# Cycles of Intervals in 31 Tone Equal Temperament

Most musicians are familiar with the circle of fifths and corresponding circle of fourths, which are the only cycles of intervals that can cycle through all 12 notes of 12-ET. Using other intervals in 12-ET, the original note is reached again before cycling through all notes.

The number 31 is prime, which means that any musical interval can be used to cycle completely through all notes.

## Circle of Fifths in 31-ET

To figure out the circle of fifths, we begin as normal. In 12-ET, B# is the same as C, so we are back to the beginning; in 31-ET, however, it is not, so the cycle continues. When we reach double-sharps, we can proceed by identifying enharmonic equivalents:

C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#-F##-...

B# is the same as Cd (G semiflat) and F## is the same as Gd, so we proceed:

B? is the same as Cb, so:

...A?-E?-Cb-Gb-Db-Ab-Eb-Bb-F-C

We have gone full-circle. The circle is thus:

## Circle of Whole Tones

Whole tones more quickly reach unfamiliar intervals than fourths or fifths, in 31-ET, but they still start by going through some familiar intervals. The sequence begins:

C-D-E-F#-G#-A#-B#-Dd,...

The intervals (relative to the starting note) cycled through progress: major third, lesser septimal tritone, septimal minor sixth, septimal minor seventh, diesis (shy of the octave), and so forth...

## Circle of Minor Thirds

The circle of minor thirds is an interesting feature of 31-ET. Like major thirds, this is an interval that cycles through all intervals in 31-ET, but does not have this property in 12-ET. The minor third is particularly interesting in that it is a very familiar melodic interval, yet its progression in 31-ET quickly reaches intervals that are unfamiliar to 12-ET and most Western music.

C-Eb-Gb-Bbb-...

Already, cycling through only four notes, or making three modulations or chord changes has led to an interval, the septimal major sixth (12:7 ratio), that is not in 12-ET. This interval is followed by the diesis (above the starting pitch), which is followed by the neutral third above the starting pitch, and then the greater undecimal tritone above the starting pitch.

Cycling through minor thirds is thus an easy way to use a familiar interval to quickly enter new, less familiar territory.

## Cycle of Major Thirds

Like the cycle of minor thirds, the cycle of major thirds also uses a familiar interval to quickly reach unfamiliar territory in 31-ET:

C-E-G#-...

Merely stacking two major thirds yields an interval not found in 12-ET: the septimal (14:9) or undecimal (11:7) minor sixth, which is the inversion of the septimal or undecimal major third. This difference reflects the fact that major thirds in 31-ET are flatter than the exactly-one-third-of-the-octave approximation in 31-ET.

Following this interval is an interval one step (one diesis) shy of the octave, followed by a neutral third above the original note, and then the particularly awkward an unfamiliar semi-augmented fifth, which does not have a natural harmonic function in 31-ET.

I personally find the cycle of major thirds in 31-ET to be more xenharmonic than the cycle of minor thirds, in part because the quickly-reached semi-augmented fifth ends up sounding just like an awkwardly out-of-tune fifth.

## Cycling with Diatonic Semitones

Following the pattern above of adding fifths to reach new intervals, the next logical cycle is cycling through diatonic semitones (or their inversion, diatonic major sevenths).

This cycle quickly leads to new intervals: a septimal whole tone, followed by a neutral third, a semi-diminished fourth, lesser septimal tritone, perfect fifth, then a minor sixth, septimal major sixth, neutral seventh, and then one diesis short of an octave, and continuing through more intervals.

I find this pattern interesting in that it uses a moderately familiar interval (i.e. one that most people familiar with 12-ET will hear as a basic "half-step") but quickly achieves things that cannot be done in 12-ET, going through unfamiliar territory only to reach a perfect fifth by stacking six half-steps, and jumping back into more unfamiliar territory.