Skip fretting: Difference between revisions

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Because the frets on a fretted instrument get closer together toward the bridge - at the first octave they are twice as dense, and at the second octave, four times -- it could be reasonable to include all the frets near the nut, and then switch to a skip-fretting system somewhere for the high notes. To this author's knowledge a partially skip-fretted instrument has not yet been made.

== ~~Some~~ notation ==

== (edo, divisor, gap) notation ==

Skip-fretting systems can be "isomorphic", with the same distance between every pair of adjacent strings, but they don't have to be. An isomorphic skip-fretting system can be described with three numbers: The EDO it allows one to play, the fraction of that EDO's notes on any particular string, and the number of steps in the EDO between adjacent strings. So, for instance, the system described above for playing 24-edo on a 12-edo guitar could be called a "24 2 9" system (9\24 being equal to 450 cents).

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[In retrospect, I wish I had used notation like "2,13\41" instead of "(41,2,13)". Both of those represent the [[Kite guitar]] tuning equally unambiguously, but I think the first is clearer.]

== Tradeoffs inherent in skip-fretting systems ==

The ideal skip-fretting system would be one that offers the player a big range without requiring too much movement or stretching, good approximations to the just intervals they want, and convenient unison or octave equivalents to any given note. These qualities are in tension.

=== ease of reach vs. frequency range ===

The smaller the interval between adjacent strings, the easier it becomes to reach all the notes of interest in a given octave, but this reduces the total range of the instrument.

The narrow 11\41 and (standard) wide 13\41 Kite guitar tunings illustrate this tradeoff. In the narrow tuning, intervals based on the 7th and 13th harmonic are much easier to play, but the interval from the first string to the sixth is 1609 cents. In the wider tuning, by contrast, it is 1902 cents.

=== 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.

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

In skip-fretting system `(edo, div, gap)`, the unison to any note lies `div` strings and `gap` frets away.

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` a whole number, there is an octave 1 string and 14 frets away. And since `1 = (41 - 3*13)/2`, there is another octave 3 strings and 1 fret away.

== Some skip-fretting systems ==

@@ Line 10: / Line 10: @@
 Because the frets on a fretted instrument get closer together toward the bridge - at the first octave they are twice as dense, and at the second octave, four times -- it could be reasonable to include all the frets near the nut, and then switch to a skip-fretting system somewhere for the high notes. To this author's knowledge a partially skip-fretted instrument has not yet been made.
-== Some notation ==
+== (edo, divisor, gap) notation ==
 Skip-fretting systems can be "isomorphic", with the same distance between every pair of adjacent strings, but they don't have to be. An isomorphic skip-fretting system can be described with three numbers: The EDO it allows one to play, the fraction of that EDO's notes on any particular string, and the number of steps in the EDO between adjacent strings. So, for instance, the system described above for playing 24-edo on a 12-edo guitar could be called a "24 2 9" system (9\24 being equal to 450 cents).
@@ Line 16: / Line 16: @@
 [In retrospect, I wish I had used notation like "2,13\41" instead of "(41,2,13)". Both of those represent the [[Kite guitar]] tuning equally unambiguously, but I think the first is clearer.]
+== Tradeoffs inherent in skip-fretting systems ==
+The ideal skip-fretting system would be one that offers the player a big range without requiring too much movement or stretching, good approximations to the just intervals they want, and convenient unison or octave equivalents to any given note. These qualities are in tension.
+=== ease of reach vs. frequency range ===
+The smaller the interval between adjacent strings, the easier it becomes to reach all the notes of interest in a given octave, but this reduces the total range of the instrument.
+The narrow 11\41 and (standard) wide 13\41 Kite guitar tunings illustrate this tradeoff. In the narrow tuning, intervals based on the 7th and 13th harmonic are much easier to play, but the interval from the first string to the sixth is 1609 cents. In the wider tuning, by contrast, it is 1902 cents.
+=== 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.
+== Finding unisons and octaves in a skip-fretting system ==
+In skip-fretting system `(edo, div, gap)`, the unison to any note lies `div` strings and `gap` frets away.
+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` a whole number, there is an octave 1 string and 14 frets away. And since `1 = (41 - 3*13)/2`, there is another octave 3 strings and 1 fret away.
 == Some skip-fretting systems ==