User:Vector/Vector's intro to 22edo/Chains of intervals
In 12edo, there are only the following chains which cycle through all intervals: the chain of fifths and the chain of fourths, which are basically the same thing and close as circles of length 12. All other circles of intervals (besides just stepping up every note in order) return to the original note before reaching all the others.
This is a result of the fact that only 1, 5, 7, and 11 are coprime to 12, meaning that there is only a single meaningful circle. However, 22edo has as its coprimes 1, 3, 5, 7, 9, 13, 15, 17, 19, and 21. Once 1 and 21 are removed, this means that there are a total of four distinct chains of intervals to examine.
Note that for reasons specified in User:Vector/Vector's intro to 22edo/The diatonic scale, note names have not been provided here, and will generally be avoided outside of designated notation sections. Instead, interval names following ADIN will be used.
Circle of fifths (9, 13)
| Edostep | Interval | Edostep | Interval | ||
|---|---|---|---|---|---|
| -11 | 11 | Tritone | 0 | 0 | Unison/Octave |
| -10 | 2 | Pentaminor 2nd | 1 | 13 | Perfect 5th |
| -9 | 15 | Pentaminor 6th | 2 | 4 | Supermajor 2nd |
| -8 | 6 | Pentaminor 3rd | 3 | 17 | Supermajor 6th |
| -7 | 19 | Pentaminor 7th | 4 | 8 | Supermajor 3rd |
| -6 | 10 | Penta4th | 5 | 21 | Supermajor 7th |
| -5 | 1 | Subminor 2nd | 6 | 12 | Penta5th |
| -4 | 14 | Subminor 6th | 7 | 3 | Pentamajor 2nd |
| -3 | 5 | Subminor 3rd | 8 | 16 | Pentamajor 6th |
| -2 | 18 | Subminor 7th | 9 | 7 | Pentamajor 3rd |
| -1 | 9 | Perfect 4th | 10 | 20 | Pentamajor 7th |
| 0 | 0 | Unison/Octave | 11 | 11 | Tritone |
This is the basis of the Pythagorean diatonic scale.
Circle of Pentamajor 3rds (7, 15)
| Edostep | Interval | Edostep | Interval | ||
|---|---|---|---|---|---|
| -11 | 11 | Tritone | 0 | 0 | Unison/Octave |
| -10 | 18 | Subminor 7th | 1 | 7 | Pentamajor 3rd |
| -9 | 3 | Pentamajor 2nd | 2 | 14 | Subminor 6th |
| -8 | 10 | Penta4th | 3 | 21 | Supermajor 7th |
| -7 | 17 | Supermajor 6th | 4 | 6 | Pentaminor 3rd |
| -6 | 2 | Pentaminor 2nd | 5 | 13 | Perfect 5th |
| -5 | 9 | Perfect 4th | 6 | 20 | Pentamajor 7th |
| -4 | 16 | Pentamajor 6th | 7 | 5 | Subminor 3rd |
| -3 | 1 | Subminor 2nd | 8 | 12 | Penta5th |
| -2 | 8 | Supermajor 3rd | 9 | 19 | Pentaminor 7th |
| -1 | 15 | Pentaminor 6th | 10 | 4 | Supermajor 2nd |
| 0 | 0 | Unison/Octave | 11 | 11 | Tritone |
The circle of pentamajor 3rds reaches the perfect fifth after 5 steps, but it also has a notable tendency to "cluster" around the major third and minor sixth, forming cycles of 3 with a slight offset each time.
Circle of Subminor 3rds (5, 17)
| Edostep | Interval | Edostep | Interval | ||
|---|---|---|---|---|---|
| -11 | 11 | Tritone | 0 | 0 | Unison/Octave |
| -10 | 16 | Pentamajor 6th | 1 | 5 | Subminor 3rd |
| -9 | 21 | Supermajor 7th | 2 | 10 | Penta4th |
| -8 | 4 | Supermajor 2nd | 3 | 15 | Pentaminor 6th |
| -7 | 9 | Perfect 4th | 4 | 20 | Pentamajor 7th |
| -6 | 14 | Subminor 6th | 5 | 3 | Pentamajor 2nd |
| -5 | 19 | Pentaminor 7th | 6 | 8 | Supermajor 3rd |
| -4 | 2 | Pentaminor 2nd | 7 | 13 | Perfect 5th |
| -3 | 7 | Pentamajor 3rd | 8 | 18 | Subminor 7th |
| -2 | 12 | Penta5th | 9 | 1 | Subminor 2nd |
| -1 | 17 | Supermajor 6th | 10 | 6 | Pentaminor 3rd |
| 0 | 0 | Unison/Octave | 11 | 11 | Tritone |
The circle of subminor thirds reaches the perfect fifth after 7 steps, first reaching a sharp "superfourth" after 2 steps.
Circle of Pentamajor 2nds (3, 19)
| Edostep | Interval | Edostep | Interval | ||
|---|---|---|---|---|---|
| -11 | 11 | Tritone | 0 | 0 | Unison/Octave |
| -10 | 14 | Subminor 6th | 1 | 3 | Pentamajor 2nd |
| -9 | 17 | Supermajor 6th | 2 | 6 | Pentaminor 3rd |
| -8 | 20 | Pentamajor 7th | 3 | 9 | Perfect 4th |
| -7 | 1 | Subminor 2nd | 4 | 12 | Penta5th |
| -6 | 4 | Supermajor 2nd | 5 | 15 | Pentaminor 6th |
| -5 | 7 | Pentamajor 3rd | 6 | 18 | Subminor 7th |
| -4 | 10 | Penta4th | 7 | 21 | Supermajor 7th |
| -3 | 13 | Perfect 5th | 8 | 2 | Pentaminor 2nd |
| -2 | 16 | Pentamajor 6th | 9 | 5 | Subminor 3rd |
| -1 | 19 | Pentaminor 7th | 10 | 8 | Supermajor 3rd |
| 0 | 0 | Unison/Octave | 11 | 11 | Tritone |
This can be seen as one of the two counterparts to the whole-tone scale in 22edo (but see User:Vector/Vector's intro to 22edo/The diatonic scale). Note that this can also be seen as a circle of chromatic semitones, i.e. the intervals reached by adding successive sharps. As can be seen here, the classical minor third is equivalent to a double-sharp.
