Meantone family - Revision history

Sintel: Undo revision 229909 by Xenllium (talk)

2026-05-09T11:18:23Z

@@ Line 115: / Line 115: @@
 [[Badness]] (Sintel): 0.347
 === Undecimal meantone (huygens) ===
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 * [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{dead link}}
 * [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 ''Twinkle canon – 50 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
 ==== Tridecimal meanpop ====

Xenllium: /* Septimal meantone */added reasonable extensions

2026-05-09T06:52:49Z

Septimal meantone: added reasonable extensions

@@ Line 115: / Line 115: @@
 [[Badness]] (Sintel): 0.347
 === Undecimal meantone (huygens) ===
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 * [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{dead link}}
 * [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 ''Twinkle canon – 50 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
 ==== Tridecimal meanpop ====

Godtone: /* Mohajira */ missing info; noticed due to discussion with fumica

2026-04-07T12:48:36Z

Mohajira: missing info; noticed due to discussion with fumica

← Older revision		Revision as of 12:48, 7 April 2026
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	{{Main\| Mohajira }}		{{Main\| Mohajira }}

	Mohajira can be viewed as derived from [[mohaha]] which maps the interval half a [[chromatic semitone\|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap\| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31.		Mohajira can be viewed as derived from [[mohaha]] which maps the interval half a [[chromatic semitone\|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap\| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31. Note that while 24 + 31 = [[55edo]] doesn't apear in the optimal ET sequence, it is a [[patent val]] tuning and recommendable if you prefer a light meantone tempering.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

Overthink: /* Meanpop */ point to new title

2026-03-21T01:55:21Z

Meanpop: point to new title

← Older revision		Revision as of 01:55, 21 March 2026
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	=== Meanpop ===		=== Meanpop ===
	{{See also\| ~~Meantone~~ vs meanpop }}		{{See also\| Huygens vs meanpop }}

	Meanpop<ref name="meantone & meanpop 2003"/><ref name="meantone & meanpop 2004"/> maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop maps the 13/8 to the double-augmented fifth (C–G𝄪), tempering out 144/143 like in grosstone. Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second.		Meanpop<ref name="meantone & meanpop 2003"/><ref name="meantone & meanpop 2004"/> maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop maps the 13/8 to the double-augmented fifth (C–G𝄪), tempering out 144/143 like in grosstone. Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second.

Xenllium at 23:03, 21 February 2026

2026-02-21T23:03:10Z

← Older revision		Revision as of 23:03, 21 February 2026
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	Badness (Sintel): 1.28		Badness (Sintel): 1.28

			=== Migration ===
			See [[Rastmic clan #Migration\|Rastmic clan]].

	== Flattone ==		== Flattone ==
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	Badness (Sintel): 1.12		Badness (Sintel): 1.12

	=== 13-limit ===		==== 13-limit ====
	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

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	Badness (Sintel): 0.920		Badness (Sintel): 0.920

			=== Ptolemy ===
			See [[Rastmic clan #Ptolemy\|Rastmic clan]].

	== Dominant ==		== Dominant ==
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	Badness (Sintel): 0.864		Badness (Sintel): 0.864

			=== Neutrominant ===
			See [[Rastmic clan #Neutrominant\|Rastmic clan]].

	== Flattertone ==		== Flattertone ==

Overthink: /* 19-limit */ fix optimal tunings

2026-01-19T05:09:56Z

19-limit: fix optimal tunings

← Older revision		Revision as of 05:09, 19 January 2026
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~2 = 1199.~~655~~{{c}}, ~3/2 = 698.~~295~~{{c}}		* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}}
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.~~488~~{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}}

	{{Optimal ET sequence\|legend=1\| 12f, 55f, 67 }}		{{Optimal ET sequence\|legend=1\| 12f, 55f, 67 }}

Overthink: /* Mildtone */ Extend to the 19-limit

2026-01-19T05:03:17Z

Mildtone: Extend to the 19-limit

← Older revision		Revision as of 05:03, 19 January 2026
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	[[Badness]] (Sintel): 2.67		[[Badness]] (Sintel): 2.67

			=== 11-limit ===

			[[Subgroup]]: 2.3.5.7.11

			[[Comma list]]: 81/80, 176/175, 7058/6875

			{{Mapping\|legend=1\| 1 0 -4 -32 \| 0 1 4 22 30}}

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}

			{{Optimal ET sequence\|legend=1\| 12, 43de, 55, 67 }}

			[[Badness]] (Sintel): 2.15

			=== 13-limit ===

			[[Subgroup]]: 2.3.5.7.11.13

			[[Comma list]]: 81/80, 176/175, 196/195, 832/825

			{{Mapping\|legend=1\| 1 0 -4 -32 -44 \| 0 1 4 22 30}}

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}

			{{Optimal ET sequence\|legend=1\| 12f, 55f, 67 }}

			[[Badness]] (Sintel): 2.04

			=== 17-limit ===

			[[Subgroup]]: 2.3.5.7.11.13.17

			[[Comma list]]: 81/80, 176/175, 189/197, 196/195, 832/825

			{{Mapping\|legend=1\| 1 0 -4 -32 -44 12\| 0 1 4 22 30 -5}}

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}

			{{Optimal ET sequence\|legend=1\| 12f, 55f, 67 }}

			[[Badness]] (Sintel): 1.98

			=== 19-limit ===

			[[Subgroup]]: 2.3.5.7.11.13.19

			[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825

			{{Mapping\|legend=1\| 1 0 -4 -32 -44 12 9\| 0 1 4 22 30 -5 -3}}

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}

			{{Optimal ET sequence\|legend=1\| 12f, 55f, 67 }}

			[[Badness]] (Sintel): 1.95

			{{Todo\|unify precision\|review}}

	== Supermean ==		== Supermean ==

Overthink: /* Mohajira */ link

2026-01-11T00:25:57Z

Mohajira: link

← Older revision		Revision as of 00:25, 11 January 2026
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	{{Main\| Mohajira }}		{{Main\| Mohajira }}

	Mohajira can be viewed as derived from mohaha which maps the interval half a [[chromatic semitone\|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap\| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31.		Mohajira can be viewed as derived from [[mohaha]] which maps the interval half a [[chromatic semitone\|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap\| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

Overthink: /* 17-limit */ the other extension has been removed

2025-12-25T08:19:03Z

17-limit: the other extension has been removed

← Older revision		Revision as of 08:19, 25 December 2025
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	===== 17-limit =====		===== 17-limit =====
	This extension ~~is perhaps the most practical, as it~~ maps 17/16 to the minor second (C–D♭), and 19/16 to the minor third (C–E♭), suitable for a system generated by a mildly tempered fifth.		This extension maps 17/16 to the minor second (C–D♭), and 19/16 to the minor third (C–E♭), suitable for a system generated by a mildly tempered fifth.

	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.13.17

FloraC: /* Septimal meantone */ move certain things here from meantone article

2025-12-23T12:09:03Z

Septimal meantone: move certain things here from meantone article

← Older revision		Revision as of 12:09, 23 December 2025
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	{{Wikipedia\| Septimal meantone temperament }}		{{Wikipedia\| Septimal meantone temperament }}

	In septimal meantone, ten fifths get to the interval class for 7, so that [[7/4]] is an augmented sixth (C–A♯), [[7/6]] is an augmented second (C–D♯), [[7/5]] is an augmented fourth (C–F♯), and [[21/16]] is an augmented third (C–E♯). ~~Septimal meantone tempers out~~ the ~~common 7-limit commas~~ [[126/125]], [[225/224]], and [[3136/3125]] ~~and in~~ fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 ~~and~~ 225/224.		In septimal meantone, ten fifths get to the interval class for 7, so that [[7/4]] is an augmented sixth (C–A♯), [[7/6]] is an augmented second (C–D♯), [[7/5]] is an augmented fourth (C–F♯), and [[21/16]] is an augmented third (C–E♯). This mapping is rationalized by the fact that 81/80 factors as ([[126/125]])⋅([[225/224]]), and septimal meantone tempers out both of these commas as well as their difference, [[3136/3125]]. In fact it can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125, 225/224, and 3136/3125.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7