Skip fretting: Difference between revisions

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(Note: Despite it's name, skip-fretting is relevant not only to fretted stringed instruments, but to the layout of other two-dimensional grid instruments like the Lumatone and the monome.)

(Note: Despite it's name, skip-fretting is relevant not only to fretted stringed instruments, but to the layout of other two-dimensional grid instruments like the Lumatone and the monome, where it is called skip-key, skip-keyed, etc.)

== Introduction ==

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=== Ease of reach vs. harmonic accuracy ===

The relationship is not linear, but as a loose rule, higher EDOs provide closer ~~approxiamtions~~ to the harmonic series. However, skip-frettings for higher EDOs provide fewer unisons and octaves. For instance, [[Skip fretting system 63 3 17]] is in general more faithful than 41-edo is to the harmonic series, but unisons lie 17 frets apart on a guitar with 21 frets per octave. That's equivalent to a stretch of 9.7 frets on a standard 12-edo guitar. By contrast, on the Kite guitar, which uses 41-edo, the distance between unisons is only 13 frets on a 20.5-fret guitar, equivalent to about 7.6 frets on a 12-edo guitar.

The relationship is not linear, but as a loose rule, higher EDOs provide closer approximations to the harmonic series. However, skip-frettings for higher EDOs provide fewer unisons and octaves. For instance, [[Skip fretting system 63 3 17]] is in general more faithful than 41-edo is to the harmonic series, but unisons lie 17 frets apart on a guitar with 21 frets per octave. That's equivalent to a stretch of 9.7 frets on a standard 12-edo guitar. By contrast, on the Kite guitar, which uses 41-edo, the distance between unisons is only 13 frets on a 20.5-fret guitar, equivalent to about 7.6 frets on a 12-edo guitar.

== Finding unisons and octaves in a skip-fretting system ==

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This author has yet to find or see a formula for determining the octaves. However, the following procedure does the job: Let `n` be a number of strings. If `f = (edo - n*gap) / div` is a whole number, then an octave can be found `n` strings and `f` frets away.

For instance, for the standard Kite tuning, `(edo, div, gap)` = `(41,2,13)`. Since `14 = (41 - 1*13)/2` is a whole number, there is an octave 1 string and 14 frets away. And since `1 = (41 - 3*13)/2` is another whole number, there is another octave 3 strings and 1 fret away.

For instance, for the standard Kite guitar tuning, `(edo, div, gap)` = `(41,2,13)`. Since `14 = (41 - 1*13)/2` is a whole number, there is an octave 1 string and 14 frets away. And since `1 = (41 - 3*13)/2` is another whole number, there is another octave 3 strings and 1 fret away.

However, if the divisor of the system is coprime with the edo, alternating strings have no unisons or octaves. For example, 41edo has no divisors other than itself, so octaves can be found regardless of what interval the strings are tuned to or how many frets are skipped each time as long as the fretboard is wide enough. 46edo can be divided by two, so alternating strings on a 23edo guitar never have unisons or octaves, making systems like that harder to tune by ear or normal guitar tuners and use in live performance unless tuning to notes that they have in common with 12edo.

== Some skip-fretting systems ==

In addition to the layouts listed below, every [[:Category:Lumatone mappings|Lumatone mapping]] can also be interpreted as a skip fretting. As three distinct ones, in fact, depending on which of the three hexagonal axes is mapped to the two rectangular axes on the fretboard. (Nothing guarantees that it will be a *good* layout on a guitar, but it will at least often be feasible.)

* [[Skip fretting system 27 2 9]]: 27-edo on a 13.5-edo guitar.

* [[Skip fretting system 31 2 9]]: Make the higher register of a 31-edo guitar easier to play by omitting every other fret, while keeping all the frets in the lower register.

* [[Skip fretting system 31 3 7]]: 31-edo on a 10.33-edo guitar.

* [[Skip fretting system 34 2 9]]: 34-edo on a 17-edo guitar.

* [[Skip fretting system 34 2 11]]: 34-edo on a 17-edo guitar, very much like the [[Kite guitar]].

* [[Skip fretting system 38 2 11]]: 38-edo on a 19-edo guitar.

* [[Skip fretting system 41 2 11]]: Same fret layout as the Kite guitar, with narrower string gaps and easier-to-reach higher-limit intervals.

* [[The Kite Guitar|The Kite Guitar, system 41 2 13]]: 41-edo on a 20.5-edo guitar.

* [[Skip fretting system 43 2 9]]: 43-edo on a 21.5-edo guitar.

* [[Skip fretting system 44 2 11]]: 44-edo on a 22-edo guitar.

* [[Skip fretting system 46 2 11]]: 46-edo on a 23-edo guitar.

* [[Skip fretting system 48 2 13]]: 48-edo on a 24-edo guitar.

* [[Skip fretting system 53 3 14]]: 53-edo on a 17.66-edo or [[28edt]] guitar.

* [[Skip fretting system 53 3 17]]: 53-edo on a 17.66-edo guitar, with an unusually wide range.

* [[Skip fretting system 58 2 15]]: 58-edo on a 29-edo guitar.

* [[Skip fretting system 58 4 15]]: 58-edo on a 14.5-edo guitar.

* [[Skip fretting system 58 2 17]]: 58-edo on a 29-edo guitar.

* [[Skip fretting system 60 2 29]]: 60-edo on a 30-edo guitar.

* [[Skip fretting system 60 3 19]]: 60-edo on a 20-edo guitar.

* [[Skip fretting system 60 4 17]]: 60-edo on a 15-edo guitar.

* [[Skip fretting system 63 3 17]]: 63-edo on a 21-edo guitar.

* [[Skip fretting system 72 2 27, a.~~k.a. Isomorphic grid layout 27\72 x~~ 2~~\72~~]]: 72-edo on a 36-edo guitar.

* [[Skip fretting system 72 2 27]]: 72-edo on a 36-edo guitar.

* [[Skip fretting system 87 2 17]]: 87-edo on a 43.5-edo guitar.

* [[Skip fretting system 90 5 17]]: 90-edo on an 18-edo guitar.

* [[~~Isomorphic grid layout~~ 7~~\94 x~~ 16~~\94~~]]: 94-edo on a 13.429-edo guitar.

* [[Skip fretting system 94 7 16]]: 94-edo on a 13.429-edo guitar.

* [[Skip fretting system 140 8 37]]: 140edo on a 17.5-edo guitar.

[[Category:Guitar]]

[[Category:Skip fretting| ]]

@@ Line 1: / Line 1: @@
-(Note: Despite it's name, skip-fretting is relevant not only to fretted stringed instruments, but to the layout of other two-dimensional grid instruments like the Lumatone and the monome.)
+(Note: Despite it's name, skip-fretting is relevant not only to fretted stringed instruments, but to the layout of other two-dimensional grid instruments like the Lumatone and the monome, where it is called skip-key, skip-keyed, etc.)
 == Introduction ==
@@ Line 35: / Line 35: @@
 === Ease of reach vs. harmonic accuracy ===
-The relationship is not linear, but as a loose rule, higher EDOs provide closer approxiamtions to the harmonic series. However, skip-frettings for higher EDOs provide fewer unisons and octaves. For instance, [[Skip fretting system 63 3 17]] is in general more faithful than 41-edo is to the harmonic series, but unisons lie 17 frets apart on a guitar with 21 frets per octave. That's equivalent to a stretch of 9.7 frets on a standard 12-edo guitar. By contrast, on the Kite guitar, which uses 41-edo, the distance between unisons is only 13 frets on a 20.5-fret guitar, equivalent to about 7.6 frets on a 12-edo guitar.
+The relationship is not linear, but as a loose rule, higher EDOs provide closer approximations to the harmonic series. However, skip-frettings for higher EDOs provide fewer unisons and octaves. For instance, [[Skip fretting system 63 3 17]] is in general more faithful than 41-edo is to the harmonic series, but unisons lie 17 frets apart on a guitar with 21 frets per octave. That's equivalent to a stretch of 9.7 frets on a standard 12-edo guitar. By contrast, on the Kite guitar, which uses 41-edo, the distance between unisons is only 13 frets on a 20.5-fret guitar, equivalent to about 7.6 frets on a 12-edo guitar.
 == Finding unisons and octaves in a skip-fretting system ==
@@ Line 42: / Line 42: @@
 This author has yet to find or see a formula for determining the octaves. However, the following procedure does the job: Let `n` be a number of strings. If `f = (edo - n*gap) / div` is a whole number, then an octave can be found `n` strings and `f` frets away.
-For instance, for the standard Kite tuning, `(edo, div, gap)` = `(41,2,13)`. Since `14 = (41 - 1*13)/2` is a whole number, there is an octave 1 string and 14 frets away. And since `1 = (41 - 3*13)/2` is another whole number, there is another octave 3 strings and 1 fret away.
+For instance, for the standard Kite guitar tuning, `(edo, div, gap)` = `(41,2,13)`. Since `14 = (41 - 1*13)/2` is a whole number, there is an octave 1 string and 14 frets away. And since `1 = (41 - 3*13)/2` is another whole number, there is another octave 3 strings and 1 fret away.
+However, if the divisor of the system is coprime with the edo, alternating strings have no unisons or octaves. For example, 41edo has no divisors other than itself, so octaves can be found regardless of what interval the strings are tuned to or how many frets are skipped each time as long as the fretboard is wide enough. 46edo can be divided by two, so alternating strings on a 23edo guitar never have unisons or octaves, making systems like that harder to tune by ear or normal guitar tuners and use in live performance unless tuning to notes that they have in common with 12edo.
 == Some skip-fretting systems ==
+In addition to the layouts listed below, every [[:Category:Lumatone mappings|Lumatone mapping]] can also be interpreted as a skip fretting. As three distinct ones, in fact, depending on which of the three hexagonal axes is mapped to the two rectangular axes on the fretboard. (Nothing guarantees that it will be a *good* layout on a guitar, but it will at least often be feasible.)
+* [[Skip fretting system 27 2 9]]: 27-edo on a 13.5-edo guitar.
 * [[Skip fretting system 31 2 9]]: Make the higher register of a 31-edo guitar easier to play by omitting every other fret, while keeping all the frets in the lower register.
+* [[Skip fretting system 31 3 7]]: 31-edo on a 10.33-edo guitar.
 * [[Skip fretting system 34 2 9]]: 34-edo on a 17-edo guitar.
+* [[Skip fretting system 34 2 11]]: 34-edo on a 17-edo guitar, very much like the [[Kite guitar]].
 * [[Skip fretting system 38 2 11]]: 38-edo on a 19-edo guitar.
 * [[Skip fretting system 41 2 11]]: Same fret layout as the Kite guitar, with narrower string gaps and easier-to-reach higher-limit intervals.
 * [[The Kite Guitar|The Kite Guitar, system 41 2 13]]: 41-edo on a 20.5-edo guitar.
+* [[Skip fretting system 43 2 9]]: 43-edo on a 21.5-edo guitar.
 * [[Skip fretting system 44 2 11]]: 44-edo on a 22-edo guitar.
+* [[Skip fretting system 46 2 11]]: 46-edo on a 23-edo guitar.
 * [[Skip fretting system 48 2 13]]: 48-edo on a 24-edo guitar.
+* [[Skip fretting system 53 3 14]]: 53-edo on a 17.66-edo or [[28edt]] guitar.
+* [[Skip fretting system 53 3 17]]: 53-edo on a 17.66-edo guitar, with an unusually wide range.
 * [[Skip fretting system 58 2 15]]: 58-edo on a 29-edo guitar.
+* [[Skip fretting system 58 4 15]]: 58-edo on a 14.5-edo guitar.
+* [[Skip fretting system 58 2 17]]: 58-edo on a 29-edo guitar.
+* [[Skip fretting system 60 2 29]]: 60-edo on a 30-edo guitar.
+* [[Skip fretting system 60 3 19]]: 60-edo on a 20-edo guitar.
+* [[Skip fretting system 60 4 17]]: 60-edo on a 15-edo guitar.
 * [[Skip fretting system 63 3 17]]: 63-edo on a 21-edo guitar.
-* [[Skip fretting system 72 2 27, a.k.a. Isomorphic grid layout 27\72 x 2\72]]: 72-edo on a 36-edo guitar.
+* [[Skip fretting system 72 2 27]]: 72-edo on a 36-edo guitar.
+* [[Skip fretting system 87 2 17]]: 87-edo on a 43.5-edo guitar.
 * [[Skip fretting system 90 5 17]]: 90-edo on an 18-edo guitar.
-* [[Isomorphic grid layout 7\94 x 16\94]]: 94-edo on a 13.429-edo guitar.
+* [[Skip fretting system 94 7 16]]: 94-edo on a 13.429-edo guitar.
+* [[Skip fretting system 140 8 37]]: 140edo on a 17.5-edo guitar.
 [[Category:Guitar]]
 [[Category:Skip fretting| ]] <!-- main category -->