# Skip fretting system 53 3 14

A good way to play 53-edo on a 17.66-edo guitar (effectively 28edt, or an 18edo guitar with the intonation on the bridge shifted forwards so the third harmonic and the 28th fret sound the same note.) is to tune each pair of adjacent strings 14\43 apart. (That's 317 cents, a bit sharp of 6/5.) This is probably best accomplished by ear via fretting a string at the 13th fret and then tuning it to an octave above the one below.

Among the possible skip fretting systems for 53-edo, the (53,3,14) system is especially effective because every ratio in the 2.3.5.13 subgroup can be reached with a stretch of 4 frets or less. Conveniently, this is also the subgroup 53edo is best tuned in, with less than 3 cents error, audibly indistinguishable from just intonation, making the most in tune harmonies also the easiest to play. For this reason, it could also be called a catakleismic or Trinidad guitar, as it's optimised for playing music in those temperaments.

Here is where all the prime intervals lie.

note fretboard position
0 steps = 1 % 1 string 0 fret 0
53 steps = 2 % 1 string 4 fret -1
31 steps = 3 % 2 string 2 fret 1
17 steps = 5 % 4 string 1 fret 1
43 steps = 7 % 4 string 2 fret 5
24 steps = 11 % 8 string 3 fret -7
37 steps = 13 % 8 string 2 fret 3
5 steps = 17 % 16 string 1 fret -3
13 steps = 19 % 16 string 2 fret -5
28 steps = 23 % 16 string 2 fret 0
45 steps = 29 % 16 string 3 fret 1
51 steps = 31 % 16 string 3 fret 3

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).