Skip fretting system 46 2 11

A good way to play in 46-edo on a stringed instrument is with a 23-edo fretboard and strings tuned 11\46 apart.

layout

In the following diagram, the headstock is toward the top, the bridge toward the bottom, and the lower (thicker) strings are on the left side. Each nuber represents a harmonic modulo the octave -- so 3 represents 3:2, 15 represents 15:8, etc. The harmonics drawn are the ones that 46-edo approximates best. A trailing _ indicates that the harmonic lies an octave lower than it would otherwise (so, e.g., 15:16 rather than 15:8), and a trailing ' indicates the reverse (so, e.g., 17:8 rather than 17:16).

  -- -- -- 3  --
  15_-- -- -- --
  31_-- -- -- 15
  1  -- -- -- 31
  -- -- 23 -- 2
  17 5  -- 7  --
  -- -- -- -- 17'
  9  -- -- -- --
  -- 11 13 -- 9'

An important aspect of this diagram is that each string carries a substantial number of harmonics. For instance, since 1, 3 and 5 all lie on different strings, close-position major chords are playable.

Since 11\46 is small, some intervals that look unplayable can in fact be played. for instance, 7:6 looks like it can't, because 3 and 7 lie on the same string. However, for each haronic drawn, the same note can be played two strings up and eleven frets down. Thus 7:6 can be played by reaching across two strings and down 6 frets (equivalent to 3.1 frets of 12-edo).

pros, cons, and comparison to the Kite guitar

46-edo is harmonically exceptional. Among other virtues, it is the first edo consistent in the 13-limit. See 46edo.

The thirds in 46-edo can be easier for a listener used to 12-edo to accept than those in 41-edo. (In 46-edo, thirds are 5c sharp; in 12-edo they are 14c sharp; and in 41-edo they are 6c flat.)

The Kite tuning is more economical with strings. If the root is at string 0 fret 0, then the octave in the Kite system lies on string 3 fret 1, whereas in this system it lies at string 4 fret 1. Thus 6 open strings in the Kite system spans 1902 cents (a root and a fifth), whereas in this one they span 1435 cents (a root and a septimal second).

The most difficult 15-limit ratios (12:11 and 13:12) to play span 8 frets of 23-edo, which is equivalent to 4.2 frets of 12-edo (since 8*12/23 = 4.2). This compares favorably to the equivalent figure for the Kite tuning, which is 4.6 frets of 12 edo.