Skip fretting system 58 2 15
One way to play 58-edo on a 29-edo guitar is to tune each pair of adjacent strings 15\58 apart. That's about 310.3 cents, or 5.3 cents flat of 6:5.
Among the possible skip fretting systems for 58-edo, the (58,2,15) system is especially convenient in that every 7-limit interval spans at most 3 frets, and every interval in the 2.3.5.7.13.23 subgroupspans at most 4 frets. As it makes it particularly easy to play music composed using myna temperament, it could also be called a myna guitar.
Where the first primes intervals 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 on this side
- - - - 13 - - - - 1 - -
13 - - - - 1 - - - - - -
- 1 - - - - - - 7 - 5 3
- - - - 7 - 5 3 - - - -
7 - 5 3 - - - - - - - -
- - - - - - - - - 9 - -
- - - - - 9 - - - - 11 -
bass - 9 - - - - 11 - - - - - treble
strings - - 11 - - - - - - - - - strings
- - - - - - - - - - - -
- - - - - - - - - - - -
- - - - - - - - - - - -
- - - - - - - - - - - -
- - - - - - - - - - 13 -
- - - - - - 13 - - - - 1
bridge on this side
As a table
| note | fretboard position |
|---|---|
| 0 steps = 1 % 1 | string 0 fret 0 |
| 58 steps = 2 % 1 | string 4 fret - 1 |
| 34 steps = 3 % 2 | string 2 fret 2 |
| 19 steps = 5 % 4 | string 1 fret 2 |
| 47 steps = 7 % 4 | string 3 fret 1 |
| 27 steps = 11 % 8 | string 1 fret 6 |
| 41 steps = 13 % 8 | string 3 fret - 2 |
| 5 steps = 17 % 16 | string - 1 fret 10 |
| 14 steps = 19 % 16 | string 0 fret 7 |
| 30 steps = 23 % 16 | string 2 fret 0 |
| 50 steps = 29 % 16 | string 2 fret 10 |
| 55 steps = 31 % 16 | string 3 fret 5 |
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.
Adaptation to Mystery Temperament
The same guitar with 29-edo frets can be played in an arbitrary mystery temperament by retuning adjacent strings to alternating tunings. For example, 315 cents as the approximate 6:5 and 306 cents as the residue to make skip strings 15 steps of 29-edo. Such a tuning gives improved approximations to just intonation at the expense of half of all intervals and half of all chords being incorrect and more out of tune then 58-edo.
Comparison to 31-edo
A 29-edo guitar is not much easier to play than a 31-edo guitar. In particular you'll need to use your fingernail to fret the highest notes. A skip-fretting system is substantially more confusing than one that includes every note. While 58-edo is more faithful to the harmonic series, 31-edo is nonetheless exceptionally good.
Comparison to the Kite tuning
The Kite Guitar uses 20.5 frets per octave to make a 41-edo instrument, and its frets are wide enough to be quite playable. If a standard guitar's scale length is 25", then an instrument using the 15\58 x 2\58 tuning described in this article would have a minimum fret width equal to the Kite Guitar's if it had a scale length of 25" * 29 / 20.5 = 35.4 inches. That is, while a 29-edo guitar might be difficult to play, a long-scale tapping instrument in 29-edo would be as playable as a Kite Guitar. (The most well-known tapping instrument, the Chapman Stick, has a scale length of 36".)