Lumatone mapping for 89edo: Difference between revisions
Hotcrystal0 (talk | contribs) meantone → diatonic |
Did analysis of Sensipent-related temperament 8L 3s mapping to find common consonances (both in-scale and out-of-scale); move this, myna, and orgone out of "Other" |
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{{Lumatone EDO mapping|n=89|start=2|xstep=15|ystep=-8}} | {{Lumatone EDO mapping|n=89|start=2|xstep=15|ystep=-8}} | ||
== Myna == | |||
The [[myna]] mapping is quite lopsided, with its [[4L 7s]] scale having a 17:3 step ratio, but 89edo is the optimal patent val for the temperament and efficiently gives access to the full 11-limit. The generator is a mildly flat classic minor third ~[[6/5]] (23\89). 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]] (however, not ~[[13/16]] unless 89f is used); 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]]. 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). | |||
The [[myna]] mapping is quite lopsided, but is the optimal patent val for the temperament and efficiently gives access to the full 11-limit. The generator is a mildly flat classic minor third ~[[6/5]] (23\89). 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]] (however, not ~[[13/16]] unless 89f is used); 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]]. 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). | |||
{{Lumatone EDO mapping|n=89|start=58|xstep=3|ystep=14}} | {{Lumatone EDO mapping|n=89|start=58|xstep=3|ystep=14}} | ||
== Orgone == | |||
The [[Orgone]] 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 | The [[Orgone]] mapping also has a [[4L 7s]] scale, but with a 10:7 step ratio; it 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. 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 fewer than in the myna mapping. | ||
{{Lumatone EDO mapping|n=89|start=76|xstep=7|ystep=3}} | {{Lumatone EDO mapping|n=89|start=76|xstep=7|ystep=3}} | ||
== Sensipent-related temperament == | |||
It is possible to map [[89edo]] by stacking near-just (only slightly sharp) tricesimoprimal ultramajor thirds ~[[31/24]] (33\89), as in a 2.3.5.7.17.19.29.31 subgroup version of [[sensipent]] temperament, but not tempering out the [[sensipent comma]] 78732/78125 (which is instead tempered to -2\89). Stacking two tricesimoprimal ultramajor thirds yields a mildly sharp classic major sixth ~[[5/3]] (easily accessed by moving moving two keys right plus one key down-right from the root note). After octave-reduction, three of them make a mildly flat lesser tridecimal neutral second ~[[13/12]] (which is notable for being just one key right). Unlike actual sensipent or extensions thereof, five of these generators miss the classic major seventh ~[[9/5]] (which would be 75\89, two keys down from an octave above the root note), instead yielding 76\89 functioning as a near-just tricesoprimal supermajor sixth ~[[56/31]], a mildly flat vicesimononal supraminor seventh ~[[29/16]], and a slightly flat undevicesimal supraminor seventh ~[[38/21]]. On the other hand, after another octave-reduction, six of them yield a slightly sharp septimal minor third ~[[7/6]] (easily reached by moving two keys right from the root note). Unlike actual sensipent, seven of them miss the fifth ~[[3/2]] (which would be 52/89, one key down and three keys left from an octave above the root note), instead yielding 53\89 which is a near-just septendecimal wide fifth ~[[68/45]]. After another octave-reduction, nine generators (for which actual sensipent would yield a classic major third ~[[5/4]]) yield the near-just Boethian major third ~[[24/19]] (at 30\89, and easily reached by moving three keys right from the root note, and a very good substitute for ~5/4). Twenty generators are needed to reach the near-just fourth ~[[4/3]] (at 37\89), putting it out of reach of the 8L 3s scale, but easily reached by moving three keys right plus one key up, while the fifth ~[[3/2]] is reached by moving five keys right plus one key down-right from the root note). The [[8L 3s]] (10:3 step ratio) scale is much less lopsided than the myna mapping and minimizes note loss at the left and right edges. [[Bryan Deister]] has demonstrated this mapping in [https://www.youtube.com/watch?v=Rjx1zr3HIE0 ''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. | |||
{{Lumatone EDO mapping|n=89|start=5|xstep=10|ystep=-7}} | {{Lumatone EDO mapping|n=89|start=5|xstep=10|ystep=-7}} | ||
Latest revision as of 04:38, 2 November 2026
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 diatonic 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.
Myna
The myna mapping is quite lopsided, with its 4L 7s scale having a 17:3 step ratio, but 89edo is the optimal patent val for the temperament and efficiently gives access to the full 11-limit. The generator is a mildly flat classic minor third ~6/5 (23\89). 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 (however, not ~13/16 unless 89f is used); 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. 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).
Orgone
The Orgone mapping also has a 4L 7s scale, but with a 10:7 step ratio; it 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. 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 fewer than in the myna mapping.
Sensipent-related temperament
It is possible to map 89edo by stacking near-just (only slightly sharp) tricesimoprimal ultramajor thirds ~31/24 (33\89), as in a 2.3.5.7.17.19.29.31 subgroup version of sensipent temperament, but not tempering out the sensipent comma 78732/78125 (which is instead tempered to -2\89). Stacking two tricesimoprimal ultramajor thirds yields a mildly sharp classic major sixth ~5/3 (easily accessed by moving moving two keys right plus one key down-right from the root note). After octave-reduction, three of them make a mildly flat lesser tridecimal neutral second ~13/12 (which is notable for being just one key right). Unlike actual sensipent or extensions thereof, five of these generators miss the classic major seventh ~9/5 (which would be 75\89, two keys down from an octave above the root note), instead yielding 76\89 functioning as a near-just tricesoprimal supermajor sixth ~56/31, a mildly flat vicesimononal supraminor seventh ~29/16, and a slightly flat undevicesimal supraminor seventh ~38/21. On the other hand, after another octave-reduction, six of them yield a slightly sharp septimal minor third ~7/6 (easily reached by moving two keys right from the root note). Unlike actual sensipent, seven of them miss the fifth ~3/2 (which would be 52/89, one key down and three keys left from an octave above the root note), instead yielding 53\89 which is a near-just septendecimal wide fifth ~68/45. After another octave-reduction, nine generators (for which actual sensipent would yield a classic major third ~5/4) yield the near-just Boethian major third ~24/19 (at 30\89, and easily reached by moving three keys right from the root note, and a very good substitute for ~5/4). Twenty generators are needed to reach the near-just fourth ~4/3 (at 37\89), putting it out of reach of the 8L 3s scale, but easily reached by moving three keys right plus one key up, while the fifth ~3/2 is reached by moving five keys right plus one key down-right from the root note). The 8L 3s (10:3 step ratio) scale is much less lopsided than the myna mapping and 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.
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.