Lumatone mapping for 86edo - Revision history

Lucius Chiaraviglio: /* Unnamed subgroup 2.3.5♭.5♯.7.11♭.17♭ rank-3 temperament */ Add 13/8

2026-08-30T22:26:44Z

Unnamed subgroup 2.3.5♭.5♯.7.11♭.17♭ rank-3 temperament: Add 13/8

← Older revision		Revision as of 22:26, 30 August 2026
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	== Unnamed subgroup 2.3.5♭.5♯.7.11♭.17♭ rank-3 temperament ==		== Unnamed subgroup 2.3.5♭.5♯.7.11♭.17♭ rank-3 temperament ==
	[[Bryan Deister]] has demonstrated a [[6L 2s]] (13:4 step ratio) mapping for [[86edo]] in [https://www.youtube.com/shorts/BHHlnlEvelI ''86edo groove''] (2026). This mapping requires use of the 86ceg val in combination with the 86eg val (dual 5th harmonic, subgroup 2.3.5♭.5♯.7.11♭.17♭), in which several large errors come close to canceling each other out. The rightward generator 13\86 functions as a near-just ptolemaic whole tone ~[[10/9\|10♭/9]]; the down-right generator 4\86 functions as a sharp large septendecimal quartertone ~[[34/33]]; the MOS chroma is a flat tridecimal 2/3-tone ~[[14/13]]; and the semi-octave is mapped as a near-just [[99/70\|99/70♭]] ~ [[140/99\|140♭/99]] (the kalisma [[9801/9800]] is tempered out). Backing down from the semioctave by one rightward generator makes a near-just undecimal (pentacircle) major third ~[[14/11]]; backing down by two of these from the semioctave (or going up by one of these plus one down-right generator from the root note) makes a sharp septimal whole tone ~[[8/7]]~~; in~~ the forward direction, four of the rightward generators make a near-just undevicesimal diminished sixth ~[[38/25]] constituted as 38/(5♭×5♯) before semioctave-reduction; semioctave-reduction turns this into the aforementioned MOS chroma. Going up by three down-right generators make a near-just classic diatonic semitone ~[[16/15\|16/15♯]]; seven of them make the sharp classic major third ~[[5/4\|5♯/4]]; nine down-right generators (usually incurring a vertical wraparound) or four MOS chromas make the sharp fourth ~[[4/3]]. The range is a bit over four octaves, which slope slightly upwards, but this mapping misses a considerable number of notes in each octave.		[[Bryan Deister]] has demonstrated a [[6L 2s]] (13:4 step ratio) mapping for [[86edo]] in [https://www.youtube.com/shorts/BHHlnlEvelI ''86edo groove''] (2026). This mapping requires use of the 86ceg val in combination with the 86eg val (dual 5th harmonic, subgroup 2.3.5♭.5♯.7.11♭.17♭), in which several large errors come close to canceling each other out. The rightward generator 13\86 functions as a near-just ptolemaic whole tone ~[[10/9\|10♭/9]]; the down-right generator 4\86 functions as a sharp large septendecimal quartertone ~[[34/33]]; the MOS chroma is a flat tridecimal 2/3-tone ~[[14/13]]; and the semi-octave is mapped as a near-just [[99/70\|99/70♭]] ~ [[140/99\|140♭/99]] (the kalisma [[9801/9800]] is tempered out). Backing down from the semioctave by one rightward generator makes a near-just undecimal (pentacircle) major third ~[[14/11]]; backing down by two of these from the semioctave (or going up by one of these plus one down-right generator from the root note) makes a sharp septimal whole tone ~[[8/7]]. Backing down by two of these from the octave instead of the semioctave makes a moderately flat tridecimal neutral sixth ~[[13/8]]. In the forward direction from the root note, four of the rightward generators make a near-just undevicesimal diminished sixth ~[[38/25]] constituted as 38/(5♭×5♯) before semioctave-reduction; semioctave-reduction turns this into the aforementioned MOS chroma. Going up by three down-right generators make a near-just classic diatonic semitone ~[[16/15\|16/15♯]]; seven of them make the sharp classic major third ~[[5/4\|5♯/4]]; nine down-right generators (usually incurring a vertical wraparound) or four MOS chromas make the sharp fourth ~[[4/3]]. The range is a bit over four octaves, which slope slightly upwards, but this mapping misses a considerable number of notes in each octave.
	{{Lumatone EDO mapping\|n=86\|start=15\|xstep=13\|ystep=-9}}		{{Lumatone EDO mapping\|n=86\|start=15\|xstep=13\|ystep=-9}}

	{{Navbox Lumatone}}		{{Navbox Lumatone}}

Lucius Chiaraviglio: Minor reorganization; add Bryan Deister's Unnamed subgroup 2.3.5♭.5♯.7.11♭.17♭ rank-3 temperament mapping

2026-08-30T22:17:25Z

Minor reorganization; add Bryan Deister's Unnamed subgroup 2.3.5♭.5♯.7.11♭.17♭ rank-3 temperament mapping

← Older revision		Revision as of 22:17, 30 August 2026
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	{{Lumatone mapping intro}} ~~You can use the b val, but it is very sharp and will not cover all the notes due to the size of the edo.~~		{{Lumatone mapping intro}}

	== Diatonic ==		== Diatonic ==
			You can use the b val, but it is very sharp and will not cover all the notes due to the size of the edo.
	{{Lumatone EDO mapping\|n=86\|start=20\|xstep=16\|ystep=-13}}		{{Lumatone EDO mapping\|n=86\|start=20\|xstep=16\|ystep=-13}}

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	[[Bryan Deister]] has demonstrated a [[9L 1s]] mapping for [[86edo]] in [https://www.youtube.com/shorts/5_W323Iea18 ''microtonal improvisation in 86edo''] (2025). The rightwards generator 9\86 is a slightly flat tridecimal supraminor second (~[[14\13]]), which is a quarter of the somewhat sharp perfect fourth (~[[4/3]]), as befits Tesseract temperament ([[28812/28561]] is tempered out); unfortunately, the infill extension of Tesseract to include the 5th harmonic (as three of these generators) reaches the second-best approximation of it for 86 (86c val); this also puts this mapping on the spectrum for [[Negri]] temperament in the 5-limit, although not in the 7-limit, since 86edo does not temper ~[[8/7]], ~[[7/6]], and ~[[15/13]] to the same interval (instead, it represents them all as distinct intervals). It is possible to reach all of the notes in each octave with sufficient stacking of only the rightward generator; but for convenience, it may be preferable to add as a second generator the upwards generator 4\86, a chromatic quarter-tone that functions primarily as the al-Farabi quarter tone ~[[33/32]], but also ~[[30/29]], ~[[31/30]], and ~[[32/31]]. The range is just a bit over three octaves with no missed notes, and the octaves slope up substantially, hence the placement of the first note 0 in the lower left corner.		[[Bryan Deister]] has demonstrated a [[9L 1s]] mapping for [[86edo]] in [https://www.youtube.com/shorts/5_W323Iea18 ''microtonal improvisation in 86edo''] (2025). The rightwards generator 9\86 is a slightly flat tridecimal supraminor second (~[[14\13]]), which is a quarter of the somewhat sharp perfect fourth (~[[4/3]]), as befits Tesseract temperament ([[28812/28561]] is tempered out); unfortunately, the infill extension of Tesseract to include the 5th harmonic (as three of these generators) reaches the second-best approximation of it for 86 (86c val); this also puts this mapping on the spectrum for [[Negri]] temperament in the 5-limit, although not in the 7-limit, since 86edo does not temper ~[[8/7]], ~[[7/6]], and ~[[15/13]] to the same interval (instead, it represents them all as distinct intervals). It is possible to reach all of the notes in each octave with sufficient stacking of only the rightward generator; but for convenience, it may be preferable to add as a second generator the upwards generator 4\86, a chromatic quarter-tone that functions primarily as the al-Farabi quarter tone ~[[33/32]], but also ~[[30/29]], ~[[31/30]], and ~[[32/31]]. The range is just a bit over three octaves with no missed notes, and the octaves slope up substantially, hence the placement of the first note 0 in the lower left corner.
	{{Lumatone EDO mapping\|n=86\|start=82\|xstep=9\|ystep=-4}}		{{Lumatone EDO mapping\|n=86\|start=82\|xstep=9\|ystep=-4}}

			== Unnamed subgroup 2.3.5♭.5♯.7.11♭.17♭ rank-3 temperament ==
			[[Bryan Deister]] has demonstrated a [[6L 2s]] (13:4 step ratio) mapping for [[86edo]] in [https://www.youtube.com/shorts/BHHlnlEvelI ''86edo groove''] (2026). This mapping requires use of the 86ceg val in combination with the 86eg val (dual 5th harmonic, subgroup 2.3.5♭.5♯.7.11♭.17♭), in which several large errors come close to canceling each other out. The rightward generator 13\86 functions as a near-just ptolemaic whole tone ~[[10/9\|10♭/9]]; the down-right generator 4\86 functions as a sharp large septendecimal quartertone ~[[34/33]]; the MOS chroma is a flat tridecimal 2/3-tone ~[[14/13]]; and the semi-octave is mapped as a near-just [[99/70\|99/70♭]] ~ [[140/99\|140♭/99]] (the kalisma [[9801/9800]] is tempered out). Backing down from the semioctave by one rightward generator makes a near-just undecimal (pentacircle) major third ~[[14/11]]; backing down by two of these from the semioctave (or going up by one of these plus one down-right generator from the root note) makes a sharp septimal whole tone ~[[8/7]]; in the forward direction, four of the rightward generators make a near-just undevicesimal diminished sixth ~[[38/25]] constituted as 38/(5♭×5♯) before semioctave-reduction; semioctave-reduction turns this into the aforementioned MOS chroma. Going up by three down-right generators make a near-just classic diatonic semitone ~[[16/15\|16/15♯]]; seven of them make the sharp classic major third ~[[5/4\|5♯/4]]; nine down-right generators (usually incurring a vertical wraparound) or four MOS chromas make the sharp fourth ~[[4/3]]. The range is a bit over four octaves, which slope slightly upwards, but this mapping misses a considerable number of notes in each octave.
			{{Lumatone EDO mapping\|n=86\|start=15\|xstep=13\|ystep=-9}}

	{{Navbox Lumatone}}		{{Navbox Lumatone}}

Yourmusic Productions: b val diatonic mapping

2025-07-22T08:51:00Z

b val diatonic mapping

← Older revision		Revision as of 08:51, 22 July 2025
Line 1:		Line 1:
	{{Lumatone mapping intro}}		{{Lumatone mapping intro}} You can use the b val, but it is very sharp and will not cover all the notes due to the size of the edo.

			== Diatonic ==
			{{Lumatone EDO mapping\|n=86\|start=20\|xstep=16\|ystep=-13}}

	== Tesseract + Negri ==		== Tesseract + Negri ==

Lucius Chiaraviglio: Create Lumatone mapping for 86edo, starting with Bryan Deister's Lumatone mapping for 86edo

2025-07-21T09:05:06Z

Create Lumatone mapping for 86edo, starting with Bryan Deister's Lumatone mapping for 86edo

New page

{{Lumatone mapping intro}}

== Tesseract + Negri ==
[[Bryan Deister]] has demonstrated a [[9L 1s]] mapping for [[86edo]] in [https://www.youtube.com/shorts/5_W323Iea18 ''microtonal improvisation in 86edo''] (2025). The rightwards generator 9\86 is a slightly flat tridecimal supraminor second (~[[14\13]]), which is a quarter of the somewhat sharp perfect fourth (~[[4/3]]), as befits Tesseract temperament ([[28812/28561]] is tempered out); unfortunately, the infill extension of Tesseract to include the 5th harmonic (as three of these generators) reaches the second-best approximation of it for 86 (86c val); this also puts this mapping on the spectrum for [[Negri]] temperament in the 5-limit, although not in the 7-limit, since 86edo does not temper ~[[8/7]], ~[[7/6]], and ~[[15/13]] to the same interval (instead, it represents them all as distinct intervals). It is possible to reach all of the notes in each octave with sufficient stacking of only the rightward generator; but for convenience, it may be preferable to add as a second generator the upwards generator 4\86, a chromatic quarter-tone that functions primarily as the al-Farabi quarter tone ~[[33/32]], but also ~[[30/29]], ~[[31/30]], and ~[[32/31]]. The range is just a bit over three octaves with no missed notes, and the octaves slope up substantially, hence the placement of the first note 0 in the lower left corner.
{{Lumatone EDO mapping|n=86|start=82|xstep=9|ystep=-4}}

{{Navbox Lumatone}}