Minortone family - Revision history

FloraC: /* Seminar */ review

2026-07-02T07:54:12Z

Seminar: review

← Older revision		Revision as of 07:54, 2 July 2026
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	Badness (Sintel): 0.886		Badness (Sintel): 0.886

	==== 2.3.5.7.11.17 ====		==== 2.3.5.7.11.17 subgroup ====
	Subgroup: 2.3.5.7.11.17		Subgroup: 2.3.5.7.11.17

	Comma list: ~~3025~~/~~3024~~, ~~4375~~/~~4374~~, ~~2100875~~/~~2097152~~, ~~1225~~/~~1224~~		Comma list: 1089/1088, 1225/1224, 1701/1700, 262395/262144

	Mapping: {{mapping\| 2 -2 -6 12 13 4\| 0 17 35 -21 -20 6}}		Mapping: {{mapping\| 2 -2 -6 12 13 4 \| 0 17 35 -21 -20 6 }}

	Optimal tunings:		Optimal tunings:
	* WE: ~99/70 = 600.~~037~~{{c}}, ~10/9 = 182.~~473~~{{c}}		* WE: ~99/70 = 600.0371{{c}}, ~10/9 = 182.4699{{c}}
	* CWE: ~99/70 = 600.~~000~~{{c}}, ~10/9 = 182.~~459~~{{c}}		* CWE: ~99/70 = 600.0000{{c}}, ~10/9 = 182.4589{{c}}

			{{Optimal ET sequence\|legend=0\| 46, 250, 296, 342, 1414de }}

	Badness (Sintel): 0.831		Badness (Sintel): 0.831

Hkm: /* Seminar */

2026-07-01T21:35:28Z

Seminar

← Older revision		Revision as of 21:35, 1 July 2026
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	Badness (Sintel): 0.886		Badness (Sintel): 0.886

			==== 2.3.5.7.11.17 ====
			Subgroup: 2.3.5.7.11.17

			Comma list: 3025/3024, 4375/4374, 2100875/2097152, 1225/1224

			Mapping: {{mapping\| 2 -2 -6 12 13 4\| 0 17 35 -21 -20 6}}

			Optimal tunings:
			* WE: ~99/70 = 600.037{{c}}, ~10/9 = 182.473{{c}}
			* CWE: ~99/70 = 600.000{{c}}, ~10/9 = 182.459{{c}}

			Badness (Sintel): 0.831

	== Domain ==		== Domain ==

FloraC: Hemidomain -> semidomain; 13-limit ore -> orefield (see talk)

2026-06-20T12:53:45Z

Hemidomain -> semidomain; 13-limit ore -> orefield (see talk)

← Older revision		Revision as of 12:53, 20 June 2026
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	Mitonic takes advantage of the fact that [[32/21]] is only a [[ragisma]] shy of (10/9)<sup>4</sup>, and so this 7-limit interpretation where 21 generators gives a ~8/7, if not quite so super-accurate, is more or less inevitable. While [[559edo\|559]] or [[730edo\|730]] are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. Generator chains of 46 or 79 notes can be used for this temperament.		Mitonic takes advantage of the fact that [[32/21]] is only a [[ragisma]] shy of (10/9)<sup>4</sup>, and so this 7-limit interpretation where 21 generators gives a ~8/7, if not quite so super-accurate, is more or less inevitable. While [[559edo\|559]] or [[730edo\|730]] are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. Generator chains of 46 or 79 notes can be used for this temperament.

	Extending mitonic to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: mineral (46 & 171) and ore (46 & ~~125~~). Mineral tempers out 441/440 and 16384/16335 in the 11-limit. Ore tempers out 385/384 and 1331/1323 in the 11-limit, and maps [[11/8]] to three generators. ~~In the 17-limit~~, ~~both mineral~~ and ~~ore temper out 833~~/~~832, 1225~~/~~1224, 1701/1700~~, and ~~4096/4095~~ (~~2.3.5.7.~~13.17 ~~commas)~~. The word ''mineral'' is related to ''mine'' ~~(an excavation from which ore or solid minerals are taken)~~ and ''miner'' ~~(a person who works in a mine~~, ~~also as~~ a pun on ''minor'').		Extending mitonic to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: mineral (46 & 171) and ore (46 & 171e). Mineral tempers out 441/440 and 16384/16335 in the 11-limit. Ore tempers out 385/384 and 1331/1323 in the 11-limit, and maps [[11/8]] to three generators. The latter further branches to goldmine (46 & 125f), which equates [[16/13]] with a stack of two [[10/9]]'s and tempers together [[13/12]] and [[14/13]], and orefield (46 & 125), which uses the same mapping for 13 as mineral. All have the same mapping for 17, at -6 generators. The word ''mineral'' is related to ''mine'' and ''miner'', a pun on ''minor''.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	Badness (Sintel): 1.77		Badness (Sintel): 1.77

	==== ~~13-limit~~ ====		==== Goldmine ====
	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

	Comma list: ~~352~~/~~351~~, 385/384, 1331/1323~~, 3267/3250~~		Comma list: 169/168, 325/324, 385/384, 1331/1323

	Mapping: {{mapping\| 1 -1 -3 6 3 11 \| 0 17 35 -21 3 -48 }}		Mapping: {{mapping\| 1 -1 -3 6 3 4 \| 0 17 35 -21 3 -2 }}

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1200.~~1737~~{{c}}, ~10/9 = 182.~~4966~~{{c}}		* WE: ~2 = 1200.5647{{c}}, ~10/9 = 182.5224{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.~~4705~~{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4409{{c}}

	{{Optimal ET sequence\|legend=0\| 46, ~~125~~, ~~171e~~ }}		{{Optimal ET sequence\|legend=0\| 46, 79, 125f, 171ef }}

	Badness (Sintel): 1.91		Badness (Sintel): 1.62

	===== 17-limit =====		===== 17-limit =====
	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.13.17

	Comma list: ~~352~~/~~351~~, ~~385~~/~~384~~, ~~561~~/~~560~~, ~~715~~/~~714~~, ~~1452~~/~~1445~~		Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323

	Mapping: {{mapping\| 1 -1 -3 6 3 11 5 \| 0 17 35 -21 3 -48 -6 }}		Mapping: {{mapping\| 1 -1 -3 6 3 4 5 \| 0 17 35 -21 3 -2 -6 }}

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1200.~~1403~~{{c}}, ~10/9 = 182.~~4924~~{{c}}		* WE: ~2 = 1200.4506{{c}}, ~10/9 = 182.5124{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.~~4712~~{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4465{{c}}

	{{Optimal ET sequence\|legend=0\| 46, ~~125~~, ~~171e~~ }}		{{Optimal ET sequence\|legend=0\| 46, 125f, 171ef }}

	Badness (Sintel): 1.45		Badness (Sintel): 1.40

	~~==== Goldmine ====~~
	~~The goldmine temperament (46 & 79) is another 13-limit extension of ore, equating [[13/12]] with [[14/13]] and [[16/13]] with two [[10/9]]s.~~

			==== Orefield ====
	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

	Comma list: ~~169~~/~~168, 325/324~~, 385/384, 1331/1323		Comma list: 352/351, 385/384, 1331/1323, 3267/3250

	Mapping: {{mapping\| 1 -1 -3 6 3 4 \| 0 17 35 -21 3 -2 }}		Mapping: {{mapping\| 1 -1 -3 6 3 11 \| 0 17 35 -21 3 -48 }}

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1200.~~5647~~{{c}}, ~10/9 = 182.~~5224~~{{c}}		* WE: ~2 = 1200.1737{{c}}, ~10/9 = 182.4966{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.~~4409~~{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4705{{c}}

	{{Optimal ET sequence\|legend=0\| 46, 79, ~~125f, 171ef~~ }}		{{Optimal ET sequence\|legend=0\| 46, 125, 171e }}

	Badness (Sintel): 1.62		Badness (Sintel): 1.91

	===== 17-limit =====		===== 17-limit =====
	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.13.17

	Comma list: ~~169~~/~~168~~, ~~273~~/~~272~~, ~~325~~/~~324~~, ~~385~~/~~384~~, ~~1331~~/~~1323~~		Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445

	Mapping: {{mapping\| 1 -1 -3 6 3 4 5 \| 0 17 35 -21 3 -2 -6 }}		Mapping: {{mapping\| 1 -1 -3 6 3 11 5 \| 0 17 35 -21 3 -48 -6 }}

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1200.~~4506~~{{c}}, ~10/9 = 182.~~5124~~{{c}}		* WE: ~2 = 1200.1403{{c}}, ~10/9 = 182.4924{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.~~4465~~{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4712{{c}}

	{{Optimal ET sequence\|legend=0\| 46, ~~125f~~, ~~171ef~~ }}		{{Optimal ET sequence\|legend=0\| 46, 125, 171e }}

	Badness (Sintel): 1.40		Badness (Sintel): 1.45

	=== Seminar ===		=== Seminar ===
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	[[Badness]] (Sintel): 0.354		[[Badness]] (Sintel): 0.354

	=== ~~Hemidomain~~ ===		=== Semidomain ===
	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

FloraC: Switch to Sintel's badness, WE & CWE tunings. Improve intro to each entry

2026-06-19T15:09:52Z

Switch to Sintel's badness, WE & CWE tunings. Improve intro to each entry

Show changes

FloraC: Units & misc. cleanup

2026-06-18T08:01:05Z

Units & misc. cleanup

Show changes

FloraC: FloraC moved page Minortonic family to Minortone family: The temp's name is minortone

2026-06-18T07:47:46Z

FloraC moved page Minortonic family to Minortone family: The temp's name is *minortone*

← Older revision	Revision as of 07:47, 18 June 2026
(No difference)

FloraC: Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

2025-11-27T12:20:53Z

Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

← Older revision		Revision as of 12:20, 27 November 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Minortonic family\| ]] <!-- main article -->		[[Category:Minortonic family\| ]] <!-- main article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

Lériendil: Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

2025-06-24T00:42:14Z

Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

← Older revision		Revision as of 00:42, 24 June 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Minortonic family\| ]] <!-- main article -->		[[Category:Minortonic family\| ]] <!-- main article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

VectorGraphics at 04:12, 12 June 2025

2025-06-12T04:12:42Z

← Older revision		Revision as of 04:12, 12 June 2025
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	== Mitonic ==		== Mitonic ==
	As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, {{monzo\| -16 35 -17 }}. ~~Flipping that gives the 5-limit wedgie {{multival\| 17 35 16 }}, which tells us that~~ 10/9 can be taken as the generator, with 17 of them giving a ~6, 18 of them a ~20/3, and 35 of them giving a ~40. The generator should be tuned about 1/16 of a cent flat, with 6<sup>1/17</sup> being 0.06423 cents flat and 40<sup>1/35</sup> being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.		As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, {{monzo\| -16 35 -17 }}. 10/9 can be taken as the generator, with 17 of them giving a ~6, 18 of them a ~20/3, and 35 of them giving a ~40. The generator should be tuned about 1/16 of a cent flat, with 6<sup>1/17</sup> being 0.06423 cents flat and 40<sup>1/35</sup> being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.

	However, as noted before, 32/21 is only a ragisma shy of (10/9)<sup>4</sup>, and so a 7-limit interpretation, if not quite so super-accurate, is more or less inevitable. While 559 or 730 are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. 21 generators gives a ~64/7. [[Mos scale]]s of size 20, 33, 46 or 79 notes can be used for mitonic.		However, as noted before, 32/21 is only a ragisma shy of (10/9)<sup>4</sup>, and so a 7-limit interpretation, if not quite so super-accurate, is more or less inevitable. While 559 or 730 are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. 21 generators gives a ~64/7. [[Mos scale]]s of size 20, 33, 46 or 79 notes can be used for mitonic.
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	{{Mapping\|legend=1\| 1 -1 -3 6 \| 0 17 35 -21 }}		{{Mapping\|legend=1\| 1 -1 -3 6 \| 0 17 35 -21 }}

	~~{{Multival\|legend=1\| 17 35 -21 16 -81 -147 }}~~

	[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 182.458		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 182.458
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	{{Mapping\|legend=1\| 3 -3 -9 -8 \| 0 17 35 36 }}		{{Mapping\|legend=1\| 3 -3 -9 -8 \| 0 17 35 36 }}

	~~{{Multival\|legend=1\| 51 105 108 48 28 -44 }}~~

	[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467		[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467

FloraC: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"

2025-04-13T13:50:53Z

Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"

← Older revision		Revision as of 13:50, 13 April 2025
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	{{Technical data page}}~~<br><br>~~		{{Technical data page}}
	The '''minortonic family''' tempers out the minortone comma (also known as "minortonma"), {{monzo\| -16 35 -17 }}. The head of this family is 5-limit minortone temperament, with generator a minor tone.		The '''minortonic family''' tempers out the minortone comma (also known as "minortonma"), {{monzo\| -16 35 -17 }}. The head of this family is 5-limit minortone temperament, with generator a minor tone.