Musical Symmetry Revisited

Symmetric 8-note scale

In a blog post last year I talked about symmetric musical sets – such as the tritone, the augmented triad, the diminished 7th, the whole-tone scale and the chromatic scale – which divide the octave into 2, 3, 4, 6 and 12 equal parts respectively. For various reasons musicians dislike these groupings, so they’re used very sparingly in classical music and virtually never in pop music. But as someone who’s always been more into maths than music, I’m fascinated by any kind of symmetry.

Traditionally the octave is divided into 12 semitones, so the symmetric sets I just mentioned are the only possible ones. But what if you wanted to divide the octave into 8 equal parts? That seems an obvious choice, because it’s what the word octave implies. But to do it we need to invoke quarter tones. There are 24 of these in an octave, and 24 divided by 8 is 3, so we’re looking for notes 3 quarter tones (or one and a half semitones) apart.

Writing music in quarter tones isn’t easy, because the MIDI format defines pitch as an integer number of semitones. But it does allow something called “pitch bending” (presumably to simulate bending the string of a guitar), and with a bit of patience you can use that feature to raise the necessary notes by a quarter tone.

Here’s a short (1 minute) piece I wrote to see what it would sound like. It’s basically a random composition using the 8 equally spaced notes shown in the diagram above.