Skip fretting system 46 2 11: Difference between revisions
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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 good way to play in 46-edo on a stringed instrument is with a 23-edo fretboard and strings tuned 11\46 apart. | ||
=== | === 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 -- | -- -- -- 3 -- | ||
15_-- -- -- -- | 15_-- -- -- -- | ||
-- -- -- -- 15 | |||
1 -- -- -- | 1 19 -- -- -- | ||
-- -- 23 -- 2 | -- -- 23 -- 2 treble strings | ||
17 5 -- 7 -- | 17 5 -- 7 -- on this side | ||
-- -- -- -- 17' | -- -- -- -- 17' | ||
9 -- -- -- -- | 9 -- -- -- -- | ||
-- 11 13 -- 9' | -- 11 13 -- 9' | ||
bridge on this side | |||
An | 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 | 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. Among other virtues, it is the first edo consistent in the 13-limit. See [[46edo]]. | 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 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. | 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). | ||
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 | 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. | ||
Revision as of 03:35, 16 February 2023
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: 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. 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. 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).
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.