Father–3 equivalence continuum - Revision history

FloraC: Remove the k-continuum since no one is actively arguing for it. Also remove the 3 & 33c temp, which is unenlighted result of looking at the continuum that way

2026-06-22T14:43:51Z

Remove the k-continuum since no one is actively arguing for it. Also remove the 3 & 33c temp, which is unenlighted result of looking at the continuum that way

@@ Line 110: / Line 110: @@
 | 11/4 = 2.75 || 11/7 = 1.{{overline|571428}} || [[Magus]] || {{monzo| 24 1 -11 }}
 |}
 == Mutt (5-limit) ==

FloraC at 14:40, 22 June 2026

2026-06-22T14:40:04Z

← Older revision		Revision as of 14:40, 22 June 2026
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	! ''n'' !! ''m''!! Temperament \|\| Comma		! ''n'' !! ''m''!! Temperament \|\| Comma
	\|-		\|-
	\| 7/3 = 2.{{overline\|3}} \|\| 7/4 = 1.75 \|\| [[Wesley]] \|\| {{monzo\| 13 2 -7 }}		\| 7/3 = 2.{{overline\|3}} \|\| 7/4 = 1.75 \|\| [[Wesley]] \|\| {{monzo\| -13 -2 7 }}
	\|-		\|-
	\| 5/2 = 2.5 \|\| 5/3 = 1.{{overline\|6}} \|\| [[Magic]] \|\| {{monzo\| 10 1 -5 }}		\| 5/2 = 2.5 \|\| 5/3 = 1.{{overline\|6}} \|\| [[Magic]] \|\| {{monzo\| -10 -1 5 }}
			\|-
			\| 21/8 = 2.625 \|\| 21/13 = 1.{{overline\|615384}} \|\| [[Mutt]] \|\| {{monzo\| -44 -3 21 }}
	\|-		\|-
	\| 29/11 = 2.{{overline\|63}} \|\| 29/18 = 1.6{{overline\|1}} \|\| [[Squarschmidt]] \|\| {{monzo\| 61 4 -29 }}		\| 29/11 = 2.{{overline\|63}} \|\| 29/18 = 1.6{{overline\|1}} \|\| [[Squarschmidt]] \|\| {{monzo\| 61 4 -29 }}

FloraC: Move 5-limit mutt here

2026-06-22T14:32:57Z

Move 5-limit mutt here

← Older revision		Revision as of 14:32, 22 June 2026
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	[[Badness]] (Sintel): 16.0		[[Badness]] (Sintel): 16.0

			== Mutt (5-limit) ==
			{{Main\| Mutt }}
			: ''For extensions, see [[Horwell temperaments #Mutt]].''

			[[Subgroup]]: 2.3.5

			[[Comma list]]: {{monzo\| -44 -3 21 }}

			{{Mapping\|legend=1\| 3 -2 6 \| 0 7 1 }}
			: mapping generators: ~98304/78125, ~5/4

			[[Optimal tuning]]s:
			* [[WE]]: ~98304/78125 = 400.0227{{c}}, ~5/4 = 386.0017{{c}} (~393216/390625 = 14.0210{{c}})
			: [[error map]]: {{val\| +0.068 +0.012 -0.176 }}
			* [[CWE]]: ~98304/78125 = 400.0000{{c}}, ~5/4 = 385.9858{{c}} (~393216/390625 = 14.0142{{c}})
			: error map: {{val\| 0.000 -0.055 -0.328 }}

			{{Optimal ET sequence\|legend=1\| 84, 87, 171, 771, 942, 1113, 1284, 1455, 4194cc, 5649cc }}

			[[Badness]] (Sintel): 3.81

	== Isnes ==		== Isnes ==

FloraC: Actual fix

2026-06-11T07:43:47Z

Actual fix

← Older revision		Revision as of 07:43, 11 June 2026
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	The '''father–3 equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equate a number of [[16/15\|classical diatonic semitones (16/15)]] with the [[32/27\|Pythagorean minor third (32/27)]].		The '''father–3 equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equate a number of [[16/15\|classical diatonic semitones (16/15)]] with the [[32/27\|Pythagorean minor third (32/27)]].

	Note that because 3et is a record equal temperament in the 2.5 subgroup, the continuum can be conceptualized as the [[Father–3 equivalence continuum/Godtone's approach]\|''augmented–dicot equivalence continuum'']], which Godtone argues is easier to understand, with characteristic 2.5-subgroup comma [[128/125]] as the interval with a single factor of 3 is [[25/24]].		Note that because 3et is a record equal temperament in the [[2.5 subgroup]], the continuum can be conceptualized as the [[Father–3 equivalence continuum/Godtone's approach\|''augmented–dicot equivalence continuum'']], which Godtone argues is easier to understand, with characteristic 2.5-subgroup [[comma]] [[128/125]] as the interval with a single factor of 3 is [[25/24]].

	All temperaments in the continuum satisfy {{nowrap\|(16/15)<sup>''n''</sup> ~ 32/27}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[father]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] 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 2.63252…, and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(16/15)<sup>''n''</sup> ~ 32/27}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[father]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] 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 2.63252…, and temperaments having ''n'' near this value tend to be the most accurate ones.
Line 109:		Line 109:
	\|}		\|}

	Because 3et is a record equal temperament in ~~the 2.3 subgroup and (especially)~~ the 2.5 subgroup, there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is [[128/125]], and the interval with a single factor of 3 is [[25/24]]. As such, Godtone has conceptualized this continuum as ''augmented–dicot equivalence continuum''. See [[{{PAGENAME}}/Godtone's approach]].		Because 3et is a record equal temperament in the 2.5 subgroup, there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is 128/125, and the interval with a single factor of 3 is 25/24. As such, Godtone has conceptualized this continuum as ''augmented–dicot equivalence continuum''. See [[{{PAGENAME}}/Godtone's approach]].

	Others prefer conceptualizing this continuum in terms of {{nowrap\| ''k'' {{=}} {{sfrac\|1\|''n'' − 2}} }} such that temperaments satisfy {{nowrap\|(25/24)<sup>''k''</sup> {{=}} 16/15}}. This gives rise to the name ''chromatic–diatonic equivalence continuum'', where both ''chromatic'' and ''diatonic'' refer to the classical versions of semitones. The just value of ''k'' is approximately 1.58097…		Others prefer conceptualizing this continuum in terms of {{nowrap\| ''k'' {{=}} {{sfrac\|1\|''n'' − 2}} }} such that temperaments satisfy {{nowrap\|(25/24)<sup>''k''</sup> {{=}} 16/15}}. This gives rise to the name ''chromatic–diatonic equivalence continuum'', where both ''chromatic'' and ''diatonic'' refer to the classical versions of semitones. The just value of ''k'' is approximately 1.58097…

Godtone: fix linkage

2026-06-10T19:31:31Z

fix linkage

← Older revision		Revision as of 19:31, 10 June 2026
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	The '''father–3 equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equate a number of [[16/15\|classical diatonic semitones (16/15)]] with the [[32/27\|Pythagorean minor third (32/27)]].		The '''father–3 equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equate a number of [[16/15\|classical diatonic semitones (16/15)]] with the [[32/27\|Pythagorean minor third (32/27)]].

	Note that because 3et is a record equal temperament in the [[2.5 subgroup]], the continuum can be conceptualized as the [[~~{{PAGENAME}}~~/Godtone's approach]\|''augmented–dicot equivalence continuum'']], which Godtone argues is easier to understand, with characteristic 2.5-subgroup comma [[128/125]] as the interval with a single factor of 3 is [[25/24]].		Note that because 3et is a record equal temperament in the 2.5 subgroup, the continuum can be conceptualized as the [[Father–3 equivalence continuum/Godtone's approach]\|''augmented–dicot equivalence continuum'']], which Godtone argues is easier to understand, with characteristic 2.5-subgroup comma [[128/125]] as the interval with a single factor of 3 is [[25/24]].

	All temperaments in the continuum satisfy {{nowrap\|(16/15)<sup>''n''</sup> ~ 32/27}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[father]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] 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 2.63252…, and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(16/15)<sup>''n''</sup> ~ 32/27}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[father]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] 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 2.63252…, and temperaments having ''n'' near this value tend to be the most accurate ones.
Line 109:		Line 109:
	\|}		\|}

	Because 3et is a record equal temperament in the 2.3 subgroup and (especially) the [[2.5 subgroup]], there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is [[128/125]], and the interval with a single factor of 3 is [[25/24]]. As such, Godtone has conceptualized this continuum as ''augmented–dicot equivalence continuum''. See [[{{PAGENAME}}/Godtone's approach]].		Because 3et is a record equal temperament in the 2.3 subgroup and (especially) the 2.5 subgroup, there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is [[128/125]], and the interval with a single factor of 3 is [[25/24]]. As such, Godtone has conceptualized this continuum as ''augmented–dicot equivalence continuum''. See [[{{PAGENAME}}/Godtone's approach]].

	Others prefer conceptualizing this continuum in terms of {{nowrap\| ''k'' {{=}} {{sfrac\|1\|''n'' − 2}} }} such that temperaments satisfy {{nowrap\|(25/24)<sup>''k''</sup> {{=}} 16/15}}. This gives rise to the name ''chromatic–diatonic equivalence continuum'', where both ''chromatic'' and ''diatonic'' refer to the classical versions of semitones. The just value of ''k'' is approximately 1.58097…		Others prefer conceptualizing this continuum in terms of {{nowrap\| ''k'' {{=}} {{sfrac\|1\|''n'' − 2}} }} such that temperaments satisfy {{nowrap\|(25/24)<sup>''k''</sup> {{=}} 16/15}}. This gives rise to the name ''chromatic–diatonic equivalence continuum'', where both ''chromatic'' and ''diatonic'' refer to the classical versions of semitones. The just value of ''k'' is approximately 1.58097…

Godtone: i will at least make it more clearly linked to

2026-06-10T19:30:22Z

i will at least make it more clearly linked to

← Older revision		Revision as of 19:30, 10 June 2026
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	The '''father–3 equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equate a number of [[16/15\|classical diatonic semitones (16/15)]] with the [[32/27\|Pythagorean minor third (32/27)]].		The '''father–3 equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equate a number of [[16/15\|classical diatonic semitones (16/15)]] with the [[32/27\|Pythagorean minor third (32/27)]].

			Note that because 3et is a record equal temperament in the [[2.5 subgroup]], the continuum can be conceptualized as the [[{{PAGENAME}}/Godtone's approach]\|''augmented–dicot equivalence continuum'']], which Godtone argues is easier to understand, with characteristic 2.5-subgroup comma [[128/125]] as the interval with a single factor of 3 is [[25/24]].

	All temperaments in the continuum satisfy {{nowrap\|(16/15)<sup>''n''</sup> ~ 32/27}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[father]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] 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 2.63252…, and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(16/15)<sup>''n''</sup> ~ 32/27}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[father]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] 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 2.63252…, and temperaments having ''n'' near this value tend to be the most accurate ones.
Line 107:		Line 109:
	\|}		\|}

	Because 3et is a record equal temperament in ~~both~~ 2.3 and [[2.5 subgroup]]s, there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is [[128/125]], and the interval with a single factor of 3 is [[25/24]]. As such, Godtone has conceptualized this continuum as ''augmented–dicot equivalence continuum''. See [[{{PAGENAME}}/Godtone's approach]].		Because 3et is a record equal temperament in the 2.3 subgroup and (especially) the [[2.5 subgroup]], there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is [[128/125]], and the interval with a single factor of 3 is [[25/24]]. As such, Godtone has conceptualized this continuum as ''augmented–dicot equivalence continuum''. See [[{{PAGENAME}}/Godtone's approach]].

	Others prefer conceptualizing this continuum in terms of {{nowrap\| ''k'' {{=}} {{sfrac\|1\|''n'' − 2}} }} such that temperaments satisfy {{nowrap\|(25/24)<sup>''k''</sup> {{=}} 16/15}}. This gives rise to the name ''chromatic–diatonic equivalence continuum'', where both ''chromatic'' and ''diatonic'' refer to the classical versions of semitones. The just value of ''k'' is approximately 1.58097…		Others prefer conceptualizing this continuum in terms of {{nowrap\| ''k'' {{=}} {{sfrac\|1\|''n'' − 2}} }} such that temperaments satisfy {{nowrap\|(25/24)<sup>''k''</sup> {{=}} 16/15}}. This gives rise to the name ''chromatic–diatonic equivalence continuum'', where both ''chromatic'' and ''diatonic'' refer to the classical versions of semitones. The just value of ''k'' is approximately 1.58097…

FloraC: + position of squarschmidt in the continuum. + link

2026-04-28T13:12:16Z

+ position of squarschmidt in the continuum. + link

← Older revision		Revision as of 13:12, 28 April 2026
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	\|-		\|-
	\| 5/2 = 2.5 \|\| 5/3 = 1.{{overline\|6}} \|\| [[Magic]] \|\| {{monzo\| 10 1 -5 }}		\| 5/2 = 2.5 \|\| 5/3 = 1.{{overline\|6}} \|\| [[Magic]] \|\| {{monzo\| 10 1 -5 }}
			\|-
			\| 29/11 = 2.{{overline\|63}} \|\| 29/18 = 1.6{{overline\|1}} \|\| [[Squarschmidt]] \|\| {{monzo\| 61 4 -29 }}
	\|-		\|-
	\| 8/3 = 2.{{overline\|6}} \|\| 8/5 = 1.6 \|\| [[Würschmidt]] \|\| {{monzo\| 17 1 -8 }}		\| 8/3 = 2.{{overline\|6}} \|\| 8/5 = 1.6 \|\| [[Würschmidt]] \|\| {{monzo\| 17 1 -8 }}
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	== Squarschmidt (5-limit) ==		== Squarschmidt (5-limit) ==
	A generator for the ~~squarschimidt~~ temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents.		: ''For extensions, see [[Hemimage temperaments #Squarschmidt]].''

			A generator for the squarschmidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents.

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

FloraC: Move classical squarschmidt here

2026-04-28T13:04:05Z

Move classical squarschmidt here

@@ Line 199: / Line 199: @@
 [[Badness]] (Sintel): 30.4
 [[Category:3edo]]
 [[Category:Equivalence continua]]

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

2026-04-28T13:01:02Z

Switch to Sintel's badness, WE & CWE tunings

← Older revision		Revision as of 13:01, 28 April 2026
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	== 3 & 33c ==		== 3 & 33c ==
	This low-accuracy high-complexity temperament corresponds to {{nowrap\|''n'' {{=}} 9/4}} and {{nowrap\|''m'' {{=}} 9/5}}.		This low-accuracy high-complexity temperament corresponds to {{nowrap\| ''n'' {{=}} 9/4 }} and {{nowrap\| ''m'' {{=}} 9/5 }}.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	{{Mapping\|legend=1\| 3 2 6 \| 0 3 1 }}		{{Mapping\|legend=1\| 3 2 6 \| 0 3 1 }}

	: mapping generators: ~125/96, ~5/4		: mapping generators: ~125/96, ~5/4

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~125/96 = ~~1\3~~, ~5/4 = ~~368~~.~~2534~~ (~25/24 = 31.~~7466~~)		* [[WE]]: ~125/96 = 401.2633{{c}}, ~5/4 = 367.0585{{c}} (~25/24 = 34.2047{{c}})
	* [[CWE]]: ~125/96 = ~~1\3~~, ~5/4 = 366.8103 (~25/24 = 33.1897)		: [[error map]]: {{val\| +3.790 +1.747 -11.676 }}
			* [[CWE]]: ~125/96 = 400.0000{{c}}, ~5/4 = 366.8103{{c}} (~25/24 = 33.1897{{c}})
			: error map: {{val\| 0.000 -1.524 -19.503 }}

	{{Optimal ET sequence\|legend=1\| 3, …, 33c, 36c, 69cc }}		{{Optimal ET sequence\|legend=1\| 3, …, 33c, 36c, 69cc }}
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	== Isnes ==		== Isnes ==
	Isnes is so called because the generator is half of a [[8/5]] ~~minor sixth~~, in a similar way that [[sensi]] has a generator of half a [[5/3]]. This corresponds to {{nowrap\|''n'' {{=}} 19/7}} and ''m'' {{=}} 19/12}}.		Isnes is so called because the generator is half of a [[5/2]] major tenth, in a similar way that [[sensi]] has a generator of half a [[5/3]] major sixth. This corresponds to {{nowrap\|''n'' {{=}} 19/7 }} and {{nowrap\| ''m'' {{=}} 19/12 }}.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	[[Comma list]]: {{monzo\| 41 2 -19 }}		[[Comma list]]: {{monzo\| 41 2 -19 }}

	{{Mapping\|legend=1\| 1 ~~8 3~~ \| 0 -19 -2 }}		{{Mapping\|legend=1\| 1 -11 1 \| 0 19 2 }}
			: mapping generators: ~2, ~3145728/1953125
	: mapping generators: ~2, ~1953125~~/1572864~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~2 = ~~1\1~~, ~1953125~~/1572864~~ = ~~405~~.~~1689~~		* [[WE]]: ~2 = 1199.2782{{c}}, ~3145728/1953125 = 794.4174{{c}}
	* [[CWE]]: ~2 = ~~1\1~~, ~1953125~~/1572864~~ = ~~405~~.~~1272~~		: [[error map]]: {{val\| -0.722 -0.090 +1.799 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3145728/1953125 = 794.8728{{c}}
			: error map: {{val\| 0.000 +0.628 +3.432 }}

	{{Optimal ET sequence\|legend=1\| 3, 71b, 74, 77, 157, 548ccc }}		{{Optimal ET sequence\|legend=1\| 3, 71b, 74, 77, 157, 548ccc }}

FloraC: Chromatic -> dicot

2026-04-28T11:54:28Z

Chromatic -> dicot

← Older revision		Revision as of 11:54, 28 April 2026
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	\|}		\|}

	Because 3et is a record equal temperament in both 2.3 and [[2.5 subgroup]]s, there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is [[128/125]], and the interval with a single factor of 3 is [[25/24]]. As such, Godtone has conceptualized this continuum as ''~~augmented–chromatic~~ equivalence continuum''~~, where ''chromatic'' refers to the classical version of the semitone~~. See [[{{PAGENAME}}/Godtone's approach]].		Because 3et is a record equal temperament in both 2.3 and [[2.5 subgroup]]s, there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is [[128/125]], and the interval with a single factor of 3 is [[25/24]]. As such, Godtone has conceptualized this continuum as ''augmented–dicot equivalence continuum''. See [[{{PAGENAME}}/Godtone's approach]].

	Others prefer conceptualizing this continuum in terms of {{nowrap\| ''k'' {{=}} {{sfrac\|1\|''n'' − 2}} }} such that temperaments satisfy {{nowrap\|(25/24)<sup>''k''</sup> {{=}} 16/15}}. This gives rise to the name ''chromatic–diatonic equivalence continuum'', where both ''chromatic'' and ''diatonic'' refer to the classical versions of semitones. The just value of ''k'' is approximately 1.58097…		Others prefer conceptualizing this continuum in terms of {{nowrap\| ''k'' {{=}} {{sfrac\|1\|''n'' − 2}} }} such that temperaments satisfy {{nowrap\|(25/24)<sup>''k''</sup> {{=}} 16/15}}. This gives rise to the name ''chromatic–diatonic equivalence continuum'', where both ''chromatic'' and ''diatonic'' refer to the classical versions of semitones. The just value of ''k'' is approximately 1.58097…