Skip fretting system 58 2 15

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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 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 lowest harmonics lie

The diagram below (or at right, or somewhere) is of a hypothetical 12-string guitar in this tuning. It shows where each of the odd harmonics through the 29th lies. 1 represents octave equivalents of the root, 3 represents octave equivalents of the 3rd harmonic (3:2, 3:1, 3:4, etc.), etc.

Where the harmonics lie in the 15\58 x 2\58 isomorphic layout

Since 58-edo is consistent in the 17-limit + 29 group, the location of any interval in that group can be determined from the corresponding harmonics. (For instance, to rise by 7:5, go up two strings and down one fret, because that takes you from harmonic 5 to harmonic 7.)

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".)