Lumatone mapping for 89edo

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Revision as of 23:33, 31 October 2026 by Lucius Chiaraviglio (talk | contribs) (→Other Mappings: Reorganize; fix wording because they are not actually maximally efficient at covering the full gamut; add Bryan Deister's Myna mapping that uses a less lopsided 8L 3s scale (10:3 step ratio))
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There are many conceivable ways to map 89edo onto the onto the Lumatone keyboard. Only one, however, agrees with the Standard Lumatone mapping for Pythagorean. Due to the size of the edo, this mapping covers less than 2/3rds of the notes.

Diatonic

Note that since 89edo is a schismatic tuning, the best approximation to ~5/4 is the diminished fourth. That said, since the fifth ~[3/2] is flattened by almost exactly one quarer of Boethius' comma 513/512, the meantone major third is almost exactly Boethius' major third ~24/19, and since one third of such a small comma is not very far off from one fourth, the corresponding minor third ~19/16 is also not too far from just. Using 89edo as a quasi-meantone tuning with the extremely accurate ~24/19 substituting for the less accurate ~5/4 will enable many chords to be played despite the missing notes, as Bryan Deister demonstrates in 89edo improv (2026-04-15), and it would be possible to switch between presets for different subsets of 89edo to enable modulation to remote keys.

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Orgone

One way to cover the whole gamut (in at least the middle of the range) covers is with an orgone 4L 7s scale (10:7 step ratio). This mapping focuses on making the 2.7.11 subgroup easy to play, but is extremely efficient at covering that and only a tenth of a cent off the minimax tuning, plus it skips or repeats fewer notes at the edges. The overall range is a bit over three octaves, with a gentle downwards slope, but the bass and treble octaves lose some notes at the left and right edges.

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Myna

It is possible to map 89edo by stacking mildly flat classic minor thirds ~6/5 (23\89), as in myna temperament, for which 89edo is the optimal patent val. Two generators make a slightly sharp greater septimal tritone ~10/7; three of them make a slightly flat septimal major sixth ~12/7; after octave-reduction, five of them make a mildly sharp undecimal neutral third ~11/9; seven of them make a slightly sharp septimal minor seventh ~7/4; nine of them make a mildly sharp classic major third ~5/4; ten of them make a near-just fifth ~3/2.

4L 7s

A 4L 7s scale (17:3 step ratio) efficiently gives access to the full 11-limit, although it is very lopsided. The overall range is a bit over three octaves, with a gentle downwards slope, but the bass and treble octaves lose many notes at the left and right edges (although the middle of the range does cover the whole gamut, even having a few repeated notes).

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8L 3s

A much less lopsided 8L 3s (10:3 step ratio) scale minimizes note loss at the left and right edges. Bryan Deister has demonstrated this mapping in 89edo improv (2026-10-28). This mapping is maximally efficient, with a range of just over three octaves (which slope upwards very gently), with no missed notes (even in the bass and treble note 0 to note 0 octaves) and no repeated notes.

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Char-based + Sesquart rank-3 temperament

Bryan Deister has demonstrated a 10L 3s (8:3 step ratio) mapping for 89edo in microtonal improvisation in 89edo (2025). The rightward generator 8\89 functions as both the 16/15 (the classical diatonic semitone) and 17/16 (the large septendecimal semitone), meaning that the charisma 256/255 is tempered out, although 89edo is not listed as a tuning of any of the temperaments on the page for 256/255. Eight of these generators make a near-just Axirabian paraminor fifth ~16/11. Although being prime, 89edo technically needs no second generator, a second generator helps with reaching consonant intervals other than ~16/11; one obvious choice is the scale chroma (upwards generator) 5\89 which functions as a somewhat sharp small tridecimal third tone ~27/26; three of these make a near-just Pythagorean whole tone ~9/8; four of them make a near-just septimal minor third ~7/6. It is also possible to combine both generators to get 13\89, a slightly sharp undevicesimal supraneutral second ~21/19; four of these generator combinations make a near-just fifth ~3/2, as in Sesquiquartififths temperament and its higher-limit Sesquart extension. The range is slightly over 2⅓ octaves with no missed notes and a few repeated notes, and the octaves slope up mildly.

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