24} on the set of major and minor chords, through transpositions and inversions. However, there is also another action of this group in which the operations are called contextual operations. Though it is the isomorphic to D_{24}, the group is then named the PLR-group. We’ll see what it does in a next post.

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