In music theory, if you start with any note on an equal-tempered scale and go upwards by intervals of a perfect fifth, you will eventually pass through every other note in the chromatic scale before ending back on the same note. For example, if you start with the note C and move upwards in perfect fifths, you will generate the following sequence of notes:

C G D A E B F♯/G♭ C♯/D♭ G♯/A♭ D♯/E♭ A♯/B♭ F C

This is called thecircle of fifths.

There are three other ways achieve a similar full circle of notes: by moving upwards in perfect-fourth, major-seventh, and minor-second intervals. No other intervals will achieve a full circle of notes. For example, moving upwards by intervals of a minor third will only produce the following notes starting with C:

C E♭ G♭ A C

The reason has to do with the group-theoretic structure of set of all chromatic intervals. Each interval can be defined by the number of semitone intervals it contains: we denote bythe interval that contains preciselyxxsemitones. For example, the semitone (minor-second) interval is1, the whole-tone (major-second) interval is2, the minor-third interval is3, the major-third interval is4, and so on, up to the major-seventh interval11. The unison interval0is also included. Intervals can be added by simply adding the individual semitones modulo 12 – e.g.3+4=7,8+6=2, etc. Then:

The set Χ = {0,1, …,11} is a cyclic group of order 12 with respect to addition of intervals.

(I use the symbol Χ (chi) for “chromatic”. ) Now we can see why the circles of fifths, fourths, major sevenths, and minor seconds work. These are the intervals7,5,11, and1. And the numbers 1, 5, 7, 11 are precisely the positive integers that are coprime with 12 (i.e. {n∊ℤ<sup>+</sup>: gcd(n,12) = 1} = {1, 5, 7, 11}) – hence these are precisely the generators of the order-12 cyclic group ℤ/12ℤ (to which Χ is isomorphic).

Finally, note that the incomplete circle of major seconds corresponds to the whole-tone scale much favoured by the composer Claude Debussy. There are just two cycles of whole-tone notes in the scale:

M2<sub>1</sub> = {C, D, E, F♯, G♯, A♯}, M2<sub>2</sub> = {C♯, D♯, F, G, A, B}

If you start with any note in either of these cycles and move upwards by whole-tone intervals, you will generate all the notes in the same cycle but none of the notes in the other cycle. Reason:The subset of Χ generated by2is a subgroup of index 2 in Χ and M2<sub>1</sub> and M2<sub>2</sub> correspond to the two cosets of the subgroup <2> in Χ.

In the same way, since <3> is a subgroup of index 3 in Χ, there are three mutually exclusive cycles of minor-third notes:

m3<sub>1</sub> = {C, E♭, G♭, A}, m3<sub>2</sub> = {D♭, E, G, B♭}, m3<sub>3</sub> = {D, F, A♭, B}