Skip fretting system 46 2 11

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A good way to play in 46-edo on a stringed instrument is with a 23-edo fretboard and strings tuned 11\46 apart, a neo-gothic approximation of 13/11.

Layout: Harmonics on the fretboard

Each number in this diagram represents a harmonic modulo octaves -- so 3 represents 3:2, 15 represents 15:8, etc. A trailing _ indicates that the harmonic lies in the octave below 1 (i.e. 15_ represents 15:16 as opposed to 15:8), and a trailing ' indicates the harmonic lies an octave above 2 (so, e.g., 17' represents 17:8 rather than 17:16).

  headstock on this side
  -- -- -- 3  --
  15_-- -- -- --
  -- -- -- -- 15
  1  19 -- -- --
  -- -- 23 -- 2     treble strings
  17 5  -- 7  --    on this side
  -- -- -- -- 17'
  9  -- -- -- --
  -- 11 13 -- 9'
  bridge on this side

An appealing aspect of this layout 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 easily playable. If they were all on the same string, that would not be the case.

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 harmonic 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 (which requires a stretch of the hand equivalent to 3.1 frets of 12-edo).

Pros, cons, and comparison to the Kite guitar

46-edo is harmonically exceptional, having one of the lowest high prime limit errors of any edo under 100, although it is only consistent up to the 13. 41-edo is consistent up to 15.

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.) But the 5th is less accurate, and sharp as opposed to 12-edo's familiar flatness. (12edo is 2c flat, 41edo is 0.5c sharp, and 46edo is 2.4c sharp.)

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. Whereas 6 open strings in the Kite system spans 1902 cents (a root and a fifth), in this one they span 1435 cents (a root and a septimal second). Without being able to play 10ths, 11ths or 12ths, chord voicings are considerably limited.

The Kite tuning is extremely efficient in that every single 9-odd-limit ratio spans at most 4 frets, and almost every ratio within that 4-fret span is in fact a 9-odd-limit ratio. The exceptions are 11/8 and 16/11. Also all minor or neutral 2nds and their octave inverses, which mathematically must be a higher odd limit. Whereas in this system, there are many remote low-odd-limit ratios and many nearby high-odd-limit ratios. For example, 6/5 is 5 frets away, but 11/9 is only 1 fret away. Likewise 7/6 is 5 frets away but 13/11 is 0 frets away. And the highly dissonant wolf 5th 678c is only 2 frets away.

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 is a little narrower (i.e. easier) than the widest 15-limit stretch in the Kite tuning, which is 4.6 frets of 12 edo.