Lumatone mapping for 89edo - Revision history

Lucius Chiaraviglio: 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"

2026-11-02T04:38:55Z

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"

← Older revision		Revision as of 04:38, 2 November 2026
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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}}

	~~== Other Mappings ==~~		== Myna ==
	~~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.~~		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).
	=== 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. 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 ===		== 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~~, 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 many notes at the left and right edges.		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 ===		== Sensipent-related temperament ==
	~~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 [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~~. However, this does not put simple ratios as close together as the previous two~~.		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}}

Hotcrystal0: meantone → diatonic

2026-11-02T02:34:19Z

meantone → diatonic

← Older revision		Revision as of 02:34, 2 November 2026
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	== Diatonic ==		== 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 [https://www.youtube.com/shorts/mU4KcJd6E7U ''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.		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 [https://www.youtube.com/shorts/mU4KcJd6E7U ''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.
	{{Lumatone EDO mapping\|n=89\|start=2\|xstep=15\|ystep=-8}}		{{Lumatone EDO mapping\|n=89\|start=2\|xstep=15\|ystep=-8}}

Lucius Chiaraviglio: /* Sensipent (2.3.5.31 version) */ Probably better put "Sensipent-related" until I get a chance to check out how many actual Sensipent intervals this hits with the 33\89 generator, since 89edo tempers the 5-limit Sensipent comma to -2\89

2026-11-01T09:27:52Z

Sensipent (2.3.5.31 version): Probably better put "Sensipent-related" until I get a chance to check out how many actual Sensipent intervals this hits with the 33\89 generator, since 89edo tempers the 5-limit Sensipent comma to -2\89

← Older revision		Revision as of 09:27, 1 November 2026
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	{{Lumatone EDO mapping\|n=89\|start=76\|xstep=7\|ystep=3}}		{{Lumatone EDO mapping\|n=89\|start=76\|xstep=7\|ystep=3}}

	=== Sensipent ~~(2.3.5.31 version)~~ ===		=== Sensipent-related temperament ===
	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 [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. However, this does not put simple ratios as close together as the previous two.		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 [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. However, this does not put simple ratios as close together as the previous two.
	{{Lumatone EDO mapping\|n=89\|start=5\|xstep=10\|ystep=-7}}		{{Lumatone EDO mapping\|n=89\|start=5\|xstep=10\|ystep=-7}}

Lucius Chiaraviglio: /* Sensipent */ 89edo doesnt temper out the 5-limit Sensipent comma, but it uses the 2.3.31 version of the generator ~31/24

2026-11-01T09:19:03Z

Sensipent: 89edo doesnt temper out the 5-limit Sensipent comma, but it uses the 2.3.31 version of the generator ~31/24

← Older revision		Revision as of 09:19, 1 November 2026
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	{{Lumatone EDO mapping\|n=89\|start=76\|xstep=7\|ystep=3}}		{{Lumatone EDO mapping\|n=89\|start=76\|xstep=7\|ystep=3}}

	=== Sensipent ===		=== Sensipent (2.3.5.31 version) ===
	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 [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. However, this does not put simple ratios as close together as the previous two.		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 [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. However, this does not put simple ratios as close together as the previous two.
	{{Lumatone EDO mapping\|n=89\|start=5\|xstep=10\|ystep=-7}}		{{Lumatone EDO mapping\|n=89\|start=5\|xstep=10\|ystep=-7}}

Lucius Chiaraviglio: Fix typo that broke the Lumatone mapping intro template

2026-11-01T09:10:10Z

Fix typo that broke the Lumatone mapping intro template

← Older revision		Revision as of 09:10, 1 November 2026
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	{Lumatone mapping intro}} Due to the size of the edo, this mapping covers less than 2/3rds of the notes.		{{Lumatone mapping intro}} Due to the size of the edo, this mapping covers less than 2/3rds of the notes.

	== Diatonic ==		== Diatonic ==

Lucius Chiaraviglio: /* 8L 3s */ Found Sensipent temperament; will have to fix the rest of this later

2026-11-01T08:52:28Z

8L 3s: Found Sensipent temperament; will have to fix the rest of this later

← Older revision		Revision as of 08:52, 1 November 2026
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	{{Lumatone EDO mapping\|n=89\|start=76\|xstep=7\|ystep=3}}		{{Lumatone EDO mapping\|n=89\|start=76\|xstep=7\|ystep=3}}

	=== ~~8L 3s~~ ===		=== Sensipent ===
	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 [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. However, this does not put simple ratios as close together as the previous two.		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 [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. However, this does not put simple ratios as close together as the previous two.
	{{Lumatone EDO mapping\|n=89\|start=5\|xstep=10\|ystep=-7}}		{{Lumatone EDO mapping\|n=89\|start=5\|xstep=10\|ystep=-7}}

Lucius Chiaraviglio: /* Other Mappings */ I misread the 8L 3s scale tree, but Sensoid is neither the correct scale name nor a temperament name found in the Sensi/Sensipent pages; also restore Myna temperament quick description text and text about notes chopped off at the left and right edges; will have to fix this up some more later

2026-11-01T08:43:30Z

Other Mappings: I misread the 8L&nbsp3s scale tree, but Sensoid is neither the correct scale name nor a temperament name found in the Sensi/Sensipent pages; also restore Myna temperament quick description text and text about notes chopped off at the left and right edges; will have to fix this up some more later

← Older revision		Revision as of 08:43, 1 November 2026
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	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.		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 ===		=== 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.		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 ===		=== 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.		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, 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 many notes at the left and right edges.
	{{Lumatone EDO mapping\|n=89\|start=76\|xstep=7\|ystep=3}}		{{Lumatone EDO mapping\|n=89\|start=76\|xstep=7\|ystep=3}}

	=== ~~Sensoid~~ ===		=== 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 [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. However, this does not put simple ratios as close together as the previous two.		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 [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. However, this does not put simple ratios as close together as the previous two.
	{{Lumatone EDO mapping\|n=89\|start=5\|xstep=10\|ystep=-7}}		{{Lumatone EDO mapping\|n=89\|start=5\|xstep=10\|ystep=-7}}

Yourmusic Productions: 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.

2026-11-01T08:06:54Z

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.

← Older revision		Revision as of 08:06, 1 November 2026
Line 1:		Line 1:
	{{Lumatone mapping intro}} Due to the size of the edo, this mapping covers less than 2/3rds of the notes.		{Lumatone mapping intro}} Due to the size of the edo, this mapping covers less than 2/3rds of the notes.

	== Diatonic ==		== Diatonic ==
Line 5:		Line 5:
	{{Lumatone EDO mapping\|n=89\|start=2\|xstep=15\|ystep=-8}}		{{Lumatone EDO mapping\|n=89\|start=2\|xstep=15\|ystep=-8}}

	== ~~Orgone~~ ==		== Other Mappings ==
	~~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.		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.
			{{Lumatone EDO mapping\|n=89\|start=58\|xstep=3\|ystep=14}}

			=== 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.
	{{Lumatone EDO mapping\|n=89\|start=76\|xstep=7\|ystep=3}}		{{Lumatone EDO mapping\|n=89\|start=76\|xstep=7\|ystep=3}}

	== ~~Myna~~ ==		=== Sensoid ===
	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]].		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 [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. However, this does not put simple ratios as close together as the previous two.

	~~=== 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).
	~~{{Lumatone EDO mapping\|n=89\|start=58\|xstep=3\|ystep=14}}~~

	~~=== 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 [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}}

Lucius Chiaraviglio: /* 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)

2026-10-31T23:33:22Z

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)

← Older revision		Revision as of 23:33, 31 October 2026
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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}}

	== ~~Other Mappings~~ ==		== Orgone ==
	~~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~~.		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.
	=== Myna ===		{{Lumatone EDO mapping\|n=89\|start=76\|xstep=7\|ystep=3}}
	~~The~~ [[myna]] ~~mapping is quite lopsided~~, ~~but~~ is the optimal patent val ~~for the temperament and~~ efficiently gives access to the full 11-limit.
			== 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).
	{{Lumatone EDO mapping\|n=89\|start=58\|xstep=3\|ystep=14}}		{{Lumatone EDO mapping\|n=89\|start=58\|xstep=3\|ystep=14}}

	=== ~~Orgone~~ ===		=== 8L 3s ===
	~~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~~.		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 [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=76\|xstep=7\|ystep=3}}		{{Lumatone EDO mapping\|n=89\|start=5\|xstep=10\|ystep=-7}}

	== Char-based + Sesquart rank-3 temperament ==		== Char-based + Sesquart rank-3 temperament ==

Lucius Chiaraviglio: /* Diatonic */ Demo video: Left out the artist name by mistake; also add more specific date because another one is coming

2026-10-29T07:08:16Z

Diatonic: Demo video: Left out the artist name by mistake; also add more specific date because another one is coming

← Older revision		Revision as of 07:08, 29 October 2026
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	== Diatonic ==		== 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 ~~demonstrated~~ in [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026), and it would be possible to switch between presets for different subsets of 89edo to enable modulation to remote keys.		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 [https://www.youtube.com/shorts/mU4KcJd6E7U ''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.
	{{Lumatone EDO mapping\|n=89\|start=2\|xstep=15\|ystep=-8}}		{{Lumatone EDO mapping\|n=89\|start=2\|xstep=15\|ystep=-8}}