23edo and octave stretching

23edo is not typically taken seriously as a tuning except by those interested in extreme xenharmony. Its fifths are significantly flat, and is neighbors 22edo and 24edo generally get more attention.

However, when using a slightly stretched octave of around 1216 cents, 23edo looks much better, and it approximates the perfect fifth (and various other intervals involving the 5th, 7th, 11th, and 13th harmonics) to within 18 cents or so. If we can tolerate errors around this size in 12edo, we can probably tolerate them in stretched-23edo as well.

The perfect fifth is sharper than it is in 7edo, and thus the width of the perfect fifth falls within the syntonic temperament's tuning range. However, stretched-23 is *not* a syntonic temperament; using the perfect fifth as generator results in a pelogic ("mavila" or "antidiatonic") scale. Because of this, stretched-23 is not an extension of or replacement for 12edo, but rather an alternative to it; its strengths tend to be 12edo's weaknesses and vice versa, so they complement each other.

Stretched-23 is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the superantidiatonic scale), since its fifth is more consonant and less "wolfish" than fifths in other pelogic-family temperaments.

The table below gives the intervals and error values assuming a stretched octave of *exactly* 1216 cents.

Stretching the octave by this much weakens (but does not eliminate) the sense of octave equivalency. It also yields some odd results; stacking two perfect fifths results in a (stretched) octave plus 3 steps. However the 9:8 whole tone is approximated by 4 rather than 3 steps. This is because the triple octave (8:1) is stretched by nearly a quarter tone, and thus this version of stretched 23edo is not consistent for intervals involving 8.

Another odd feature of this scale is that the perfect fourth (4:3) is sandwiched almost exactly between two scale degrees, thus resulting in two fourths (a major and a minor one). This might not actually be a bad thing. In common-practice music, the perfect fourth, despite having low harmonic entropy, was often classified as a dissonant interval for reasons relating to Lipps-Meyer's law. It was considered a dissonance even after thirds and sixths began to be reclassified as consonances. Thus, by splitting the fourth in two we might actually be reducing this dissonance. Because of the split fourth, 23edo is also not consistent for intervals involving the factor 4.

The octave-stretching also results is various other intervals "inverting" in unexpected ways; for instance the septimal minor seventh (7:4) is the inversion of the septimal minor third (7:6), not of the septimal whole tone (8:7)! Because of this property, this interval could also be considered a sixth rather than a seventh (depending on the context in which it occurs).

## Stretched 23edo and pianos

Pianos typically have stretched octaves due to the inharmonicity of the strings. This stretch is concentrated at the low and high octaves of the piano. As a result, while the standard 88-key piano covers over seven octaves, only the middle four octaves or so are commonly used, because the treble and bass registers have so much stretch and do not sound as good. Ordinary pianos are tuned with 12 keys per octave, and 12edo is ideal for perfect octaves, but does not work as well for stretched ones. The total amount of stretch depends on the size of the piano but typically is around 70 cents (across the tuning range of the whole piano), although this stretch is unequally distributed as stated before.

Alternatively, it might be possible to design a piano in which the stretch is evenly distributed. Stretched 23edo would be ideal for such a piano. A piano covering between four and five octaves would be ideal; such a piano would have around 100 keys (versus 88 for a standard piano) and the total amount of stretch across the entire tuning range would be about the same as on a standard piano. While the range is narrower (4-5 octaves rather than 7-8), the effective *usable* range is about the same (since the low and high ranges of a standard piano are usually avoided due to the extreme stretch and dissonance in those regions, whereas in a stretched-23 piano, the stretch is evenly distributed and the entire range sounds equally well).

## Stretched 23edo and guitars

Guitars, unlike pianos, typically do not have much stretch since the strings are not as stiff. Thus, stretched-23 is not as natural an option for them. However, it might be possible to design a guitar-like instrument using stiffer wire (more akin to piano wire). Such a "guitar" could only be played with a stiff plectrum; attempting to play it with just the fingers would be very painful. Other families of stringed instruments could be adapted in a similar manner.

## Stretched 23edo and other Western instruments

Bowed violins do *not* adapt well to stretched-23, since the bowing action results in modelocking (and as a result their spectrum must be perfectly harmonic rather than stretched).

Brass and wind instruments should work well if the air column is suitably shaped.

## Stretched 23edo and inharmonic instruments

Stretched 23edo provides a good option for those seeking to combine Western-style instruments like piano and guitar (which have nearly harmonic spectra) with more obviously inharmonic ones (idiophones) from other cultures. In particular, the natural fifth in stretched-23 is almost halfway between the very flat fifth of 9edo (which is used, for example, in Indonesian pelog) and the harmonic perfect fifth. As for slendro, it may be approximated as 5-5-4-5-4, or (if we use the *diminished* octave as our repeat unit instead) 5-4-5-4-4.

Many non-Western scales do not have octave equivalence to begin with, so stretching or squashing the octave does not present much of a problem.