Lumatone mapping for 89edo

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Revision as of 08:06, 1 November 2026 by Yourmusic Productions (talk | contribs) (The 8L 3s scale has a generator of 33\89, not 23\89, which makes it a sharp sensi variant that uses the second best fifth, not a myna one.)
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{Lumatone mapping intro}} 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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Other Mappings

The smallest mapping that covers the whole gamut is the 4L 7s one. Interestingly, there are two different mappings that fit that criteria, both good.

Myna

The myna mapping is quite lopsided, but is the optimal patent val for the temperament and efficiently gives access to the full 11-limit.

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Orgone

The Orgone mapping focusses 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.

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Sensoid

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. However, this does not put simple ratios as close together as the previous two.

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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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