Syntonic–31 equivalence continuum - Revision history

FloraC at 15:18, 25 February 2026

2026-02-25T15:18:17Z

← Older revision		Revision as of 15:18, 25 February 2026
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	== Counterwürschmidt ==		== Counterwürschmidt ==
	~~{{See also\|~~ Mirkwai clan #Grendel }}		: ''For extensions, see [[Mirkwai clan #Grendel]].''

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5

FloraC: Switch to Sintel's badness, WE & CWE tunings

2026-02-25T15:16:48Z

Switch to Sintel's badness, WE & CWE tunings

Show changes

FloraC: Units & misc. cleanup

2026-02-25T15:03:05Z

Units & misc. cleanup

Show changes

FloraC: Expand (ampersand and valentine)

2025-01-23T10:53:07Z

Expand (ampersand and valentine)

← Older revision		Revision as of 10:53, 23 January 2025
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	== Ampersand ==		== Ampersand ==
	: ''For extensions, see [[Gamelismic clan #Miracle]].''		: ''For extensions, see [[Gamelismic clan #Miracle]].''

			Ampersand is the 5-limit version of miracle, tempering out the [[ampersand comma]], which is the difference between a perfect fifth and a stack of six [[16/15\|classical diatonic semitones]]. It can be described as the 31 & 41 temperament, corresponding to ''n'' = 6.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	{{Mapping\|legend=1\| 1 1 3 \| 0 6 -7 }}		{{Mapping\|legend=1\| 1 1 3 \| 0 6 -7 }}

	[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~16/15 = 116.680		[[Optimal tuning]]s:
			* [[CTE]]: ~2 = 1200.000, ~16/15 = 116.701
			: [[error map]]: {{val\| 0.000 -1.750 -3.219 }}
			* [[CWE]]: ~2 = 1200.000, ~16/15 = 116.680
			: error map: {{val\| 0.000 -1.874 -3.075 }}

	{{Optimal ET sequence\|legend=1\| 10, 21, 31, 41, 72 }}		{{Optimal ET sequence\|legend=1\| 10, 21, 31, 41, 72 }}
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	== Valentine (5-limit) ==		== Valentine (5-limit) ==
	: ''For extensions, see [[Gamelismic clan #Valentine]].''		: ''For extensions, see [[Gamelismic clan #Valentine]].''

			The 5-limit version of valentine tempers out the [[valentine comma]], which is the difference between a perfect fifth and a stack of nine [[25/24\|classical chromatic semitones]]. It can be described as the 31 & 46 temperament, corresponding to ''n'' = 9.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5

FloraC: Move 5-limit valentine here

2025-01-23T10:42:46Z

Move 5-limit valentine here

@@ Line 158: / Line 158: @@
 * Smith: 0.165755
 * Dirichlet: 3.89
 == Quadlaleyo (31 &amp; 70c) ==

FloraC: Sorting by ploidacot; update the data for ampersand

2025-01-06T13:19:31Z

Sorting by ploidacot; update the data for ampersand

← Older revision		Revision as of 13:19, 6 January 2025
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	[http://x31eq.com/cgi-bin/rt.cgi?ets=31_12c&limit=5 The temperament finder - 5-limit 31 & 12c]		[http://x31eq.com/cgi-bin/rt.cgi?ets=31_12c&limit=5 The temperament finder - 5-limit 31 & 12c]

			== Ampersand ==
			: ''For extensions, see [[Gamelismic clan #Miracle]].''

			[[Subgroup]]: 2.3.5

			[[Comma list]]: {{monzo\| -25 7 6 }} = 34171875/33554432

			{{Mapping\|legend=1\| 1 1 3 \| 0 6 -7 }}

			[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~16/15 = 116.680

			{{Optimal ET sequence\|legend=1\| 10, 21, 31, 41, 72 }}

			[[Badness]]:
			* Smith: 0.165755
			* Dirichlet: 3.89

	== Quadlaleyo (31 & 70c) ==		== Quadlaleyo (31 & 70c) ==
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	[http://x31eq.com/cgi-bin/rt.cgi?ets=31_70c&limit=5 The temperament finder - 5-limit 31 & 70c]		[http://x31eq.com/cgi-bin/rt.cgi?ets=31_70c&limit=5 The temperament finder - 5-limit 31 & 70c]

	== ~~Ampersand~~ ==		== Lalasepbigu (31 & 13c) ==
	~~{{See also\| Gamelismic clan #Miracle }}~~		[[Subgroup]]: 2.3.5

	[[~~Subgroup~~]]: ~~2.3.5~~		[[Comma list]]: {{monzo\| -7 25 -14 }} = 847288609443/781250000000

	~~[[Comma list]]:~~ {{~~monzo~~\| -25 ~~7 6~~ }} ~~= 34171875/33554432~~		{{Mapping\|legend=1\| 1 7 12 \| 0 -14 -25 }}

	~~{{Mapping\|legend~~=~~1\| 1 1 3 \| 0 6 -7 }}~~		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~25000/19683 = 464.423

	[[Optimal ~~tuning]] ([[POTE]]): ~2~~ = ~~1200.000~~, ~~~16/15 = 116.673~~		{{Optimal ET sequence\|legend=1\| 13c, 18bc, 31, 44c, 49bc, 75c, 80bc }}

	~~{{Optimal ET sequence\|legend=1\| 10, 21, 31, 41, 72 }}~~		[[Badness]] (Smith): 2.094918

	[~~[Badness]] (Smith)~~: 0.~~165755~~		[http://x31eq.com/cgi-bin/rt.cgi?ets=31_13c&limit=5 The temperament finder - 5-limit 31 & 13c]

	== Counterwürschmidt ==		== Counterwürschmidt ==
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	: mapping generators: ~2, ~5/4		: mapping generators: ~2, ~5/4

	[[Optimal tuning]] ([[CTE]]): ~2 = 1200.~~000~~, ~5/4 = 386.8710		[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~5/4 = 386.8710

	{{Optimal ET sequence\|legend=1\| 28b, 31, 90, 121, 152, 335, 822, 1157c, 1492c }}		{{Optimal ET sequence\|legend=1\| 28b, 31, 90, 121, 152, 335, 822, 1157c, 1492c }}

	[[Badness]] (Smith): 0.420		[[Badness]] (Smith): 0.420

	~~== Lalasepbigu (31 & 13c) ==~~
	~~[[Subgroup]]: 2.3.5~~

	~~[[Comma list]]: {{monzo\| -7 25 -14 }} = 847288609443/781250000000~~

	~~{{Mapping\|legend=1\| 1 7 12 \| 0 -14 -25 }}~~

	~~[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~25000/19683 = 464.423~~

	~~{{Optimal ET sequence\|legend=1\| 13c, 18bc, 31, 44c, 49bc, 75c, 80bc }}~~

	~~[[Badness]] (Smith): 2.094918~~

	~~[http://x31eq.com/cgi-bin/rt.cgi?ets=31_13c&limit=5 The temperament finder - 5-limit 31 & 13c]~~

	[[Category:31edo]]		[[Category:31edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

FloraC: Cleanup

2025-01-06T13:16:18Z

Cleanup

Show changes

ArrowHead294 at 19:45, 17 December 2024

2024-12-17T19:45:29Z

← Older revision		Revision as of 19:45, 17 December 2024
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	The '''~~syntonic-31~~ equivalence continuum''' is a [[equivalence continuum\|continuum]] of 5-limit temperaments which equate a number of [[81/80\|syntonic commas (81/80)]] with a [[31st-octave temperaments\|31-comma ({{monzo\| -49 31 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[31edo]].		The '''syntonic–31 equivalence continuum''' is a [[equivalence continuum\|continuum]] of 5-limit temperaments which equate a number of [[81/80\|syntonic commas (81/80)]] with a [[31st-octave temperaments\|31-comma ({{monzo\| -49 31 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[31edo]].

	All temperaments in the continuum satisfy {{nowrap\|(81/80)<sup>''n''</sup> ~ {{monzo\|-49 31}}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[31edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is approximately 7.46781…, and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(81/80)<sup>''n''</sup> ~ {{monzo\|-49 31}}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[31edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is approximately 7.46781…, and temperaments having ''n'' near this value tend to be the most accurate ones.

ArrowHead294 at 19:43, 17 December 2024

2024-12-17T19:43:00Z

← Older revision		Revision as of 19:43, 17 December 2024
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	The '''syntonic-31 equivalence continuum''' is an [[equivalence continuum\|continuum]] of 5-limit temperaments which equate a number of [[81/80\|syntonic commas (81/80)]] with a [[31st-octave temperaments\|31-comma ({{monzo\| -49 31 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[31edo]].		The '''syntonic-31 equivalence continuum''' is a [[equivalence continuum\|continuum]] of 5-limit temperaments which equate a number of [[81/80\|syntonic commas (81/80)]] with a [[31st-octave temperaments\|31-comma ({{monzo\| -49 31 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[31edo]].

	All temperaments in the continuum satisfy {{nowrap\|(81/80)<sup>''n''</sup> ~ {{monzo\|-49 31}}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[31edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is approximately 7.46781…, and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(81/80)<sup>''n''</sup> ~ {{monzo\|-49 31}}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[31edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is approximately 7.46781…, and temperaments having ''n'' near this value tend to be the most accurate ones.

ArrowHead294 at 16:59, 17 December 2024

2024-12-17T16:59:08Z

← Older revision		Revision as of 16:59, 17 December 2024
Line 1:		Line 1:
	The '''syntonic-31 equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[81/80\|syntonic commas (81/80)]] with a [[31st-octave temperaments\|31-comma ({{monzo\| -49 31 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[31edo]].		The '''syntonic-31 equivalence continuum''' is an [[equivalence continuum\|continuum]] of 5-limit temperaments which equate a number of [[81/80\|syntonic commas (81/80)]] with a [[31st-octave temperaments\|31-comma ({{monzo\| -49 31 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[31edo]].

	All temperaments in the continuum satisfy {{nowrap\|(81/80)<sup>''n''</sup> ~ {{monzo\|-49 31}}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[31edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is approximately 7.46781…, and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(81/80)<sup>''n''</sup> ~ {{monzo\|-49 31}}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[31edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is approximately 7.46781…, and temperaments having ''n'' near this value tend to be the most accurate ones.