Skip fretting system 31 2 9

Guitars with all notes of 31-edo exist and are playable, especially in the lower registers. However, in the higher registers one must use a fingernail to fret, which makes speed difficult, and things like sweep-picking impossible. Using the (31,2,9) skip-fretting system in the higher registers could alleviate that problem.

The system involves including only every other fret of 31-edo, with each pair of adjacent strings tuned 9\31 apart. That's 348.4 cents, a very neutral third. Among the other possible skip fretting systems for 31-edo, the (31,2,9) system is especially convenient in that every interval in the 3.7.11.13.23.29 group spans at most 4 frets. Intervals involving the fifth harmonic are not extremely convenient, because the major third spans four frets and lies on the same string as the perfect fifth. This layout restricts certain harmonic possibilities -- but if frets were only skipped in the higher registers, those harmonic possibilities would remain in the lower registers, which is arguably the only place they can be played anyway.

Where the notes lie

As a diagram

In the folowing 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.

            headstock this side
            -  -  5  -  -  -  -
            7  -  -  -  -  -  -
            -  9 11  -  -  5  -
            -  -  -  7  -  -  -
    bass    1  -  3  -  9 11  -  treble
   strings  -  -  -  -  -  -  7  strings
            -  - 13  1  -  3  -
            -  -  -  -  -  -  -
            -  -  -  -  - 13  1
            5  -  -  -  -  -  -
            -  -  -  -  -  -  -
             bridge this side

As a table

note	fretboard position
0 steps = 1 % 1	string 0 fret 0
31 steps = 2 % 1	string 3 fret 2
18 steps = 3 % 2	string 2 fret 0
10 steps = 5 % 4	string 2 fret - 4
25 steps = 7 % 4	string 3 fret - 1
14 steps = 11 % 8	string 2 fret - 2
22 steps = 13 % 8	string 2 fret 2
3 steps = 17 % 16	string - 1 fret 6
8 steps = 19 % 16	string 0 fret 4
16 steps = 23 % 16	string 2 fret - 1
27 steps = 29 % 16	string 3 fret 0
30 steps = 31 % 16	string 2 fret 6

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.