Superpyth–22 equivalence continuum - Revision history

FloraC: Update link

2026-04-05T12:21:17Z

Update link

← Older revision		Revision as of 12:21, 5 April 2026
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	! ''n'' !! ''m'' !! Temperament !! Comma		! ''n'' !! ''m'' !! Temperament !! Comma
	\|-		\|-
	\| 11/5 = 2.2 \|\| 11/6 = 1.8{{overline\|3}} \|\| [[Hendecatonic]] \|\| {{monzo\| 43 -11 -11 }}		\| 11/5 = 2.2 \|\| 11/6 = 1.8{{overline\|3}} \|\| [[Hendecatonic (temperament)\|Hendecatonic]] \|\| {{monzo\| 43 -11 -11 }}
	\|-		\|-
	\| 9/4 = 2.25 \|\| 9/5 = 1.8 \|\| [[Escapade]] \|\| {{monzo\| 32 -7 -9 }}		\| 9/4 = 2.25 \|\| 9/5 = 1.8 \|\| [[Escapade]] \|\| {{monzo\| 32 -7 -9 }}

FloraC: - doublewide and comic (addressed in misc. 5-limit temps) since this is now a page of mathematical interest

2025-08-23T11:20:02Z

- doublewide and comic (addressed in misc. 5-limit temps) since this is now a page of mathematical interest

@@ Line 122: / Line 122: @@
 | 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Magic]] || {{monzo| -10 -1 5 }}
 |}
 [[Category:22edo]]
 [[Category:Equivalence continua]]

FloraC: - CTE & POTE tunings

2025-08-23T11:17:04Z

- CTE & POTE tunings

← Older revision		Revision as of 11:17, 23 August 2025
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	* [[CWE]]: ~625/432 = 600.0000{{c}}, ~6/5 = 325.8147{{c}} (~144/125 = 274.185{{c}})		* [[CWE]]: ~625/432 = 600.0000{{c}}, ~6/5 = 325.8147{{c}} (~144/125 = 274.185{{c}})
	: error map: {{val\| 0.000 +1.304 -8.869 }}		: error map: {{val\| 0.000 +1.304 -8.869 }}
	~~<!-- * [[CTE]]: ~625/432 = 600.000{{c}}, ~6/5 = 326.170{{c}} (~144/125 = 273.830{{c}})~~
	~~: error map: {{val\| 0.000 +2.727 -7.802 }} -->~~

	{{Optimal ET sequence\|legend=1\| 4, 14b, 18, 22, 48, 70c, 118c }}		{{Optimal ET sequence\|legend=1\| 4, 14b, 18, 22, 48, 70c, 118c }}
Line 162:		Line 160:
	* [[CWE]]: ~2187/1600 = 600.0000{{c}}, ~40/27 = 655.0201{{c}} (~81/80 = 55.0201{{c}})		* [[CWE]]: ~2187/1600 = 600.0000{{c}}, ~40/27 = 655.0201{{c}} (~81/80 = 55.0201{{c}})
	: error map: {{val\| 0.000 +8.085 -1.173 }}		: error map: {{val\| 0.000 +8.085 -1.173 }}
	~~<!-- * [[CTE]]: ~2187/1600 = 600.000{{c}}, ~40/27 = 654.560{{c}} (~81/80 = 54.560{{c}})~~
	~~: error map: {{val\| 0.000 +7.165 -4.393 }}~~
	* [[POTE]]: ~2187/1600 = 600.000{{c}}, ~40/27 = 655.382{{c}} (~81/80 = 55.382{{c}})
	~~: error map: {{val\| 0.000 +8.809 +1.361 }} -->~~

	{{Optimal ET sequence\|legend=1\| 20c, 22, 86b, 108b, 130b }}		{{Optimal ET sequence\|legend=1\| 20c, 22, 86b, 108b, 130b }}

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

2025-08-23T11:16:45Z

Switch to Sintel's badness, WE & CWE tunings, per community consensus

← Older revision		Revision as of 11:16, 23 August 2025
Line 135:		Line 135:

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~625/432 = 600.~~000~~, ~6/5 = ~~326~~.~~170~~ (~144/125 = ~~273~~.~~830~~)		* [[WE]]: ~625/432 = 600.6814{{c}}, ~6/5 = 325.8892{{c}} (~144/125 = 274.7921{{c}})
	: [[error map]]: {{val\| 0.~~000~~ +2.~~727~~ -7.~~802~~ }}		: [[error map]]: {{val\| +1.363 +2.283 -6.602 }}
	* [[CWE]]: ~625/432 = 600.~~000~~, ~6/5 = 325.~~815~~ (~144/125 = 274.185)		* [[CWE]]: ~625/432 = 600.0000{{c}}, ~6/5 = 325.8147{{c}} (~144/125 = 274.185{{c}})
	: error map: {{val\| 0.000 +1.304 -8.869 }}		: error map: {{val\| 0.000 +1.304 -8.869 }}
			<!-- * [[CTE]]: ~625/432 = 600.000{{c}}, ~6/5 = 326.170{{c}} (~144/125 = 273.830{{c}})
			: error map: {{val\| 0.000 +2.727 -7.802 }} -->

	{{Optimal ET sequence\|legend=1\| 4, 14b, 18, 22, 48, 70c, 118c }}		{{Optimal ET sequence\|legend=1\| 4, 14b, 18, 22, 48, 70c, 118c }}

	[[Badness]]:		[[Badness]] (Sintel): 5.32
	* Smith: 0.227
	* Dirichlet: 5.32

	== Comic (5-limit) ==		== Comic (5-limit) ==
Line 158:		Line 158:

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~2187/1600 = 600.000, ~40/27 = 654.560 (~81/80 = 54.560)		* [[WE]]: ~2187/1600 = 598.9674{{c}}, ~40/27 = 654.2542{{c}} (~81/80 = 55.2868{{c}})
	: [[error map]]: {{val\| 0.000 +7.165 -4.393 }}		: [[error map]]: {{val\| -2.065 +5.521 -3.436 }}
	* [[POTE]]: ~2187/1600 = 600.000, ~40/27 = 655.382 (~81/80 = 55.382)		* [[CWE]]: ~2187/1600 = 600.0000{{c}}, ~40/27 = 655.0201{{c}} (~81/80 = 55.0201{{c}})
	: error map: {{val\| 0.000 +8.809 +1.361 }}		: error map: {{val\| 0.000 +8.085 -1.173 }}
			<!-- * [[CTE]]: ~2187/1600 = 600.000{{c}}, ~40/27 = 654.560{{c}} (~81/80 = 54.560{{c}})
			: error map: {{val\| 0.000 +7.165 -4.393 }}
			* [[POTE]]: ~2187/1600 = 600.000{{c}}, ~40/27 = 655.382{{c}} (~81/80 = 55.382{{c}})
			: error map: {{val\| 0.000 +8.809 +1.361 }} -->

	{{Optimal ET sequence\|legend=1\| 20c, 22, 86b, 108b, 130b }}		{{Optimal ET sequence\|legend=1\| 20c, 22, 86b, 108b, 130b }}

	[[Badness]] (~~Smith~~): 0.~~491181~~		[[Badness]] (Sintel): 11.5

	[[Category:22edo]]		[[Category:22edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

FloraC: Style

2025-08-23T11:07:04Z

Style

← Older revision		Revision as of 11:07, 23 August 2025
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	All temperaments in the continuum satisfy {{nowrap\| (20480/19683)<sup>''n''</sup> ~ 250/243 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to 5-limit [[superpyth]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by 22edo due to it being the unique equal temperament that [[tempering out\|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 2.284531…, and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\| (20480/19683)<sup>''n''</sup> ~ 250/243 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to 5-limit [[superpyth]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by 22edo due to it being the unique equal temperament that [[tempering out\|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 2.284531…, and temperaments having ''n'' near this value tend to be the most accurate ones.

	The 22-comma is the characteristic 3-limit comma tempered out in 22edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1\|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1\|harmonic 3]] in the generator chain. For an ''n'' that is not coprime with 22, however, the corresponding temperament splits the [[octave]] into gcd(''n'',~~ ~~22) parts, and splits the interval class of 3 into ''n''/gcd(''n'',~~ ~~22). For example:		The 22-comma is the characteristic 3-limit comma tempered out in 22edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1\|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1\|harmonic 3]] in the generator chain. For an ''n'' that is not coprime with 22, however, the corresponding temperament splits the [[octave]] into {{nowrap\| gcd(''n'', 22) }} parts, and splits the interval class of 3 into {{nowrap\| ''n''/gcd(''n'', 22) }}. For example:
	* [[Quasisuper]] ({{nowrap\|''n'' {{=}} 1}}) is generated by a fifth with an unsplit octave;		* [[Quasisuper]] ({{nowrap\| ''n'' {{=}} 1 }}) is generated by a fifth with an unsplit octave;
	* [[Diaschismic]] ({{nowrap\|''n'' {{=}} 2}}) splits the octave in two, as 2 divides 22;		* [[Diaschismic]] ({{nowrap\| ''n'' {{=}} 2 }}) splits the octave in two, as 2 divides 22;
	* [[Porcupine]] ({{nowrap\|''n'' {{=}} 3}}) splits the fourth in three, as 3 is coprime with 22;		* [[Porcupine]] ({{nowrap\| ''n'' {{=}} 3 }}) splits the fourth in three, as 3 is coprime with 22;
	* Etc.		* Etc.

	{\| class="wikitable center-1"		{\| class="wikitable center-1"
	\|+ style="font-size: 105%;" \| Temperaments ~~in the continuum~~		\|+ style="font-size: 105%;" \| Temperaments with integer ''n''
	\|-		\|-
	! rowspan="2" \| ''n''		! rowspan="2" \| ''n''
Line 62:		Line 62:
	\|}		\|}

	We may also invert the continuum by setting ''m'' such that {{nowrap\|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''~~quasisuper-22~~ equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.778495… The quasisuper comma is both larger and more complex than the superpyth comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.		We may also invert the continuum by setting ''m'' such that {{nowrap\| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''quasisuper–22 equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.778495…. The quasisuper comma is both larger and more complex than the superpyth comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.

	{\| class="wikitable center-1"		{\| class="wikitable center-1"
	\|+ style="font-size: 105%;" \| Temperaments ~~in the continuum~~		\|+ style="font-size: 105%;" \| Temperaments with integer ''m''
	\|-		\|-
	! rowspan="2" \| ''m''		! rowspan="2" \| ''m''

FloraC: Mark as mathematical interest for lack of justification

2025-08-13T14:42:04Z

Mark as mathematical interest for lack of justification

← Older revision		Revision as of 14:42, 13 August 2025
Line 1:		Line 1:
	~~The '''superpyth-22 equivalence continuum''' is a [[equivalence continuum\|continuum]] of 5-limit [[regular temperament\|temperaments]] which equate a number of [[superpyth comma]]s,~~ {{~~nowrap\|20480/19683 {{=}} {{monzo\| 12 -9 1 }}}}, with the 22-comma, {{monzo\| 35 -22~~ }}~~. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[22edo]].~~		{{Mathematical interest}}

	All temperaments in the continuum satisfy {{nowrap\|(20480/19683)<sup>''n''</sup> ~ 250/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to 5-limit [[superpyth]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by 22edo due to it being the unique equal temperament that [[tempering out\|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 2.284531…, and temperaments having ''n'' near this value tend to be the most accurate ones.		The '''superpyth-22 equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equate a number of [[superpyth comma]]s, 20480/19683 ({{monzo\| 12 -9 1 }}), with the 22-comma, {{monzo\| 35 -22 }}. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[22edo]].

			All temperaments in the continuum satisfy {{nowrap\| (20480/19683)<sup>''n''</sup> ~ 250/243 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to 5-limit [[superpyth]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by 22edo due to it being the unique equal temperament that [[tempering out\|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 2.284531…, and temperaments having ''n'' near this value tend to be the most accurate ones.

	The 22-comma is the characteristic 3-limit comma tempered out in 22edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1\|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1\|harmonic 3]] in the generator chain. For an ''n'' that is not coprime with 22, however, the corresponding temperament splits the [[octave]] into gcd(''n'', 22) parts, and splits the interval class of 3 into ''n''/gcd(''n'', 22). For example:		The 22-comma is the characteristic 3-limit comma tempered out in 22edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1\|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1\|harmonic 3]] in the generator chain. For an ''n'' that is not coprime with 22, however, the corresponding temperament splits the [[octave]] into gcd(''n'', 22) parts, and splits the interval class of 3 into ''n''/gcd(''n'', 22). For example:

Lériendil: 5-limit interp

2025-07-23T03:09:46Z

5-limit interp

← Older revision		Revision as of 03:09, 23 July 2025
Line 133:		Line 133:

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~625/432 = 600.000, ~6/5 = 326.170 (~7/6 = 273.830)		* [[CTE]]: ~625/432 = 600.000, ~6/5 = 326.170 (~144/125 = 273.830)
	: [[error map]]: {{val\| 0.000 +2.727 -7.802 }}		: [[error map]]: {{val\| 0.000 +2.727 -7.802 }}
	* [[CWE]]: ~625/432 = 600.000, ~6/5 = 325.815 (~7/6 = 274.185)		* [[CWE]]: ~625/432 = 600.000, ~6/5 = 325.815 (~144/125 = 274.185)
	: error map: {{val\| 0.000 +1.304 -8.869 }}		: error map: {{val\| 0.000 +1.304 -8.869 }}

FloraC: +link to wizard

2025-01-27T11:54:46Z

+link to wizard

← Older revision		Revision as of 11:54, 27 January 2025
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	\|-		\|-
	\| 7/3 = 2.{{overline\|3}} \|\| 7/4 = 1.75 \|\| [[Orson]] \|\| {{monzo\| -21 3 7 }}		\| 7/3 = 2.{{overline\|3}} \|\| 7/4 = 1.75 \|\| [[Orson]] \|\| {{monzo\| -21 3 7 }}
			\|-
			\| 12/5 = 2.4 \|\| 12/7 = 1.{{overline\|714285}} \|\| [[Wizard]] \|\| {{monzo\| -31 2 12 }}
	\|-		\|-
	\| 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 }}

FloraC: -superpyth (more properly addressed in the syntonic--diatonic continuum)

2025-01-27T11:47:50Z

-superpyth (more properly addressed in the syntonic--diatonic continuum)

@@ Line 118: / Line 118: @@
 | 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Magic]] || {{monzo| -10 -1 5 }}
 |}
 == Doublewide (5-limit) ==

FloraC: Complete

2024-12-25T10:18:15Z

Complete

← Older revision		Revision as of 10:18, 25 December 2024
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	== Superpyth (5-limit) ==		== Superpyth (5-limit) ==
	: ''For extensions, see [[Archytas clan #Superpyth]] and [[Jubilismic clan #Bipyth]].''		: ''For extensions, see [[Archytas clan #Superpyth]] and [[Jubilismic clan #Bipyth]].''

			In the 5-limit, superpyth tempers out [[20480/19683]]. It has a fifth generator of ~3/2 = ~710¢ and ~5/4 is found at +9 generator steps, as an augmented second (C–D#).

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	{{Mapping\|legend=1\| 1 0 -12 \| 0 1 9 }}		{{Mapping\|legend=1\| 1 0 -12 \| 0 1 9 }}

			: mapping generators: ~2, ~3

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	* Dirichlet: 5.32		* Dirichlet: 5.32

	== Comic ==		== Comic (5-limit) ==
	: ''For extensions, see [[Jubilismic clan #Comic]].''		: ''For extensions, see [[Jubilismic clan #Comic]].''

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	{{Mapping\|legend=1\| 2 1 -3 \| 0 2 7 }}		{{Mapping\|legend=1\| 2 1 -3 \| 0 2 7 }}

			: mapping generators: ~2187/1600, ~40/27

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[POTE]]: ~600.000, ~81/80 = 55.382		* [[CTE]]: ~2187/1600 = 600.000, ~40/27 = 654.560 (~81/80 = 54.560)
			: [[error map]]: {{val\| 0.000 +7.165 -4.393 }}
			* [[POTE]]: ~2187/1600 = 600.000, ~40/27 = 655.382 (~81/80 = 55.382)
			: error map: {{val\| 0.000 +8.809 +1.361 }}

	{{Optimal ET sequence\|legend=1\| 20c, 22, 86b, 108b, 130b }}		{{Optimal ET sequence\|legend=1\| 20c, 22, 86b, 108b, 130b }}

← Older revision		Revision as of 11:07, 23 August 2025
Line 5:		Line 5:
	All temperaments in the continuum satisfy {{nowrap\| (20480/19683)<sup>''n''</sup> ~ 250/243 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to 5-limit [[superpyth]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by 22edo due to it being the unique equal temperament that [[tempering out\|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 2.284531…, and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\| (20480/19683)<sup>''n''</sup> ~ 250/243 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to 5-limit [[superpyth]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by 22edo due to it being the unique equal temperament that [[tempering out\|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 2.284531…, and temperaments having ''n'' near this value tend to be the most accurate ones.

	The 22-comma is the characteristic 3-limit comma tempered out in 22edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1\|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1\|harmonic 3]] in the generator chain. For an ''n'' that is not coprime with 22, however, the corresponding temperament splits the [[octave]] into gcd(''n'',~~ ~~22) parts, and splits the interval class of 3 into ''n''/gcd(''n'',~~ ~~22). For example:		The 22-comma is the characteristic 3-limit comma tempered out in 22edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1\|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1\|harmonic 3]] in the generator chain. For an ''n'' that is not coprime with 22, however, the corresponding temperament splits the [[octave]] into {{nowrap\| gcd(''n'', 22) }} parts, and splits the interval class of 3 into {{nowrap\| ''n''/gcd(''n'', 22) }}. For example:
	* [[Quasisuper]] ({{nowrap\|''n'' {{=}} 1}}) is generated by a fifth with an unsplit octave;		* [[Quasisuper]] ({{nowrap\| ''n'' {{=}} 1 }}) is generated by a fifth with an unsplit octave;
	* [[Diaschismic]] ({{nowrap\|''n'' {{=}} 2}}) splits the octave in two, as 2 divides 22;		* [[Diaschismic]] ({{nowrap\| ''n'' {{=}} 2 }}) splits the octave in two, as 2 divides 22;
	* [[Porcupine]] ({{nowrap\|''n'' {{=}} 3}}) splits the fourth in three, as 3 is coprime with 22;		* [[Porcupine]] ({{nowrap\| ''n'' {{=}} 3 }}) splits the fourth in three, as 3 is coprime with 22;
	* Etc.		* Etc.

	{\| class="wikitable center-1"		{\| class="wikitable center-1"
	\|+ style="font-size: 105%;" \| Temperaments ~~in the continuum~~		\|+ style="font-size: 105%;" \| Temperaments with integer ''n''
	\|-		\|-
	! rowspan="2" \| ''n''		! rowspan="2" \| ''n''
Line 62:		Line 62:
	\|}		\|}

	We may also invert the continuum by setting ''m'' such that {{nowrap\|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''~~quasisuper-22~~ equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.778495… The quasisuper comma is both larger and more complex than the superpyth comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.		We may also invert the continuum by setting ''m'' such that {{nowrap\| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''quasisuper–22 equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.778495…. The quasisuper comma is both larger and more complex than the superpyth comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.

	{\| class="wikitable center-1"		{\| class="wikitable center-1"
	\|+ style="font-size: 105%;" \| Temperaments ~~in the continuum~~		\|+ style="font-size: 105%;" \| Temperaments with integer ''m''
	\|-		\|-
	! rowspan="2" \| ''m''		! rowspan="2" \| ''m''

← Older revision		Revision as of 14:42, 13 August 2025
Line 1:		Line 1:
	~~The '''superpyth-22 equivalence continuum''' is a [[equivalence continuum\|continuum]] of 5-limit [[regular temperament\|temperaments]] which equate a number of [[superpyth comma]]s,~~ {{~~nowrap\|20480/19683 {{=}} {{monzo\| 12 -9 1 }}}}, with the 22-comma, {{monzo\| 35 -22~~ }}~~. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[22edo]].~~		{{Mathematical interest}}

	All temperaments in the continuum satisfy {{nowrap\|(20480/19683)<sup>''n''</sup> ~ 250/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to 5-limit [[superpyth]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by 22edo due to it being the unique equal temperament that [[tempering out\|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 2.284531…, and temperaments having ''n'' near this value tend to be the most accurate ones.		The '''superpyth-22 equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equate a number of [[superpyth comma]]s, 20480/19683 ({{monzo\| 12 -9 1 }}), with the 22-comma, {{monzo\| 35 -22 }}. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[22edo]].

			All temperaments in the continuum satisfy {{nowrap\| (20480/19683)<sup>''n''</sup> ~ 250/243 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to 5-limit [[superpyth]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by 22edo due to it being the unique equal temperament that [[tempering out\|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 2.284531…, and temperaments having ''n'' near this value tend to be the most accurate ones.

	The 22-comma is the characteristic 3-limit comma tempered out in 22edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1\|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1\|harmonic 3]] in the generator chain. For an ''n'' that is not coprime with 22, however, the corresponding temperament splits the [[octave]] into gcd(''n'', 22) parts, and splits the interval class of 3 into ''n''/gcd(''n'', 22). For example:		The 22-comma is the characteristic 3-limit comma tempered out in 22edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1\|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1\|harmonic 3]] in the generator chain. For an ''n'' that is not coprime with 22, however, the corresponding temperament splits the [[octave]] into gcd(''n'', 22) parts, and splits the interval class of 3 into ''n''/gcd(''n'', 22). For example: