Harmony

Tried to learn a little about harmony, and got distracted by math again. Maybe if I wrote some down I could stop going in circles (nyuk nyuk).

Let's talk notes and their relationships. Consider the twelve even tempered tones of the western scale, from C up to D#, ignoring what octave they are in. (In math language, I'd say we are identifying pitches separated by an octave; WP says they are now a pitch class, and that I'm now talking about what a psycho-acoustics person might call chroma rather than note. Musically, we're just ignoring the inversions, to keep things simpler for now.) We can arrange these notes around a circle in two ways: with adjacent notes separated by a semitone, ie the chromatic circle, or separated by a fifth, in the circle of fifths. If you take steps around the notes by semitones or fifths, you end up back where you started after twelve steps. In math language, we can say that these notes, and the intervals connecting them, are isomorphic to addition modulo 12, and that the notes form a cyclic group. The semitone and the fifth can each be considered generators of the group, ie a step that will take us all the way around. Their inverses are the reverse steps, which for the semitone we can call -1 (if a semitone is +1) or equivalently +11 (these are the same, modulo 12, just like a major 7th up and a half step down take you to the same note, ignoring the octave). The inverse of a fifth is a fourth, ie a fifth up = +7 semitones is cancelled by a fourth up = +5 semitones = -7. These are the only generators of the whole group, which for complicated reasons Wikipedia wants to call Z/12Z, or maybe C₁₂. What happens if you try iterating one of the other intervals?

Well, the whole group is cyclic, so you're going to get something cyclic, it just won't be the whole group. We can call what you do get a subgroup. For example, a minor 3rd = 3 generates the following group starting from 0: 0, 3, 6, 9; which is a dim7 chord, the tetrad consisting of only minor 3rds. This subgroup is C₄. Musically, we have three of these depending on which note you begin on. The other intervals also generate subgroups: C₆, the whole tone scale, generated by a whole step; there are two of these; C₃, the augmented triad generated by a major 3rd, there are four of these, and C₂, the tritone dyad generated by the tritone; there are six of these. And that's it: that's all the patterns we can make by fixed step sizes among the twelve tones.