Skip fretting

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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, where it is called skip-key, skip-keyed, etc.)

Introduction

Skip fretting allows a player of a fretted stringed instrument to play in a higher EDO than would otherwise be possible or convenient. In most skip-fretting systems, the guitar skips every other fret, so each string has only half of the notes.

The most familiar skip-fretting systems allow someone with an ordinary 12-edo guitar to tune to 24-edo by retuning their guitar, for instance by tuning 450 cents between every pair of adjacent strings. 350 cents, 550 cents etc. would all work too. The even strings have half the notes, and the odd strings have the other half.

Partial skip-fretting

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.

(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).

[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.]

The relevance to keyboard players of skip-fretting

Any skip-fretting system can be used on any two-dimensional grid instrument, such as the Lumatone or the monome. Whereas for a string player the numbers `div` and `gap` in an `(edo, div, gap)` system have different meanings, for a keyboardist they don't: Both `div` and `gap` describe the amount by which the pitch changes between two keys adjacent on a particular axis.

For example, the Kite skip-fretting system (41,2,13) involves keeping only every 2nd fret from 41-edo and putting 13\41 between the strings. The reverse, keeping every 13th fret and putting 2\41 between every pair of strings, would be ridiculous on a guitar, but makes just as much sense on a keyboard, and in fact results in the same system, just swapping the two axes.

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 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

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 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 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.)

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