Archytas–diatonic equivalence continuum - Revision history

Godtone: relink

2026-05-01T13:45:43Z

relink

← Older revision		Revision as of 13:45, 1 May 2026
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	\|-		\|-
	\| 5		\| 5
	\| [[Obscenity]]		\| [[5th-octave temperaments#Obscenity\|Obscenity]]
	\| [[4194304/4084101]]		\| [[4194304/4084101]]
	\| {{Monzo\| 22 -5 -5 }}		\| {{Monzo\| 22 -5 -5 }}

FloraC: + m-continuum and separate fractional values

2026-05-01T12:52:12Z

+ m-continuum and separate fractional values

@@ Line 6: / Line 6: @@
 {| class="wikitable center-1"
-|+ style="font-size: 105%;" | Temperaments in the continuum
+|+ style="font-size: 105%;" | Temperaments with integer ''n''
 |-
 ! rowspan="2" | ''n''
@@ Line 16: / Line 16: @@
 |-
 | 0
-| [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=off&weights=weil&target=&edos=5+%26+5d&submit_edo=submit&commas= 2.3.7 blackwood]
+| [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=off&weights=weil&target=&edos=5+%26+5d&submit_edo=submit&commas= Blackwood variant]
 | [[256/243]]
 | {{Monzo| 8 -5 }}
@@ Line 29: / Line 29: @@
 | [[49/48]]
 | {{Monzo| -4 -1 2 }}
 |-
 | 3
@@ Line 39: / Line 34: @@
 | [[1029/1024]]
 | {{Monzo| -10 1 3 }}
 |-
 | 4
@@ Line 68: / Line 48: @@
 | …
 | …
+| …
 |-
 | ∞
@@ Line 74: / Line 54: @@
 | [[64/63]]
 | {{Monzo| 6 -2 -1 }}
 |}
 [[Category:5edo]]
 [[Category:Equivalence continua]]

FloraC: Cleanup

2026-05-01T12:40:07Z

Cleanup

← Older revision		Revision as of 12:40, 1 May 2026
Line 1:		Line 1:
	The '''Archytas–diatonic equivalence continuum''', or '''septimal–diatonic equivalence continuum''', is a [[equivalence continuum\|continuum]] of [[2.3.7 subgroup]] temperaments which equate a number of [[64/63\|Archytas commas (64/63)]] with the [[256/243\|limma (256/243)]].		The '''Archytas–diatonic equivalence continuum''', or '''septimal–diatonic equivalence continuum''', is a [[equivalence continuum\|continuum]] of [[2.3.7 subgroup]] temperaments which equate a number of [[64/63\|Archytas' commas (64/63)]] with the [[256/243\|Pythagorean limma (256/243)]]. This continuum is theoretically interesting in that these are all [[2.3.7 subgroup\|2.3.7-subgroup]] temperaments [[support]]ed by [[5edo]].

	All temperaments in the continuum satisfy {{nowrap\|(64/63)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by [[5edo~~]] (~~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 3.3093…, and temperaments near this tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(64/63)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by 5edo 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 3.3093…, and temperaments near this tend to be the most accurate ones.

	256/243 is the characteristic [[3-limit]] comma tempered out in [[5edo]]. In each case, ~~we notice that~~ ''n'' equals the order of harmonic 7 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the ~~MOS scale~~.		256/243 is the characteristic [[3-limit]] comma tempered out in 5edo, and has many advantages as a target. In each case, ''n'' equals the order of harmonic 7 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain.

	{\| class="wikitable center-1 ~~center-2~~"		{\| class="wikitable center-1"
	\|+ style="font-size: 105%;" \| Temperaments in the continuum		\|+ style="font-size: 105%;" \| Temperaments in the continuum
	\|-		\|-
Line 13:		Line 13:
	\|-		\|-
	! Ratio		! Ratio
	! ~~Monzo~~		! S. monzo
	\|-		\|-
	\| 0		\| 0
	\| [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=off&weights=weil&target=&edos=5+%26+5d&submit_edo=submit&commas= 2.3.7 blackwood]		\| [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=off&weights=weil&target=&edos=5+%26+5d&submit_edo=submit&commas= 2.3.7 blackwood]
	\| [[256/243]]		\| [[256/243]]
	\| {{~~monzo~~\| 8 -5 }}		\| {{Monzo\| 8 -5 }}
	\|-		\|-
	\| 1		\| 1
	\| [[~~Trienstonic clan #No-fives trienstonic\|No-fives trienstonic~~]]		\| [[Trienstonian]]
	\| [[28/27]]		\| [[28/27]]
	\| {{~~monzo~~\| 2 -3 1 }}		\| {{Monzo\| 2 -3 1 }}
	\|-		\|-
	\| 2		\| 2
	\| [[Semaphore]]		\| [[Semaphore]]
	\| [[49/48]]		\| [[49/48]]
	\| {{~~monzo~~\| -4 -1 2 }}		\| {{Monzo\| -4 -1 2 }}
	\|-		\|-
	\| 2.5		\| 2.5
	\| [[Cloudy]]		\| [[Cloudy]] retraction
	\| [[16807/16384]]		\| [[16807/16384]]
	\| {{~~monzo~~\| -14 0 5 }}		\| {{Monzo\| -14 0 5 }}
	\|-		\|-
	\| 3		\| 3
	\| [[Slendric]]		\| [[Slendric]]
	\| [[1029/1024]]		\| [[1029/1024]]
	\| {{~~monzo~~\| -10 1 3 }}		\| {{Monzo\| -10 1 3 }}
	\|-		\|-
	\| 3.3		\| 3.3
	\| 5 & 436		\| 5 & 436
	\| ~~<abbr title="s">[very long]</abbr>~~		\| (72 digits)
	\| {{~~monzo~~\| 118 -16 -33 }}		\| {{Monzo\| 118 -16 -33 }}
	\|-		\|-
	\| 10/3		\| 10/3
	\| [[~~Slendric schisma\|~~Slendroschismic]]*		\| [[Slendroschismic]]
	\| 68719476736/68641485507		\| 68719476736/68641485507
	\| {{~~monzo~~\| 36 -5 0 -10 }}		\| {{Monzo\| 36 -5 -10 }}
	\|-		\|-
	\| 3.5		\| 3.5
	\| [[Septiness]]		\| [[Septiness]] restriction
	\| 67108864/66706983		\| 67108864/66706983
	\| {{~~monzo~~\| 26 -4 -7 }}		\| {{Monzo\| 26 -4 -7 }}
	\|-		\|-
	\| 4		\| 4
	\| [[Buzzard]]		\| [[Buzzard]]
	\| [[65536/64827]]		\| [[65536/64827]]
	\| {{~~monzo~~\| 16 -3 -4 }}		\| {{Monzo\| 16 -3 -4 }}
	\|-		\|-
	\| 5		\| 5
	\| [[~~5th-octave temperaments#Obscenity\|~~Obscenity]]		\| [[Obscenity]]
	\| 4194304/4084101		\| [[4194304/4084101]]
	\| {{~~monzo~~\| 22 -5 -5 }}		\| {{Monzo\| 22 -5 -5 }}
	\|-		\|-
	\| …		\| …
Line 71:		Line 71:
	\|-		\|-
	\| ∞		\| ∞
	\| [[~~Archytas clan#Archy\|~~Archy]]		\| [[Archy]]
	\| [[64/63]]		\| [[64/63]]
	\| {{~~monzo~~\| 6 -2 -1 }}		\| {{Monzo\| 6 -2 -1 }}
	\|}		\|}
	<nowiki />* The name "slendroschismic" may be changed to minimise confusion with confusingly named temperaments, or those temperaments may be changed instead, either way consensus would first need to be formed.

	[[Category:5edo]]		[[Category:5edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

Overthink at 23:52, 5 October 2025

2025-10-05T23:52:55Z

← Older revision		Revision as of 23:52, 5 October 2025
Line 61:		Line 61:
	\|-		\|-
	\| 5		\| 5
	\| [~~https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=off&weights=weil&target=&edos=5+%26+70&submit_edo=submit&commas= 5 & 75~~]		\| [[5th-octave temperaments#Obscenity\|Obscenity]]
	\| 4194304/4084101		\| 4194304/4084101
	\| {{monzo\| 22 -5 -5 }}		\| {{monzo\| 22 -5 -5 }}

Lériendil: Text replacement - "lendrism" to "lendroschism"

2025-06-11T12:34:06Z

Text replacement - "lendrism" to "lendroschism"

← Older revision		Revision as of 12:34, 11 June 2025
Line 46:		Line 46:
	\|-		\|-
	\| 10/3		\| 10/3
	\| [[Slendric schisma\|~~Slendrismic~~]]*		\| [[Slendric schisma\|Slendroschismic]]*
	\| 68719476736/68641485507		\| 68719476736/68641485507
	\| {{monzo\| 36 -5 0 -10 }}		\| {{monzo\| 36 -5 0 -10 }}
Line 75:		Line 75:
	\| {{monzo\| 6 -2 -1 }}		\| {{monzo\| 6 -2 -1 }}
	\|}		\|}
	<nowiki />* The name "~~slendrismic~~" may be changed to minimise confusion with confusingly named temperaments, or those temperaments may be changed instead, either way consensus would first need to be formed.		<nowiki />* The name "slendroschismic" may be changed to minimise confusion with confusingly named temperaments, or those temperaments may be changed instead, either way consensus would first need to be formed.

	[[Category:5edo]]		[[Category:5edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

Godtone: remove nonstandard obfuscatory form; that form is easy to deduce from the more informative and useful one that lets you know how to calculate the number more clearly

2025-01-15T01:45:13Z

remove nonstandard obfuscatory form; that form is easy to deduce from the more informative and useful one that lets you know how to calculate the number more clearly

← Older revision		Revision as of 01:45, 15 January 2025
Line 45:		Line 45:
	\| {{monzo\| 118 -16 -33 }}		\| {{monzo\| 118 -16 -33 }}
	\|-		\|-
	\| ~~3+{{frac\|1\|3}} =~~ 10/3		\| 10/3
	\| [[Slendric schisma\|Slendrismic]]*		\| [[Slendric schisma\|Slendrismic]]*
	\| 68719476736/68641485507		\| 68719476736/68641485507

Godtone: correct & add 10/3 equivalence to obfuscatory 3+1/3 form

2025-01-15T01:42:19Z

correct & add 10/3 equivalence to obfuscatory 3+1/3 form

← Older revision		Revision as of 01:42, 15 January 2025
Line 41:		Line 41:
	\|-		\|-
	\| 3.3		\| 3.3
	\| [[Slendric schisma\|Slendrismic]]*		\| 5 & 436
	\| <abbr title="~~332793929704998566053507268688789447/332306998946228968225951765070086144~~">[very long]</abbr>		\| <abbr title="s">[very long]</abbr>
	\| {{monzo\| 118 -16 -33 }}		\| {{monzo\| 118 -16 -33 }}
	\|-		\|-
	\| 3{{frac\|1\|3}}		\| 3+{{frac\|1\|3}} = 10/3
	\|		\| [[Slendric schisma\|Slendrismic]]*
	\| 68719476736/68641485507		\| 68719476736/68641485507
	\| {{monzo\| 36 -5 0 -10 }}		\| {{monzo\| 36 -5 0 -10 }}

ArrowHead294 at 13:27, 31 December 2024

2024-12-31T13:27:01Z

← Older revision		Revision as of 13:27, 31 December 2024
Line 42:		Line 42:
	\| 3.3		\| 3.3
	\| [[Slendric schisma\|Slendrismic]]*		\| [[Slendric schisma\|Slendrismic]]*
	\| [very long]		\| <abbr title="332793929704998566053507268688789447/332306998946228968225951765070086144">[very long]</abbr>
	\| {{monzo\| 118 -16 -33 }}		\| {{monzo\| 118 -16 -33 }}
			\|-
			\| 3{{frac\|1\|3}}
			\|
			\| 68719476736/68641485507
			\| {{monzo\| 36 -5 0 -10 }}
	\|-		\|-
	\| 3.5		\| 3.5

ArrowHead294 at 14:46, 18 December 2024

2024-12-18T14:46:09Z

← Older revision		Revision as of 14:46, 18 December 2024
Line 1:		Line 1:
	The '''Archytas–diatonic equivalence continuum''', or '''septimal–diatonic equivalence continuum''', is a continuum of [[2.3.7 subgroup]] temperaments which equate a number of [[64/63\|Archytas commas (64/63)]] with the [[256/243\|limma (256/243)]].		The '''Archytas–diatonic equivalence continuum''', or '''septimal–diatonic equivalence continuum''', is a [[equivalence continuum\|continuum]] of [[2.3.7 subgroup]] temperaments which equate a number of [[64/63\|Archytas commas (64/63)]] with the [[256/243\|limma (256/243)]].

	All temperaments in the continuum satisfy {{nowrap\|(64/63)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by [[5edo]] (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 3.3093…, and temperaments near this tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(64/63)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by [[5edo]] (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 3.3093…, and temperaments near this tend to be the most accurate ones.

ArrowHead294 at 14:45, 18 December 2024

2024-12-18T14:45:48Z

← Older revision		Revision as of 14:45, 18 December 2024
Line 1:		Line 1:
	The '''~~Archytas-diatonic~~ equivalence continuum''' or ~~the~~ '''~~septimal-diatonic~~ equivalence continuum''' is a continuum of [[2.3.7 subgroup]] temperaments which equate a number of [[64/63\|Archytas commas (64/63)]] with the [[256/243\|limma (256/243)]].		The '''Archytas–diatonic equivalence continuum''', or '''septimal–diatonic equivalence continuum''', is a continuum of [[2.3.7 subgroup]] temperaments which equate a number of [[64/63\|Archytas commas (64/63)]] with the [[256/243\|limma (256/243)]].

	All temperaments in the continuum satisfy {{nowrap\|(64/63)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by [[5edo]] (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 3.3093…, and temperaments near this tend to be the most accurate ones.		All temperaments in the continuum satisfy {{nowrap\|(64/63)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by [[5edo]] (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 3.3093…, and temperaments near this tend to be the most accurate ones.