Skip fretting system 140 8 37

A good way to play 140-edo on a 17.5edo guitar is to tune each pair of adjacent strings 37\140 apart. (That's 317 cents, a bit sharp of 6/5.) This is essentially an extension of the same kleismic system used in skip fretting systems such as 34 2 9 and 53 3 14 with greater sophistication. 53edo has a near perfect 3/2, 87edo has a near perfect 5/4. 140edo is their sum and falls between them, tuning both with less than a cent of error, doing the same with 7/4 and 13/8 into the bargain and being just 0.008 cents off the 15-limit minimax tuning. This makes it one of the most efficient and accurate ways of approximating just intonation with a subset of an equal temperament, while still making 5-limit chords easy to play.

It is not without limitations though, as you will need a fretboard with a full two octave range to be able to access all notes, and each of the 140 notes only appears in precisely one place, making many combinations of notes impossible to play for a single person. This also means you will need an electronic tuner to tune it, as there are no unisons to be found anywhere unless you have at least 9 strings, while octaves only appear every 4th string. You can make many notes easier to access by using a 35 note per octave fretboard instead, but this nears the limit of practicality for a physical instrument, making chords even more challenging.

Here is where all the prime intervals lie.

note	fretboard position
0 steps = 1 % 1	string 0 fret 0
140 steps = 2 % 1	string 4 fret -1
82 steps = 3 % 2	string 2 fret 1
45 steps = 5 % 4	string 1 fret 1
113 steps = 7 % 4	string 5 fret -9
64 steps = 11 % 8	string 0 fret 8
98 steps = 13 % 8	string 2 fret 3
12 steps = 17 % 16	string 4 fret -16
35 steps = 19 % 16	string -1 fret 9
73 steps = 23 % 16	string 5 fret -14
120 steps = 29 % 16	string 0 fret 15
134 steps = 31 % 16	string 6 fret -11

From these, the location of a compound intervals N can be added by vector-summing the string-fret positions of N's factors. For instance, since 3%2 lies at (string 2, fret 1) and 5%4 lies at (string 1, fret 1), their product 15%8 lies at (string 3, fret 2).