Skip fretting system 72 2 27

One way to play 72-edo on a 36-edo guitar is to tune each pair of adjacent strings 450 cents apart. The equivalent tuning on a grid controller would be for notes to rise by 2\72 (33.3 cents) along each column and by 27\72 (450 cents) along each row.

The resulting range across six strings is 2250 cents -- close to the 2-octave spread on a 12-edo guitar in standard tuning.

72-edo approximates the 11-limit astoundingly well, and is quite good in higher limits too. 36-edo, by contrast, has a 5:4 that's 14 cents sharp (identical to 12-edo's), and an 11:8 that's 15 cents sharp.

But whereas a 72-edo guitar would not be, a 36-edo guitar is playable. Neil Haverstick does it.

Among the possible skip fretting systems for 72-edo, the 27\72 x 2\72 (or equivalently, 4.5\12 x 1\36) system is especially convenient because every ratio in the 31-limit group sans 21 can be played within a block 4 strings wide by 17 frets long. (17 frets of 36-edo is shorter than 6 frets of 12-edo.)

The same advantage holds for the 29\72 x 2\72 skip-fretting system, but fourths on adjacent strings are hard to play in that system, because the player must bend one note while playing an adjacent string at the same fret.

Here is where all the primes intervals through 31 lie:

note	fretboard position
0 steps = 1 % 1	string 0 fret 0
72 steps = 2 % 1	string 2 fret 9
42 steps = 3 % 2	string 2 fret - 6
23 steps = 5 % 4	string 1 fret - 2
58 steps = 7 % 4	string 2 fret 2
12 steps = 9 % 8	string 0 fret 6
33 steps =11 % 8	string 1 fret 3
50 steps =13 % 8	string 2 fret - 2
65 steps =15 % 8	string 3 fret -8
6 steps =17 % 16	string 0 fret 3
18 steps =19 % 16	string 0 fret 9
28 steps =21 % 16	string 0 fret 14, or string 2 fret -13, or bend up 1\72 from string 1 fret 0.
38 steps =23 % 16	string 2 fret -8
46 steps =25 % 16	string 2 fret -4
54 steps =27 % 16	string 2 fret 0
62 steps =29 % 16	string 2 fret 4
69 steps =31 % 16	string 3 fret -6

From these, the location of any compound interval N can be added by vector-summing the string-fret positions of N's factors. See Skip fretting system 48 2 13 for details on how that's done.