Skip fretting system 53 3 17
This layout allows someone to play in 53-edo on a 17.666-edo guitar, by tuning the guitar in major thirds -- that is, with 17\53 between each pair of adjacent strings. It offers a big range -- very slightly wider than the Kite Guitar's -- and a playable layout, with strikingly easy 5-limit chords.
The diagram below, which could be interpreted 20 frets of a 12-string guitar, shows where each of the 15-limit harmonics lies. Since 53-edo is mostly consistent in the 15-limit (see below for the exception), these harmonics' positions imply where every interval in that group lies. For instance, to play 6:5 requires moving up (toward the treble side) one string and down (toward the nut) one fret.
The only exception to the above procedure is the ratios 11/7 (and its octave-complement 14/11). Since 11:8 is 7.9 cents flat and 7:4 is 4.8 cents sharp in 53-edo, the distance between them is 7.9 + 4.8 = 12.7 cents too wide. 12.7 cents is more than half of 53-edo's step size of 22.6 cents. Thus whereas the best approximations to 11:8 and 7:4 (and hence the diagram below) suggest that 11:7 is 34 steps wide, in fact 53-edo's best approximation to 11:7 is 35 steps wide. But both approximations are almost equally wrong, just in opposite directions.
20 frets of a hypothetical 12-string guitar tuned this way:
headstock
this side
1 5 - - - - - - - - - -
- - 13 - - - - - - - - 3
- - - - - - - - - 1 5 -
9 - 7 - 11 - - - - - - 13
- - - - - - - - - - - -
- - - - - - - - - 9 - 7
- - - - - - - 3 15 - - -
bass - - - - - 1 5 - - - - - treble
strings 11 - - - - - - 13 - - - - strings
this - - - - - - - - - - - - this
side - - - - - 9 - 7 - 11 - - side
- - - 3 15 - - - - - - -
- 1 5 - - - - - - - - -
- - - 13 - - - - - - - -
- - - - - - - - - - 1 5
- 9 - 7 - 11 - - - - - -
15 - - - - - - - - - - -
- - - - - - - - - - 9 -
- - - - - - - - 3 15 - -
- - - - - - 1 5 - - - -
The (few?) harmonically difficult ratios have easy octave counterparts
One notable drawback to this tuning that, because harmonics 3 and 7 lie on the same string, a harmonic 7:6 is difficult to play. (Doing so requires reaching back 13 frets, or 883 cents, and across three strings.) However, the ratio 7:3 (an octave wider than 7:6) is unusually easy to play, being 3 string crossings and 1 fret wide. Following the same logic, for every difficult interval R less than an octave, it can be shown that R plus an octave is easy to play.
There seem to be very few such difficult ratios in the 15-limit. I (Jeff Brown) only see five: 7:6, 13:12, 14:13, 9:8, and 11:9.