Skip fretting system 31 3 7
This fretting system follows both Orwell temperament with the interval between each string, 7\31, and Miracle temperament with the fret spacing, which is even wider than 12edo, at 3\31 = 116.13¢. While there are other skip frettings for 31edo (see 31 2 9), this one is of particular interest. Its fret spacing just happens to be extremely close to every other fret of the 41 2 9 skip fretting (117.07¢). As a result, any Kite guitar can be retuned to within a few cents of 62edo. See Tuning A Kite Guitar To 31edo or 62edo.
Where the notes lie
As a diagram
In the following the strings are vertical and the frets are horizontal. 1 represents octave equivalents of the root, 3 represents octave equivalents of the 3rd harmonic (3:2, 3:1, 3:4, etc.), etc.
nut
- - - - - - - -
- - - 9 - - - -
- - - - - 13 - 9
- - - - 3 7 - -
bass - 1 - 11 - - - - treble
strings - - 5 - - 1 - 11 strings
- - - - - - 5 -
- - - - - - - -
9 - - - - - - -
- - - - 9 - - -
- - - - - - - -
bridge
9 is shown twice because it's such an outlier. Note the awkwardness of playing do-re-mi (1-9-5).
As a table
| interval | fretboard vector
(strings, frets) |
|---|---|
| unison | +3, -7 |
| 2/1 = 31\31 | +4, +1 |
| 3/2 = 18\31 | +3, -1 |
| 5/4 = 10\31 | +1, +1 |
| 7/4 = 25\31 | +4, -1 |
| 11/8 = 14\31 | +2, 0 |
| 13/8 = 22\31 | +4, -3 |
From these, the location of any compound interval can be added by vector-summing the string-fret positions of the interval's factors. See Skip fretting system 48 2 13 for details on how that's done.