Chord Geometry, Part III

In Chord Geometry, Part II I discussed the mathematical model of musical chords proposed by Dmitri Tymoczko of Princeton in his paper "The Geometry of Musical Chords," published in Science magazine. These remarks followed upon the introduction to the topic I made in The Geometry of Chords.

Tymoczko shows in his paper how principles of contrapuntal "voice leading" à la Bach serve also to underpin the chord-based harmonies in later Western music, right up to the dissonant, atonal compositions of the modern era. In the non-Euclidean geometric spaces Tymoczko defines to house all possible musical chords made of any number of notes of any imaginable pitch class, edge boundaries and certain other geometric loci are inhabited by chords that are perfectly symmetrical in one of three possible ways. Other chords close to these loci are nearly, but not perfectly, symmetrical. Voice leadings among such nearly symmetrical chords are favored by composers even when the music they are making is not overtly contrapuntal.

In the previous post I showed how this works for one form of (near) symmetry: permutational symmetry or P-symmetry. A 3-dimensional "orbifold" composed of all possible triads has edge boundaries. They consist of triads in which all three notes have the same pitch or pitch class. (A "pitch class" consists of a pitch and all other pitches separated from it by one or more exact octaves.) Harmonies traveling near these orbifold boundaries "bounce" off them as they progress from one nearly symmetrical triad to the next. These nearly P-symmetrical chords are characterized by their dissonance and are used in (Tymoczko says) "recent atonal composition, particularly the music of Ligeti and Lutoslawski."


What about the familiar sonorities of more consonant chordal progressions? They too involve near-symmetries of chordal geometry.

Besides P-symmetry, there are two other types of chordal symmetry, exact or near. The simpler of these two types is transpositonal symmetry, or T-symmetry. A chord that precisely divides the octave into equal parts possesses exact T-symmetry. Such a chord is the C diminished seventh. It can be represented by the ordered set (C, E♭, G♭, A) or, using Tymoczko's pitch-class designations, (0, 3, 6, 9).

Imagine these numbers in red on a clock face, with the 0 where the 12 should be. The other numbers — 1, 2, 4, 5, etc. — are in black. Now imagine rotating the clock face to bring the 3 to the top, to the spot where the 0 had been. The 6 moves to where the 3 was, the 9 to where the 6 was, and the 0 to where the 9 was. The same positions as were in red before on the clock face are still in red.

This rotation of the clock face is equivalent to subtracting 3 from (or adding -3 to) each pitch class in the original chord, using modulo-12 arithmetic in which all results are forced into the range from 0 through 11. For instance, -3 becomes 9 in modulo-12 arithmetic. Thus, (0, 3, 6, 9) becomes (-3, 0, 3, 6), which becomes (9, 0, 3, 6). Given appropriate pitch names, (9, 0, 3, 6) is equivalent to (E♭, G♭, A, C), which is sometimes spoken of as the 1st inversion of the C diminished seventh chord.

Tymoczko doesn't call this sort of transformation "inversion," though. He reserves that term for the transformation I'll discuss in the next installment. This one he calls "transposition." This is because the 1st inversion of the C diminished seventh chord is exactly the same as the E♭ diminished seventh chord when it is in root position: (E♭, G♭, A, C). Mere clock-face rotation simply transposes one chord that evenly divides the octave into another chord that also evenly divides the octave, but has a different root.

The same is true for clock-face rotations (modulo-12 additions) involving chords that do not evenly divide the octave. For example, the C major triad (C, E, G) becomes D major, or (D, F#, A), by adding 2 to each of its pitch classes. In other words, (0, 4, 7) transposes to (2, 6, 9).

Such major triads only approximately divide the octave. No clock-face rotation of a major triad can put the red numerals in the exact same positions as they were in before. This is why major triads are only nearly transposition-symmetrical, but not exactly so.

Near-symmetry makes major triads and other familiar types of chords useful in voice leading and harmonic progressions. Chords that are perfectly T-symmetrical, such as the C augmented triad, (0, 4, 8), lie at the center of their orbifold. Depending on how the orbifold is presented, this central locus can be thought of as a point or a line. All perfectly T-symmetrical chords correspond to that point or lie along that line.

Hence they are mathematically close, in that moving from one perfectly T-symmetrical chord to another can be done with maximum efficiency. "The perfectly even chord at the center of the orbifold," writes Tymoczko, "can be linked to all of its transpositions by the smallest possible bijective voice leading." ("Bijective" means that each note in the first chord leads to exactly one distinct note in the next.)

Sticking to perfectly even chords like augmented triads and diminished sevenths is not, however, good compositional strategy, since "efficient voice leadings between perfectly T-symmetrical chords are typically not independent." Voices move in parallel from one chord to the next: (0, 4, 8) transposes to (2, 6, 10), for instance. This slavish movement in parallel is considered too dull and limiting.

Once you switch to chords such as major triads that are not quite T-symmetrical, you can get quite efficient — i.e., small-step — voice leadings while preserving the independent movement of the various voices that make up the chord progression. This is how coherent progressions of common chords that are highly consonant are made.

These common chords are highly consonant because their pitch classes are harmonically related in simple ways. For example, the G pitch in the C major triad is close to 3/2 times the C pitch. (Only "close to," due to the way notes are spaced in so-called "equal temperament.")

Tymoczko's chord geometries using orbiforms express the consonance or dissonance of musical chords quite nicely in terms of how close the chords are to the central point or line of perfect T-symmetry. Chords that are consonant are close to that locus. Chords that are far away from that locus are dissonant and have to rely on near P-symmetry, as discussed in the previous installment, rather than near T-symmetry to furnish their progressions with the desired coherence.

There is also another useful type of symmetry, inversional symmetry or I-symmetry, which can apply to chords distributed not just near a central locus or edge boundary, but throughout the pitch-class space of an orbifold. "I-symmetry is exploited in both tonal and atonal music," says Tymocko. "It plays a salient role in the 19th century, particularly in the music of Schubert, Wagner, and Debussy." I'll discuss I-symmetry in the next installment.