Music Theory Meets Geometry

The theory of harmony that serves to organize most of the music of Europe and America (and increasingly of the rest of the world) is built on a couple of basic assumptions — axioms, if you will. First, there are exactly 12 pitch classes (A, B-flat, B, C, C-sharp, and so on). Second, the pitches relate to one another within a one-dimensional space. That is, they’re laid out in a line, which conventionally runs from side to side with lower pitches to the left and higher pitches to the right.

The relations among pitch classes, which are what harmony theory is about, all take place within this one-dimensional matrix. Octave equivalence (transposition) and the Circle of Fifths introduce wrinkles, but the wrinkles can easily be mapped onto the one-dimensional layout.

To be sure, the frequency spectrum, in which pitches are defined by their number of vibrations per second, is one-dimensional. The fact that harmony theory defines relations in one dimension is not wrong. But it’s a limitation conceptually.

Guitarists play in two dimensions. Because of the layout of the fretboard, they sometimes discover scale and chord relationships that keyboard players fail to notice. They do it by moving geometrical patterns across from one string to another, as well as up or down the neck of the guitar.

Harmony theory in two dimensions turns out to be quite interesting. When we add the possibility that there may be more than 12 pitch classes within the octave, matters become very interesting indeed.

The ancestor of my new Z-Board is the Z-tar, a MIDI interface designed for guitarists by Harvey Starr. The Z-Board goes a bit further than the Z-tar in that it has 12 horizontal rows of keys, not just six. But the idea is similar: You can move up in pitch either by moving to the right, or by moving across from one row of keys to the next.

Scales and chord voicings are two-dimensional structures. They can be moved (transposed) vertically, horizontally, or diagonally. Of course, any movement translates into an upward or downward shift in absolute pitch. In theory, you could accomplish exactly the same thing on a one-dimensional keyboard. But you wouldn’t, because the patterns would be hard to see and too spread out to play.

Any key on the Z-Board can be assigned any MIDI note number, so the nature of the 2D pitch grid is entirely arbitrary. As a starting point, it might be laid out exactly like a guitar fretboard, with half-steps from left to right and intervals of a perfect fourth (five half-steps) vertically.

Because I’m interested in microtonal scales with more than 12 equal-tempered steps per octave, it occurred to me that a slightly different layout might be more useful. So I created a key map in which the vertical intervals are five chromatic steps (I call them chroma-steps), as on a guitar, but the left-to-right interval is two chroma-steps per key rather than one. Playing a chromatic scale on this layout requires a zigzag movement, but how often do you play chromatic scales? On a 5×2 grid, chord voicings and useful scales lie more neatly under the hand.

[Edit: Having experimented with this grid for a few days, I backed up and tried the default 5×1 grid, as on a guitar tuned in fourths. The guitar grid, I’ve found, is much easier to wrap one’s brains around, at least in 19-tone. The diagram below, then, is deprecated.]

The 19-note equal-tempered tuning has embedded within it a very familiar-sounding diatonic major scale. In this case, however, the whole-steps are three chroma-steps wide, while the half-steps (more accurately, 2/3 steps) are two chroma-steps wide. When you map this major scale onto the 5×2 grid, it looks like this:

5x2 19 maj scale

The pattern is not hard to see: Ascending whole-steps are diagonals upward and to the left, while “half-steps” are horizontal moves to the right.

But why limit ourselves to familiar major scales? The strength of two-dimensional harmony theory is that any pattern is potentially interesting. Eighty years of jazz have taught us that almost any combination of pitches within the 12-note tuning can be exciting. Jazz has largely erased the notions of consonance and dissonance, which guided earlier classical theory.

An added attraction of a 19-note equal-tempered scale is that 19 is a prime number. In the 12-note tuning we all know and love, moving up or down by any number of chroma-steps except 5 or 7 repeats at the octave, and without using all of the notes. If we move up by 3’s, we get a diminished 7th chord, because 12 is divisible by 3 — and also by 2, 4, and 6. But in 19-tone, no matter what interval our pattern uses for repetition, the pattern has to cycle through all 19 pitch classes before it returns to its starting point.

As a result, scale patterns on the 2D grid generally spiral off into space harmonically rather than repeating at the octave. Evocative? I think so.

There is, as yet, no coherent harmony theory that would describe the 19-tone tuning at all, much less describe it in terms of patterns on a two-dimensional matrix. But that’s part of the fun, isn’t it? How many entirely new chords did you discover this week? I discovered several dozen.

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