Skip fretting system 53 3 17: Difference between revisions

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This layout allows someone to play in 53-edo on a 17.666-edo guitar, by tuning the guitar in major thirds -- that is, with 17\53 between each pair of adjacent strings. It offers a big range -- very slightly wider than the [[Kite Guitar]]'s -- and a playable layout, with strikingly easy 5-limit chords.

~~An important drawback is that~~, ~~because harmonics 3 and 7 lie on the same~~ string, ~~a harmonic 7:6~~ is ~~difficult to play.~~ (~~Doing so requires reaching back 13 frets~~, ~~or 883 cents, as well as across three strings~~.~~) Perhaps counter-intuitively~~, 12:7, the ~~octave-complement of 7:6, is much easier to play, requiring a stretch of only 9 frets or 611 cents~~.

The diagram below, which could be interpreted 20 frets of a 12-string guitar, shows where each of the 15-limit harmonics lies. Since 53-edo is mostly consistent in the 15-limit (see below for the exception), these harmonics' positions imply where every interval in that group lies. For instance, to play 6:5 requires moving up (toward the treble side) one string and down (toward the nut) one fret.

The diagram below, which could be interpreted 20 frets of a 12-string guitar, shows where each of the 15-limit harmonics lies. Since 53-edo is mostly (see below) for the exception consistent in the 15-limit, these harmonics' positions imply where every interval in that group lies. (For instance, to play 7/6 you move up one string and down one fret, because that takes you from harmonic 3 to harmonic 7.) Octaves are indicated as powers of 2 (specifically 1, 2, 4 and 8).

The only exception to the above procedure is the ratios 11/7 (and its octave-complement 14/11). Since 11:8 is 7.9 cents flat and 7:4 is 4.8 cents sharp in 53-edo, the distance between them is 7.9 + 4.8 = 12.7 cents too wide. 12.7 cents is more than half of 53-edo's step size of 22.6 cents. Thus whereas the best approximations to 11:8 and 7:4 (and hence the diagram below) suggest that 11:7 is 34 steps wide, in fact 53-edo's best approximation to 11:7 is 35 steps wide. But both approximations are almost equally wrong, just in opposite directions.

~~The~~ exception to the above ~~consistency rule of thumb~~ is the ratios 11/7 (and its octave-complement 14/11). Since 11:8 is 7.9 cents flat and 7:4 is 4.8 cents sharp in 53-edo, the distance between them is 7.9 + 4.8 = 12.7 cents too wide. 12.7 cents is more than half of 53-edo's step size of 22.6 cents. Thus whereas the diagram below ~~suggests~~ that 11:7 is 34 steps wide, in fact 53-edo's best approximation to 11:7 is 35 steps wide. But ~~that best approximation is still 10 cents sharp, so~~ both approximations are ~~roughly~~ equally wrong, just in opposite directions.

20 frets of a hypothetical 12-string guitar tuned this way:

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- - - - - - 1 5 - - - -

[[Category:Skip fretting]]

== The (few?) harmonically difficult ratios have easy octave counterparts ==

One notable drawback to this tuning that, because harmonics 3 and 7 lie on the same string, a harmonic 7:6 is difficult to play. (Doing so requires reaching back 13 frets, or 883 cents, and across three strings.) However, the ratio 7:3 (an octave wider than 7:6) is unusually easy to play, being 3 string crossings and 1 fret wide. Following the same logic, for every difficult interval R less than an octave, it can be shown that R plus an octave is easy to play.

There seem to be very few such difficult ratios in the 15-limit. I (Jeff Brown) only see five: 7:6, 13:12, 14:13, 9:8, and 11:9.

[[Category:Skip fretting]] [[Category:53edo]]

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 This layout allows someone to play in 53-edo on a 17.666-edo guitar, by tuning the guitar in major thirds -- that is, with 17\53 between each pair of adjacent strings. It offers a big range -- very slightly wider than the [[Kite Guitar]]'s -- and a playable layout, with strikingly easy 5-limit chords.
-An important drawback is that, because harmonics 3 and 7 lie on the same string, a harmonic 7:6 is difficult to play. (Doing so requires reaching back 13 frets, or 883 cents, as well as across three strings.) Perhaps counter-intuitively, 12:7, the octave-complement of 7:6, is much easier to play, requiring a stretch of only 9 frets or 611 cents.
+The diagram below, which could be interpreted 20 frets of a 12-string guitar, shows where each of the 15-limit harmonics lies. Since 53-edo is mostly consistent in the 15-limit (see below for the exception), these harmonics' positions imply where every interval in that group lies. For instance, to play 6:5 requires moving up (toward the treble side) one string and down (toward the nut) one fret.
-The diagram below, which could be interpreted 20 frets of a 12-string guitar, shows where each of the 15-limit harmonics lies. Since 53-edo is mostly (see below) for the exception consistent in the 15-limit, these harmonics' positions imply where every interval in that group lies. (For instance, to play 7/6 you move up one string and down one fret, because that takes you from harmonic 3 to harmonic 7.) Octaves are indicated as powers of 2 (specifically 1, 2, 4 and 8).
+The only exception to the above procedure is the ratios 11/7 (and its octave-complement 14/11). Since 11:8 is 7.9 cents flat and 7:4 is 4.8 cents sharp in 53-edo, the distance between them is 7.9 + 4.8 = 12.7 cents too wide. 12.7 cents is more than half of 53-edo's step size of 22.6 cents. Thus whereas the best approximations to 11:8 and 7:4 (and hence the diagram below) suggest that 11:7 is 34 steps wide, in fact 53-edo's best approximation to 11:7 is 35 steps wide. But both approximations are almost equally wrong, just in opposite directions.
-The exception to the above consistency rule of thumb is the ratios 11/7 (and its octave-complement 14/11). Since 11:8 is 7.9 cents flat and 7:4 is 4.8 cents sharp in 53-edo, the distance between them is 7.9 + 4.8 = 12.7 cents too wide. 12.7 cents is more than half of 53-edo's step size of 22.6 cents. Thus whereas the diagram below suggests that 11:7 is 34 steps wide, in fact 53-edo's best approximation to 11:7 is 35 steps wide. But that best approximation is still 10 cents sharp, so both approximations are roughly equally wrong, just in opposite directions.
 frets of a hypothetical 12-string guitar tuned this way:
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               -  -  -  -  -  -  1  5  -  -  -  -
-[[Category:Skip fretting]]
+== The (few?) harmonically difficult ratios have easy octave counterparts ==
+One notable drawback to this tuning that, because harmonics 3 and 7 lie on the same string, a harmonic 7:6 is difficult to play. (Doing so requires reaching back 13 frets, or 883 cents, and across three strings.) However, the ratio 7:3 (an octave wider than 7:6) is unusually easy to play, being 3 string crossings and 1 fret wide. Following the same logic, for every difficult interval R less than an octave, it can be shown that R plus an octave is easy to play.
+There seem to be very few such difficult ratios in the 15-limit. I (Jeff Brown) only see five: 7:6, 13:12, 14:13, 9:8, and 11:9.
+[[Category:Skip fretting]] [[Category:53edo]]