Lumatone mapping for 84edo: Difference between revisions

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→Sensei: Add Bryan Deister's compressed Sensei mapping (misses notes but still works well enough for a demo video)
→Orwell: Insert Bryan Deister's Porwell mapping after this
 
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The [[Orwell]] mapping has a smaller range, but is closer to the optimal tuning for the temperament and makes it easier to play harmonics together.
The [[Orwell]] mapping has a smaller range, but is closer to the optimal tuning for the temperament and makes it easier to play harmonics together.
{{Lumatone EDO mapping|n=84|start=4|xstep=8|ystep=-5}}
{{Lumatone EDO mapping|n=84|start=4|xstep=8|ystep=-5}}
== Porwell ==
[[Bryan Deister]] has demonstrated a [[5L&nbsp;11s (4/1-equivalent)]] (16:11 step ratio) mapping for [[84edo]], in [https://www.youtube.com/shorts/R4gAWZoJWpc ''84edo improv''] (2026). The down-right generator 16\84 is a slightly sharp septimal major second ~[[8/7]], and three of these make the alternate flat fifth inherited from [[7edo]]. This generator is not co-prime with 84edo, and to get to the (near-just) patent fifth ~[[3/2]], it is necessary to use a fine adjustment second generator 1\84, which functions as ~[[126/125]] (often seen as the starling comma, but here approximated as near-just rather than being tempered out). Since (8/7)<sup>3</sup> × (126/125) / (3/2) = [[6144/6125]] (the Porwell comma, which is tempered out), that makes this a rotated Porwell mapping. This mapping lends itself to an add-23 extension of Porwell with 11\84 (one key right) being a near-just large vicesimotertial neutral second ~[[23/21]], and two of these make the near-just classic minor third ~[[6/5]]. The near-just classic major third ~[[5/4]] (at 27\84, two keys right plus one key down-right) is also in easy reach. The downsides are that octaves alternate between near and far (with an overall gentle downward slope), and no single octave MOS scales are available for most choices of root note without incurring a vertical wraparound, while with over four octaves squeezed into the space of a standard-sized Lumatone, many notes are missing, although all notes appear in each double octave if not chopped off at the corners.
{{Lumatone EDO mapping|n=84|start=68|xstep=11|ystep=5}}


== Diploid Marvel extension ==
== Diploid Marvel extension ==

Latest revision as of 15:45, 23 September 2026

There are many conceivable ways to map 84edo onto the onto the Lumatone keyboard. However, it has 7 mutually-exclusive rings of fifths, so the Standard Lumatone mapping for Pythagorean is not one of them. Due to the edos size, it would not cover the whole gamut even if it was. Neither the second, third, nor fourth-best fifths work either, and the maviloid scale generated by 47/84 is even flatter than 25edo.

Antidiatonic

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Sensei

3L 5s

It is possible to map 84edo using a supermajor third as a generator, as in sensei temperament. The generator 31\84 is two keys right plus one key down-right, and it functions as a flat ~162/125 in the 5-limit, as a near-just ~84/65 in the 13-limit, as a somewhat flat septendecimal ultramajor third ~22/17 in the 17-limit (requires use of the 84e val to map correctly), and as a slightly sharp ~49/38 in the 19-limit. Two of these generators make a near-just classic major sixth ~5/3; five of them make a somewhat flat classic minor seventh ~9/5; seven of them make a near-just (slightly flat) fifth ~3/2; and nine of them make a near-just classic major third ~5/4. Unfortunately, the 3L 5s scale (13:9 step ratio) mapping tries to get too many octaves (four) for mapping a tuning system this large on the currently available version of the Lumatone, so it misses some notes in each octave; furthermore, the octaves slope down mildly. Despite the missing notes Bryan Deister has used this mapping in 84edo groove (2026).

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8L 3s

Instead, the most efficient layout that allows access to all notes is an expanded Sensei mapping, using an 8L 3s scale with a 9:4 step ratio, this reduces the range to a little over three octaves, but they are nearly level.

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Orwell

The Orwell mapping has a smaller range, but is closer to the optimal tuning for the temperament and makes it easier to play harmonics together.

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Porwell

Bryan Deister has demonstrated a 5L 11s (4/1-equivalent) (16:11 step ratio) mapping for 84edo, in 84edo improv (2026). The down-right generator 16\84 is a slightly sharp septimal major second ~8/7, and three of these make the alternate flat fifth inherited from 7edo. This generator is not co-prime with 84edo, and to get to the (near-just) patent fifth ~3/2, it is necessary to use a fine adjustment second generator 1\84, which functions as ~126/125 (often seen as the starling comma, but here approximated as near-just rather than being tempered out). Since (8/7)3 × (126/125) / (3/2) = 6144/6125 (the Porwell comma, which is tempered out), that makes this a rotated Porwell mapping. This mapping lends itself to an add-23 extension of Porwell with 11\84 (one key right) being a near-just large vicesimotertial neutral second ~23/21, and two of these make the near-just classic minor third ~6/5. The near-just classic major third ~5/4 (at 27\84, two keys right plus one key down-right) is also in easy reach. The downsides are that octaves alternate between near and far (with an overall gentle downward slope), and no single octave MOS scales are available for most choices of root note without incurring a vertical wraparound, while with over four octaves squeezed into the space of a standard-sized Lumatone, many notes are missing, although all notes appear in each double octave if not chopped off at the corners.

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Diploid Marvel extension

Bryan Deister has demonstrated a 10L 4s (8:1 step ratio) mapping for 84edo in microtonal improvisation in 84edo (2025). The rightward generator 8\84 (shared with the Orwell mapping) functions as both ~15/14 and ~16/15, making this a Marvel temperament; two of these make a slightly flat septimal supermajor second (~8/7); four of these make a slightly flat Barbados third (~13/10). However, by itself this generator would produce a contorted mapping even with the semioctave period implied by 10L 4s, so a second generator is required. The chroma of the scale (7\84, the upwards generator) is not very convenient as a secondary generator, and also produces a contorted mapping if used alone, so it makes sense to use the down-right generator 1\84 (the small step of the scale) as the secondary generator. This functions as a near-just version of the starling chroma (~126/125), but also functions as a narrow version of the syntonic chroma (~81/80), a slightly narrow version of the animist chroma (~105/104), and as a wide version of the buzurgismic/dhanvantarismic chroma (~169/168) or the mycunumic chroma (~196/195); two of these make a near-just septimal chroma (~64/63); and adding three of these to the aforementioned Barbados third (reached with the rightward generator) yields the very accurate fourth (~4/3) at 35\84 (shared with 12edo as 5\12). (Nevertheless, sheer distance may be a problem for people who do not have long fingers — a problem hard to avoid with large EDO sizes, that force octaves to be long to avoid missing notes.) Near-just versions of common intervals are reachable using both generators in combination: three rightward generators minus two down-right generators to get the classic major third ~6/5 at 22\84, and three of each generator to get the classic major third ~5/4 at 27\84. The range would be 2½ octaves with no missed notes if note for a missed note 36 in the highest semioctave; some notes are repeated for ease in vertical transitions; and the octaves are perfectly level.

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