algebra, art, music

Mathematical Music Theory 4: Intro to Post-Tonal Methods

All right. Let’s skip from first week of first year of music theory to something that probably won’t come up until a music theory elective in your third year (yes, we’re skipping two full years of theory here).

What I’m going to describe is usually encountered in a class on “Post-tonal theory.” This can be misleading because for the most part it is an extremely useful mathematical way of thinking about music theory that doesn’t particularly have to do with atonal music or 12-tone serialism.

As we’ve already pointed out our Western 12-tone scale is essentially taking an octave and dividing it up into 12 parts. Since an octave (or 12 semitones up or down) gives the same note we can mathematically think of things more clearly by just labeling a C with 0, a C# with 1, a D with 2 and so on up to labeling a B with 11. When we back to C we “wrap around” and call it 0 again.

A great way to visualize this is to draw a 12-sided figure with all the side lengths the same (a regular dodecagon). Now if we take a C major chord: 0, 4, 7, then transposing it to a major chord 3 semitones up just amounts to adding every number by 3, i.e. 3, 7, 10. In fact, given any set of notes, we have the operation of transposition T_n (i_1, i_2, \ldots, i_k)=(i_1+n, i_2+n, \ldots, i_k +n) \ \text{mod} \ 12 where mod 12 means we add by wrapping around and consider 12=0, 13=1, 14=2, etc (because they are the same notes!!).

We can also do something called inversion. This just amounts to exactly inverting every interval. This amounts to negating every single number and then figuring out what this number is mod 12. So the inversion of the C major chord: [0, 4, 7] is [0, -4, -7]=[0, 8, 5] or if we really are considering “chords” then the order doesn’t matter so it is [0,5,8]. But this is just an f minor chord! We call this operation I for “inversion.” It can be visualized as a reflection of the dodecagon as follows (don’t make fun, I whipped this together using Google draw in a minute or so):

It is pretty clear that doing T_n for all choices of n to [0,4,7] gives you all 12 majors chords and if you do both I and T_n then you’ll get all 12 minor chords too. The operations of transpositions and inversions forms something called a group. In fact, visualizing with a regular 12-gon immediately tells us that the T/I group is what mathematicians call D_{12} the Dihedral group of symmetries of the dodecagon. It has 24 elements.

We call an unordered collection of numbers between 0 and 12 a pitch class set, and we get that D_{12} acts on the set of pitch class sets. We just proved that the orbit of [0,4,7] under this action consists of exactly the collection of major and minor triads. Note that none of the triads are sent to themselves, i.e. given a non-trivial symmetry/combination of transpositions and inversions we will always get a distinct new triad. Mathematicians might say this in a fancy way: the set of major and minor triads is a torsor under the T/I-action.

It turns out this is a “generic” phenomenon in the sense that choosing some random pitch class set you are likely (in that the probability is greater than 50%) to have chosen one that has this property. We could say that it has the property of having no T/I-symmetry. Conversely, we could call a k-chord (read: an unordered chord with k notes in it) T/I-symmetric if there is some choice of non-trivial transposition and inversion such that the chord is sent to itself.

Now even though these are more rare, it turns out that for any choice of k, there is always a k-chord with this property. These exist for rather silly reasons. For example, [0,1,2, … , k] is always an example of such a chord (exercise: why?). For less trivial examples you could take the whole-tone scale [0,2,4,6,8,10]. If you do T_2 then you certainly get the whole tone scale back again. Inversion also fixes this 6-chord. This tells us that up to inversion and transposition there are only 2 distinct whole tone scales (if you want overkill then the subgroup generated by T_2 and I has 12 elements, so the Orbit-Stabilizer Theorem tells us this fact).

Here is an interesting question from pure music theory that to my knowledge is still open (although I suspect it is fairly easy to answer and if I spent time trying to figure out the answer in place of writing this post I’d have the answer). None of this was specific to dividing up an octave into 12 notes. Suppose you invent a tonal system with n notes instead. Then you’d have an action of D_n on the k-chords. Is there a simple closed form formula for the number of k-chords that are T/I-symmetric? More importantly, for a given n, which k gives the most number of k-chords with T/I-symmetry.

I should point out that if you rule out the “silly examples” of T/I-symmetry given by a strictly chromatic scale, then there is actually utility in figuring this out. T/I-symmetry has played a great role in the history of composition. For example, the augmented triad, the French augmented sixth chord, the diminished seventh, the famous chord from Stravinsky’s Petrushka, the hexatonic scale, the whole tone scale, and the octatonic scale are all examples. So I think this is more than just a novelty problem.

Mathematical Music Theory:

Part 1: The Overtone Series

Part 2: Derivation of the Western Scale System

Part 3: Combination Tones

Part 4: Intro to Post-tonal Methods

(Part 5:) The Stack of Pitch Class Sets


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