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.