Schismic–countercommatic equivalence continuum - Revision history

FloraC: More fixes

2026-05-20T10:07:29Z

More fixes

← Older revision		Revision as of 10:07, 20 May 2026
Line 137:		Line 137:

	Examples of temperaments with fractional values of ''n'':		Examples of temperaments with fractional values of ''n'':
	* [[~~Marvel temperaments #Septimin\|~~Septimin]] ({{nowrap\|''n'' {{=}} −11/2}})		* [[Septimin]] ({{nowrap\|''n'' {{=}} −11/2}})
	* [[Shibboleth]] ({{nowrap\|''n'' {{=}} −9/2}})		* [[Shibboleth]] ({{nowrap\|''n'' {{=}} −9/2}})
	* [[~~Mirkwai clan #Pluto\|~~Pluto]] ({{nowrap\|''n'' {{=}} −7/2}})		* [[Pluto]] ({{nowrap\|''n'' {{=}} −7/2}})
	* 3737 & 5585 ({{nowrap\|''n'' {{=}} 31/3 {{=}} 10.{{overline\|3}}}})		* 3737 & 5585 ({{nowrap\|''n'' {{=}} 31/3 {{=}} 10.{{overline\|3}}}})
	* 1277 & 2513 ({{nowrap\|''n'' {{=}} 21/2}})		* 1277 & 2513 ({{nowrap\|''n'' {{=}} 21/2}})

FloraC: Sorting

2026-05-20T10:06:19Z

Sorting

@@ Line 142: / Line 142: @@
 * 3737 & 5585 ({{nowrap|''n'' {{=}} 31/3 {{=}} 10.{{overline|3}}}})
 * 1277 & 2513 ({{nowrap|''n'' {{=}} 21/2}})
 == Kwai (5-limit) ==
@@ Line 202: / Line 182: @@
 [[Badness]] (Sintel): 29.1
 == Newt (5-limit) ==

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

2026-05-20T10:06:01Z

Switch to Sintel's badness, WE & CWE tunings

← Older revision		Revision as of 10:06, 20 May 2026
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	{{Mapping\|legend=1\| 1 1 -5 \| 0 2 25 }}		{{Mapping\|legend=1\| 1 1 -5 \| 0 2 25 }}
			: mapping generators: ~2, ~655360/531441

	[[Optimal tuning]] ([[~~POTE~~]]): ~655360/531441 = 351.~~476~~{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.7047{{c}}, ~655360/531441 = 351.3898{{c}}
			: [[error map]]: {{val\| -0.295 +0.529 -0.091 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~655360/531441 = 351.4654{{c}}
			: error map: {{val\| 0.000 +0.976 +0.322 }}

	{{Optimal ET sequence\|legend=1\| 41, 58, 99, 239, 338, 915b, 1253bc }}		{{Optimal ET sequence\|legend=1\| 17c, 41, 58, 99, 239, 338, 915b, 1253bc }}

	[[Badness]] (~~Smith~~): 0.~~372848~~		[[Badness]] (Sintel): 8.75

	== Kwai (5-limit) ==		== Kwai (5-limit) ==
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	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5

	[[Comma list]]: {{monzo\| 50 -33 1 }} ~~= 5629499534213120/5559060566555523~~		[[Comma list]]: {{monzo\| 50 -33 1 }}

	{{Mapping\|legend=1\| 1 0 -50 \| 0 1 33 }}		{{Mapping\|legend=1\| 1 0 -50 \| 0 1 33 }}
			: mapping generators: ~2, ~3

	[[Optimal tuning]] ([[~~POTE~~]]): ~3/2 = 702.~~630~~{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.792{{c}}, ~3/2 = 702.5077{{c}}
			: [[error map]]: {{val\| -0.208 +0.345 -0.023 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6243{{c}}
			: error map: {{val\| 0.000 +0.669 +0.288 }}

	{{Optimal ET sequence\|legend=1\| 41, 111, 152 }}		{{Optimal ET sequence\|legend=1\| 41, 111, 152, 2017bbc, 2169bbc }}

	[[Badness]] (~~Smith~~): 0.~~636715~~		[[Badness]] (Sintel): 14.9

	== Cotoneum (5-limit) ==		== Cotoneum (5-limit) ==
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	{{Mapping\|legend=1\| 1 0 80 \| 0 1 -49 }}		{{Mapping\|legend=1\| 1 0 80 \| 0 1 -49 }}
			: mapping generators: ~2, ~3

	[[Optimal tuning]] ([[~~POTE~~]]): ~3/2 = 702.~~315~~{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.8849{{c}}, ~3/2 = 702.2471{{c}}
			: [[error map]]: {{val\| -0.115 +0.177 +0.008 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3162{{c}}
			: error map: {{val\| 0.000 +0.361 +0.190 }}

	{{Optimal ET sequence\|legend=1\| 41, 135c, 176, 217, 475, 1167, 1642, 2117b }}		{{Optimal ET sequence\|legend=1\| 41, 135c, 176, 217, 475, 1167, 1642, 2117b, 3759bbc }}

	[[Badness]] (~~Smith~~): 1~~.240078~~		[[Badness]] (Sintel): 29.1

	== Newt (5-limit) ==		== Newt (5-limit) ==
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	{{Mapping\|legend=1\| 1 1 19 \| 0 2 -57 }}		{{Mapping\|legend=1\| 1 1 19 \| 0 2 -57 }}
			: mapping generators: ~2, ~{{monzo\| 47 -28 -1 }}

	[[Optimal tuning]] ([[~~POTE~~]]): ~{{monzo\| 47 -28 -1 }} = 351.~~114~~{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.9120{{c}}, ~{{monzo\| 47 -28 -1 }} = 351.0878{{c}}
			: [[error map]]: {{val\| -0.088 +0.133 +0.010 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo\| 47 -28 -1 }} = 351.1146{{c}}
			: error map: {{val\| 0.000 +0.274 +0.152 }}

	{{Optimal ET sequence\|legend=1\| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc }}		{{Optimal ET sequence\|legend=1\| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc }}

	[[Badness]] (~~Smith~~): 1.~~528465~~		[[Badness]] (Sintel): 35.9

	[[Category:41edo]]		[[Category:41edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

FloraC: Update & various cleanup

2026-05-20T09:47:54Z

Update & various cleanup

Show changes

Lériendil at 17:25, 6 May 2025

2025-05-06T17:25:55Z

← Older revision		Revision as of 17:25, 6 May 2025
Line 138:		Line 138:
	Examples of temperaments with fractional values of ''n'':		Examples of temperaments with fractional values of ''n'':
	* [[Marvel temperaments #Septimin\|Septimin]] ({{nowrap\|''n'' {{=}} −11/2}})		* [[Marvel temperaments #Septimin\|Septimin]] ({{nowrap\|''n'' {{=}} −11/2}})
	* [[~~Shibboleth family\|~~Shibboleth]] ({{nowrap\|''n'' {{=}} −9/2}})		* [[Shibboleth]] ({{nowrap\|''n'' {{=}} −9/2}})
	* [[Mirkwai clan #Pluto\|Pluto]] ({{nowrap\|''n'' {{=}} −7/2}})		* [[Mirkwai clan #Pluto\|Pluto]] ({{nowrap\|''n'' {{=}} −7/2}})
	* 3737 & 5585 ({{nowrap\|''n'' {{=}} 31/3 {{=}} 10.{{overline\|3}}}})		* 3737 & 5585 ({{nowrap\|''n'' {{=}} 31/3 {{=}} 10.{{overline\|3}}}})

FloraC: -rodan (addressed by the syntonic-diatonic continuum)

2025-01-05T11:20:31Z

-rodan (addressed by the syntonic-diatonic continuum)

@@ Line 142: / Line 142: @@
 * 3737 & 5585 ({{nowrap|''n'' {{=}} 31/3 {{=}} 10.{{overline|3}}}})
 * 1277 & 2513 ({{nowrap|''n'' {{=}} 21/2}})
 == Hemififths (5-limit) ==

FloraC: Update links. -countercomp (duplicate data from countercomp family)

2024-12-22T11:13:08Z

Update links. -countercomp (duplicate data from countercomp family)

Show changes

ArrowHead294 at 16:54, 17 December 2024

2024-12-17T16:54:00Z

← Older revision		Revision as of 16:54, 17 December 2024
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	! Monzo		! Monzo
	\|-		\|-
	\| -7		\| −7
	\| [[Merman]]		\| [[Merman]]
	\| 1121008359375 / 1099511627776		\| 1121008359375 / 1099511627776
	\| {{monzo\| -40 15 7 }}		\| {{monzo\| -40 15 7 }}
	\|-		\|-
	\| -6		\| −6
	\| [[Syntonic-31 equivalence continuum\|Ampersand]]		\| [[Syntonic–31 equivalence continuum\|Ampersand]]
	\| 34171875 / 33554432		\| 34171875 / 33554432
	\| {{monzo\| -25 7 6 }}		\| {{monzo\| -25 7 6 }}
	\|-		\|-
	\| -5		\| −5
	\| [[Magic]]		\| [[Magic]]
	\| 3125 / 3072		\| 3125 / 3072
	\| {{monzo\| -10 -1 5 }}		\| {{monzo\| -10 -1 5 }}
	\|-		\|-
	\| -4		\| −4
	\| [[Tetracot]]		\| [[Tetracot]]
	\| 20000 / 19683		\| 20000 / 19683
	\| {{monzo\| 5 -9 4 }}		\| {{monzo\| 5 -9 4 }}
	\|-		\|-
	\| -3		\| −3
	\| [[Rodan]]		\| [[Rodan]]
	\| 131072000 / 129140163		\| 131072000 / 129140163
	\| {{monzo\| 20 -17 3 }}		\| {{monzo\| 20 -17 3 }}
	\|-		\|-
	\| -2		\| −2
	\| [[Hemififths]]		\| [[Hemififths]]
	\| 858993459200 / 847288609443		\| 858993459200 / 847288609443
	\| {{monzo\| 35 -25 2 }}		\| {{monzo\| 35 -25 2 }}
	\|-		\|-
	\| -1		\| −1
	\| [[Kwai]]		\| [[Kwai]]
	\| (32 digits)		\| (32 digits)

FloraC: Vertical fractions only have a place in display-style math

2024-12-12T11:40:48Z

Vertical fractions only have a place in display-style math

← Older revision		Revision as of 11:40, 12 December 2024
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	Examples of temperaments with fractional values of ''n'':		Examples of temperaments with fractional values of ''n'':
	* [[Marvel temperaments #Septimin\|Septimin]] ({{nowrap\|''n'' {{=}} −~~{{sfrac\|~~11\|2}}}})		* [[Marvel temperaments #Septimin\|Septimin]] ({{nowrap\|''n'' {{=}} −11/2}})
	* [[Shibboleth family\|Shibboleth]] ({{nowrap\|''n'' {{=}} −~~{{sfrac\|~~9\|2}}}})		* [[Shibboleth family\|Shibboleth]] ({{nowrap\|''n'' {{=}} −9/2}})
	* [[Mirkwai clan #Pluto\|Pluto]] ({{nowrap\|''n'' {{=}} −~~{{sfrac\|~~7\|2}}}})		* [[Mirkwai clan #Pluto\|Pluto]] ({{nowrap\|''n'' {{=}} −7/2}})
	* 3737 & 5585 ({{nowrap\|''n'' {{=}} ~~{{sfrac\|~~31\|3}} {{=}} 10.{{overline\|3}}}})		* 3737 & 5585 ({{nowrap\|''n'' {{=}} 31/3 {{=}} 10.{{overline\|3}}}})
	* 1277 & 2513 ({{nowrap\|''n'' {{=}} ~~−{{sfrac\|~~21\|2}}}})		* 1277 & 2513 ({{nowrap\|''n'' {{=}} 21/2}})

	== Rodan (5-limit) ==		== Rodan (5-limit) ==

ArrowHead294: Formatting

2024-12-11T15:56:59Z

Formatting

← Older revision		Revision as of 15:56, 11 December 2024
Line 1:		Line 1:
	The '''schismic–countercommatic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768\|schismas (32805/32768)]] with the [[41-comma\|Pythagorean countercomma ({{monzo\| 65 -41 }})]]. This continuum is theoretically interesting in that these are all [[5-limit]] [[microtemperament]]s [[support]]ed by [[41edo]].		The '''schismic–countercommatic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768\|schismas (32805/32768)]] with the [[41-comma\|Pythagorean countercomma ({{monzo\| 65 -41 }})]]. This continuum is theoretically interesting in that these are all [[5-limit]] [[microtemperament]]s [[support]]ed by [[41edo]].

	All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo\| 65 -41 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 41edo 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 10.1575233481…, and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(32805/32768)<sup>''n''</sup> ~ {{monzo\| 65 -41 }}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 41edo 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 10.1575233481…, and temperaments having ''n'' near this value tend to be the most accurate ones.

	The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, 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 example:		The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, 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 example:
	* [[Cotoneum]] (''n'' = 1) is generated by a fifth;		* [[Cotoneum]] ({{nowrap\|''n'' {{=}} 1}}) is generated by a fifth;
	* [[Newt]] (''n'' = 2) splits its fifth in two;		* [[Newt]] ({{nowrap\|''n'' {{=}} 2}}) splits its fifth in two;
	* Etc.		* Etc.

Line 11:		Line 11:

	{\| class="wikitable center-1"		{\| class="wikitable center-1"
	\|+ Temperaments of integer ''n''		\|+ style="font-size: 105%;" \| Temperaments of integer ''n''
	\|-		\|-
	! rowspan="2" \| ''n''		! rowspan="2" \| ''n''
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	Examples of temperaments with fractional values of ''n'':		Examples of temperaments with fractional values of ''n'':
	* [[Marvel temperaments #Septimin\|Septimin]] (''n'' = -11/2 ~~= -5.5~~)		* [[Marvel temperaments #Septimin\|Septimin]] ({{nowrap\|''n'' {{=}} −{{sfrac\|11\|2}}}})
	* [[Shibboleth family\|Shibboleth]] (''n'' = -9/2 ~~= -4.5~~)		* [[Shibboleth family\|Shibboleth]] ({{nowrap\|''n'' {{=}} −{{sfrac\|9\|2}}}})
	* [[Mirkwai clan #Pluto\|Pluto]] (''n'' = -7/2 ~~= -3.5~~)		* [[Mirkwai clan #Pluto\|Pluto]] ({{nowrap\|''n'' {{=}} −{{sfrac\|7\|2}}}})
	* 3737 & 5585 (''n'' = 31/3 = 10.{{overline\|3}})		* 3737 & 5585 ({{nowrap\|''n'' {{=}} {{sfrac\|31\|3}} {{=}} 10.{{overline\|3}}}})
	* 1277 & 2513 (''n'' = 21/2 ~~= 10.5~~)		* 1277 & 2513 ({{nowrap\|''n'' {{=}} −{{sfrac\|21\|2}}}})

	== Rodan (5-limit) ==		== Rodan (5-limit) ==

← Older revision		Revision as of 15:56, 11 December 2024
Line 1:		Line 1:
	The '''schismic–countercommatic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768\|schismas (32805/32768)]] with the [[41-comma\|Pythagorean countercomma ({{monzo\| 65 -41 }})]]. This continuum is theoretically interesting in that these are all [[5-limit]] [[microtemperament]]s [[support]]ed by [[41edo]].		The '''schismic–countercommatic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768\|schismas (32805/32768)]] with the [[41-comma\|Pythagorean countercomma ({{monzo\| 65 -41 }})]]. This continuum is theoretically interesting in that these are all [[5-limit]] [[microtemperament]]s [[support]]ed by [[41edo]].

	All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo\| 65 -41 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 41edo 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 10.1575233481…, and temperaments having ''n'' near this value tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(32805/32768)<sup>''n''</sup> ~ {{monzo\| 65 -41 }}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 41edo 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 10.1575233481…, and temperaments having ''n'' near this value tend to be the most accurate ones.

	The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, 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 example:		The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, 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 example:
	* [[Cotoneum]] (''n'' = 1) is generated by a fifth;		* [[Cotoneum]] ({{nowrap\|''n'' {{=}} 1}}) is generated by a fifth;
	* [[Newt]] (''n'' = 2) splits its fifth in two;		* [[Newt]] ({{nowrap\|''n'' {{=}} 2}}) splits its fifth in two;
	* Etc.		* Etc.

Line 11:		Line 11:

	{\| class="wikitable center-1"		{\| class="wikitable center-1"
	\|+ Temperaments of integer ''n''		\|+ style="font-size: 105%;" \| Temperaments of integer ''n''
	\|-		\|-
	! rowspan="2" \| ''n''		! rowspan="2" \| ''n''
Line 137:		Line 137:

	Examples of temperaments with fractional values of ''n'':		Examples of temperaments with fractional values of ''n'':
	* [[Marvel temperaments #Septimin\|Septimin]] (''n'' = -11/2 ~~= -5.5~~)		* [[Marvel temperaments #Septimin\|Septimin]] ({{nowrap\|''n'' {{=}} −{{sfrac\|11\|2}}}})
	* [[Shibboleth family\|Shibboleth]] (''n'' = -9/2 ~~= -4.5~~)		* [[Shibboleth family\|Shibboleth]] ({{nowrap\|''n'' {{=}} −{{sfrac\|9\|2}}}})
	* [[Mirkwai clan #Pluto\|Pluto]] (''n'' = -7/2 ~~= -3.5~~)		* [[Mirkwai clan #Pluto\|Pluto]] ({{nowrap\|''n'' {{=}} −{{sfrac\|7\|2}}}})
	* 3737 & 5585 (''n'' = 31/3 = 10.{{overline\|3}})		* 3737 & 5585 ({{nowrap\|''n'' {{=}} {{sfrac\|31\|3}} {{=}} 10.{{overline\|3}}}})
	* 1277 & 2513 (''n'' = 21/2 ~~= 10.5~~)		* 1277 & 2513 ({{nowrap\|''n'' {{=}} −{{sfrac\|21\|2}}}})

	== Rodan (5-limit) ==		== Rodan (5-limit) ==