Chord Geometry, Part IV

This is the fourth in a series of posts discussing the mathematical model of musical chords proposed by Dmitri Tymoczko of Princeton in his paper "The Geometry of Musical Chords," published in Science magazine. The first in the series was The Geometry of Chords. The most recent was Chord Geometry, Part III.

In that most recent installment, I went into the concept of T-symmetrical chords in some depth. T-symmetry (short for transpositional symmetry) means that a chord precisely divides the octave into equal parts. That is, the intervals between the notes of the chord, including that from the top note of the chord up to the root note one octave higher, are equal in size.

Chords that are nearly, but not precisely, T-symmetrical are much used by composers. They tend to manifest pleasing harmonic consonances, while they "can be linked to at least some of their transpositions by efficient bijective voice leadings." The latter characteristic, all-important musically, means that the separate "voices" that produce the various notes of successive chords, as those voices move independently of the other chordal voices from one chord to the next, can do so by covering relatively short (i.e., efficient) pitch distances. At bottom, that is what can give any piece of Western music its coherence.

In Chord Geometry, Part II, I explored another kind of symmetry, P- (for permutational) symmetry. Perfectly P-symmetrical chords have all their notes in the same "pitch class." For instance, a triad made of three C notes — however odd such a chord may seem — is exactly P-symmetrical. {B, C, D♭} is, on the other hand, only nearly P-symmetrical, making it and like clusters of closely spaced notes good candidates for compositions that exploit atonality and harmonic dissonance. Again, voice leadings among such chords tend to be efficient and thus musically coherent.


Tymoczko describes one more type of chordal symmetry: I-symmetry, for inversion symmetry. "A chord is inversionally symmetrical (I-symmetrical) if it is invariant under reflection in pitch-class space," he says.

To the right is a movie showing how the C major triad can be inverted to become its reflection in pitch-class space, F minor. C major is, in terms of numerical pitch classes, {0, 4, 7}. F minor is {0, 5, 8}. In the movie, the notes of the C major triad are represented as if they were "hours" on a clock face, except that there is a 0 at the top of the clock instead of a 12, and hour-numerals that are not part of the chord are not shown. When C major is inverted to F minor, the clock face is in effect flipped over by means of pivoting it around the 0-6 axis. Now the hours that are shown change from 0, 4, and 7 to 0, 5, and 8.

C major and F minor are not exactly I-symmetrical. If they were, the inversion would look just like the original. But C major and F minor qualify as being nearly I-symmetrical.

You can change the C major triad into a perfectly I-symmetrical one just by raising the middle note from pitch class 4 to pitch class 5 — E becomes E#, or F — making the chord into a C suspended triad.

Inverting C suspended yields C suspended again, so C suspended is a perfectly I-symmetrical triad.





These ideas about chord symmetry flow from musical set theory. Set theory as it applies to music, says Wikipedia, "deals with collections of pitches and pitch classes, which may be ordered or unordered, and which can be related by musical operations such as transposition, inversion, and complementation."

In the examples given above, the operation called inversion is front and center.



Here is another movie, this one of a seventh chord and its inversion. The original chord is not one with a common name. It can be called "C 0_2_4_6." The inversion has the same pattern, but now with F# as the root. You can call it "F# 6_8_10_0," since F# is of pitch class 6, where C is of pitch class 0.

"C 0_2_4_6" does not initially appear to be I-symmetrical. That's because the pivot point is 0 in the movie. Pivoting the clock face around the axis running through 0 (and also through 6) seems to give a completely different arrangement of pitches. But notice that "F# 6_8_10_0," if rotated one half turn of the clock face, looks just like "C 0_2_4_6." You do the math by adding 6 to each note in {6, 8, 10, 0}. Then, since you are using modulo-12 arithmetic, if the result is 12 or higher, subtract 12:

• (6 + 6) mod 12 = 12 mod 12 = 12 - 12 = 0
• (8 + 6) mod 12 = 14 mod 12 = 14 - 12 = 2
• (10 + 6) mod 12 = 16 mod 12 = 16 - 12 = 4
• (0 + 6) mod 12 = 6 mod 12 = 6

Mathematically, this is rotation. Musically, it is transposition. "F# 6_8_10_0" rotates/transposes to become the original chord, "C 0_2_4_6." The inversion of the original chord is identical to the original under rotation/transposition. Ergo, the original chord was I-symmetrical after all.


Below is a movie, a swatch from a longer movie of Mr. Tymoczko's, that shows why this is important:




















There are two progressions, each with two chords. The first two-chord progression is from an F# half-diminished seventh chord, {F#, A, C, E} or {6, 9, 0, 4}, to an F dominant seventh, {F, A, C, E♭} or {5, 9, 0, 3}. This example comes from Wagner's Parsifal via the Tymoczko paper.

The second two-chord progression, also cited in Tymoczko's paper, is from Debussy's Prelude à l'après-midi d'un faune. It changes from A# half-diminished seventh, {A#, C#, E, G#} or {10, 1, 4, 8}, to B♭dominant seventh, {B♭, D, F, A♭} or {10, 2, 5, 8}.

In both examples, a four-note chord that is nearly but not exactly I-symmetrical shifts to a second four-note chord that, again, is almost but not quite I-symmetrical.

In the first example, (6, 9, 0, 4) patterns with (0, 2, 5, 8), which can be thought of as its chord-structure template. (The parentheses mean the order of the numbers is significant. Curly braces would mean note order is not important.) If you add 8 to each number in (6, 9, 0, 4) — then subtract 12 if the result is 12 or higher — you get (2, 5, 8, 0). That can be reordered to (0, 2, 5, 8).

Likewise, (5, 9, 0, 3) patterns with (0, 3, 6, 8). You add 3 to each number — subtracting 12 is not necessary here — then do the proper reordering of the result: (8, 0, 3, 6) —> (0, 3, 6, 8).

It can be demonstrated that the second example likewise involves a progression between two chords whose chord-structure templates compute to (0, 2, 5, 8) and (0, 3, 6, 8), respectively.

This is why the orange ball in the diagram on the left side of the movie moves between balls labeled "0 2 5 8" and "0 3 6 8" ... in both examples!

Admittedly, it is hard to read the numbers on the various balls in the diagram; try clicking on the movie to see a larger version. The reddish ball in the center of the cluster is labeled "0 3 6 9," which means it represents all four-note chords whose chord-structure template is (0, 3, 6, 9). All (0, 3, 6, 9) chords are I-symmetrical — not just nearly so, but exactly (hence the reddish color of the ball). Here we are ignoring the fact that (0, 3, 6, 9) chords also happen to be perfectly T-symmetrical, by the way. Chords with structure (0, 2, 5, 8) and those with structure (0, 3, 6, 8) both "live right next door" to (0, 3, 6, 9) chords in Tymoczko's geometrical chord space, known as an "orbiform."

There are four flavors of (0, 2, 5, 8) chords that can be derived by minimally altering (0, 3, 6, 9), just as there are four flavors of (0, 3, 6, 8) chords ... which is why each of these chord classes is represented four times in the diagram, while the perfectly I-symmetrical (0, 3, 6, 9) appears but once. This is not tremendously important in the present context. The key thing is that the line segment between (0, 3, 6, 9) and each particular (0, 2, 5, 8) flavor is short, as is the line segment between (0, 3, 6, 9) and each particular (0, 3, 6, 8) flavor.

Accordingly, the chord progression from a (0, 2, 5, 8) chord to a (0, 3, 6, 8) chord traverses, in effect, just two of these short segments: one segment to arrive at (0, 3, 6, 9), and a second segment to arrive at (0, 3, 6, 8). Such a progression, in fact, represents an "efficient voice leading" and is therefore musically coherent.

In more practical terms, (F#, A, C, E) in the first example moves to (F, A, C, E♭) by lowering its bottom and top voices a semitone apiece while holding its second and third — or inner — voices at their original pitches. In the second example, (C#, E, G#, A#) becomes (D, F, A♭, B♭) as the lower two voices ascend one semitone apiece while the upper two voices don't change in pitch.

So the math discussed above is a roundabout way of showing that the specifics of these two actual chord progressions are equivalent in crucial ways. They both represent efficient, hence coherent, voice leadings because they both move between two chords that each live next door to the same perfectly I-symmetrical chord — namely, the one whose structure pattern is (0, 3, 6, 9).

Wagner and Debussy may not have known anything about the mathematics involved, but they clearly grasped their import.