Squares - Revision history

FloraC: Cleanup on infobox

2026-02-09T09:44:54Z

Cleanup on infobox

← Older revision		Revision as of 09:44, 9 February 2026
Line 1:		Line 1:
	{{Infobox ~~Regtemp~~		{{Infobox regtemp
	\| Title = Skwares; Squares		\| Title = Skwares; Squares
	\| Subgroups = 2.3.7, 2.3.7.11, 2.3.5.7.11		\| Subgroups = 2.3.7, 2.3.7.11, 2.3.5.7.11
	\| Comma basis = [[19683/19208]] (2.3.7); <br> [[99/98]], [[243/242]] (2.3.7.11); <br> [[81/80]], [[99/98]], [[121/120]] (11-limit)		\| Comma basis = [[19683/19208]] (2.3.7); <br> [[99/98]], [[243/242]] (2.3.7.11); <br> [[81/80]], [[99/98]], [[121/120]] (11-limit)
	\| Edo join 1 = 14c \| Edo join 2 = 17c		\| Edo join 1 = 14c \| Edo join 2 = 17c
	\| ~~Generator~~ = 9/7 \| ~~Generator~~ tuning = 426.0 \| Optimization method = CWE		\| Mapping = 1; -4 -16 -9 -10
			\| Generators = 9/7 \| Generators tuning = 426.0 \| Optimization method = CWE
	\| MOS scales = [[3L 2s]], [[3L 5s]], [[3L 8s]], [[3L 11s]], [[14L 3s]]		\| MOS scales = [[3L 2s]], [[3L 5s]], [[3L 8s]], [[3L 11s]], [[14L 3s]]
	~~\| Mapping = 1; -4 -16 -9 -10~~
	\| Pergen = (P8, P11/4)		\| Pergen = (P8, P11/4)
	\| Odd limit 1 = (2.3.7) 7 \| Mistuning 1 = 4.7 \| Complexity 1 = 11		\| Odd limit 1 = 2.3.7 7 \| Mistuning 1 = 4.7 \| Complexity 1 = 11
	\| Odd limit 2 = 11 \| Mistuning 2 = 10.8 \| Complexity 2 = 17		\| Odd limit 2 = 11 \| Mistuning 2 = 10.8 \| Complexity 2 = 17
	}}		}}

	At its most basic level, '''squares''' can be thought of as a [[2.3.7 subgroup\|2.3.7-subgroup]] temperament (sometimes called ''skwares''), generated by a flat [[~]][[9/7]] such that four of them stack to the perfect eleventh, [[8/3]], therefore [[tempering out]] the comma [[19683/19208]]. However, it is more natural to think of the temperament first as [[2.3.7.11 subgroup]], tempering out [[99/98]] so as to identify the generator with [[14/11]] in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, [[18/11]], two of which are then identified with 8/3 due to tempering out [[243/242]]. This can also be thought of as an octavization of the 3.7.11-subgroup [[mintaka]] temperament by identifying [[2/1]] with a false octave corresponding to 99/49~243/121, in a manner similar to [[sensi]]'s relation to [[BPS]].		At its most basic level, '''squares''' can be thought of as a [[2.3.7 subgroup\|2.3.7-subgroup]] temperament (sometimes called ''skwares''), generated by a flat [[~]][[9/7]] such that four of them stack to the perfect eleventh, [[8/3]], therefore [[tempering out]] the comma [[19683/19208]]. However, it is more natural to think of the temperament first as [[2.3.7.11 subgroup]], tempering out [[99/98]] so as to identify the generator with [[14/11]] in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, [[18/11]], two of which are then identified with 8/3 due to tempering out [[243/242]]. This can also be thought of as an octavization of the 3.7.11-subgroup [[mintaka]] temperament by identifying [[2/1]] with a false octave corresponding to 99/49~243/121, in a manner similar to [[sensi]]'s relation to [[BPS]].

FloraC: /* Interval chain */

2025-12-07T09:29:25Z

Interval chain

← Older revision		Revision as of 09:29, 7 December 2025
Line 40:		Line 40:
	\| 0		\| 0
	\| 0.0		\| 0.0
	\| 1/1		\| '''1/1'''
	\|		\|
	\|		\|

FloraC: /* Interval chain */ adopt CWE tuning

2025-12-07T09:29:08Z

Interval chain: adopt CWE tuning

← Older revision		Revision as of 09:29, 7 December 2025
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	== Interval chain ==		== Interval chain ==
	In the following table, ~~prime~~ harmonics and subharmonics are labelled in '''bold'''.		In the following table, harmonics and subharmonics 1–13 are labelled in '''bold'''.

	{\| class="wikitable center-1 right-2"		{\| class="wikitable center-1 right-2"
Line 46:		Line 46:
	\|-		\|-
	\| 1		\| 1
	\| 425.6		\| 425.8
	\| 9/7, 14/11		\| 9/7, 14/11
	\|		\|
Line 53:		Line 53:
	\|-		\|-
	\| 2		\| 2
	\| 851.2		\| 851.7
	\| 18/11, 33/20, 44/27		\| 18/11, 33/20, 44/27
	\| '''13/8'''		\| '''13/8'''
Line 60:		Line 60:
	\|-		\|-
	\| 3		\| 3
	\| 76.8		\| 77.5
	\| 21/20, 28/27		\| 21/20, 28/27
	\|		\|
Line 67:		Line 67:
	\|-		\|-
	\| 4		\| 4
	\| ~~502~~.4		\| 503.4
	\| '''4/3'''		\| '''4/3'''
	\|		\|
Line 74:		Line 74:
	\|-		\|-
	\| 5		\| 5
	\| ~~928~~.0		\| 929.2
	\| 12/7		\| 12/7
	\|		\|
Line 81:		Line 81:
	\|-		\|-
	\| 6		\| 6
	\| ~~153~~.6		\| 155.1
	\| 11/10, 12/11		\| 11/10, 12/11
	\| 13/12		\| 13/12
Line 88:		Line 88:
	\|-		\|-
	\| 7		\| 7
	\| ~~579~~.2		\| 580.9
	\| 7/5		\| 7/5
	\|		\|
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	\|-		\|-
	\| 8		\| 8
	\| ~~1004~~.8		\| 1006.8
	\| 9/5, 16/9		\| 9/5, '''16/9'''
	\|		\|
	\|		\|
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	\|-		\|-
	\| 9		\| 9
	\| ~~230~~.4		\| 232.6
	\| '''8/7'''		\| '''8/7'''
	\|		\|
Line 109:		Line 109:
	\|-		\|-
	\| 10		\| 10
	\| ~~656~~.0		\| 658.4
	\| '''16/11''', 22/15		\| '''16/11''', 22/15
	\| 13/9		\| 13/9
Line 116:		Line 116:
	\|-		\|-
	\| 11		\| 11
	\| ~~1081~~.6		\| 1084.3
	\| 28/15		\| 28/15
	\| 13/7		\| 13/7
Line 123:		Line 123:
	\|-		\|-
	\| 12		\| 12
	\| ~~307~~.2		\| 310.1
	\| 6/5		\| 6/5
	\| 13/11		\| 13/11
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	\|-		\|-
	\| 13		\| 13
	\| ~~732~~.8		\| 736.0
	\| 32/21		\| 32/21
	\|		\|
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	\|-		\|-
	\| 14		\| 14
	\| ~~1158~~.4		\| 1161.8
	\| 49/25, 64/33, 96/49		\| 49/25, 64/33, 96/49
	\| 52/27		\| 52/27
Line 144:		Line 144:
	\|-		\|-
	\| 15		\| 15
	\| ~~384~~.0		\| 387.7
	\| 56/45		\| 56/45
	\| 26/21		\| 26/21
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	\|-		\|-
	\| 16		\| 16
	\| ~~809~~.6		\| 813.5
	\| '''8/5'''		\| '''8/5'''
	\|		\|
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	\|-		\|-
	\| 17		\| 17
	\| 35.2		\| 39.3
	\| 36/35, 64/63		\| 36/35, 64/63
	\|		\|
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	\|		\|
	\|}		\|}
	<nowiki />* In 11-limit ~~CTE~~ tuning		<nowiki/>* In 11-limit CWE tuning

	== Scales ==		== Scales ==

Overthink: simpler edo join

2025-12-06T18:16:09Z

simpler edo join

← Older revision		Revision as of 18:16, 6 December 2025
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	\| Subgroups = 2.3.7, 2.3.7.11, 2.3.5.7.11		\| Subgroups = 2.3.7, 2.3.7.11, 2.3.5.7.11
	\| Comma basis = [[19683/19208]] (2.3.7); <br> [[99/98]], [[243/242]] (2.3.7.11); <br> [[81/80]], [[99/98]], [[121/120]] (11-limit)		\| Comma basis = [[19683/19208]] (2.3.7); <br> [[99/98]], [[243/242]] (2.3.7.11); <br> [[81/80]], [[99/98]], [[121/120]] (11-limit)
	\| Edo join 1 = 31 \| Edo join 2 = 45		\| Edo join 1 = 14c \| Edo join 2 = 17c
	\| Generator = 9/7 \| Generator tuning = 426.0 \| Optimization method = CWE		\| Generator = 9/7 \| Generator tuning = 426.0 \| Optimization method = CWE
	\| MOS scales = [[3L 2s]], [[3L 5s]], [[3L 8s]], [[3L 11s]], [[14L 3s]]		\| MOS scales = [[3L 2s]], [[3L 5s]], [[3L 8s]], [[3L 11s]], [[14L 3s]]

Overthink: Added Infobox Regtemp

2025-11-22T19:22:09Z

Added Infobox Regtemp

← Older revision		Revision as of 19:22, 22 November 2025
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			{{Infobox Regtemp
			\| Title = Skwares; Squares
			\| Subgroups = 2.3.7, 2.3.7.11, 2.3.5.7.11
			\| Comma basis = [[19683/19208]] (2.3.7); <br> [[99/98]], [[243/242]] (2.3.7.11); <br> [[81/80]], [[99/98]], [[121/120]] (11-limit)
			\| Edo join 1 = 31 \| Edo join 2 = 45
			\| Generator = 9/7 \| Generator tuning = 426.0 \| Optimization method = CWE
			\| MOS scales = [[3L 2s]], [[3L 5s]], [[3L 8s]], [[3L 11s]], [[14L 3s]]
			\| Mapping = 1; -4 -16 -9 -10
			\| Pergen = (P8, P11/4)
			\| Odd limit 1 = (2.3.7) 7 \| Mistuning 1 = 4.7 \| Complexity 1 = 11
			\| Odd limit 2 = 11 \| Mistuning 2 = 10.8 \| Complexity 2 = 17
			}}

	At its most basic level, '''squares''' can be thought of as a [[2.3.7 subgroup\|2.3.7-subgroup]] temperament (sometimes called ''skwares''), generated by a flat [[~]][[9/7]] such that four of them stack to the perfect eleventh, [[8/3]], therefore [[tempering out]] the comma [[19683/19208]]. However, it is more natural to think of the temperament first as [[2.3.7.11 subgroup]], tempering out [[99/98]] so as to identify the generator with [[14/11]] in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, [[18/11]], two of which are then identified with 8/3 due to tempering out [[243/242]]. This can also be thought of as an octavization of the 3.7.11-subgroup [[mintaka]] temperament by identifying [[2/1]] with a false octave corresponding to 99/49~243/121, in a manner similar to [[sensi]]'s relation to [[BPS]].		At its most basic level, '''squares''' can be thought of as a [[2.3.7 subgroup\|2.3.7-subgroup]] temperament (sometimes called ''skwares''), generated by a flat [[~]][[9/7]] such that four of them stack to the perfect eleventh, [[8/3]], therefore [[tempering out]] the comma [[19683/19208]]. However, it is more natural to think of the temperament first as [[2.3.7.11 subgroup]], tempering out [[99/98]] so as to identify the generator with [[14/11]] in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, [[18/11]], two of which are then identified with 8/3 due to tempering out [[243/242]]. This can also be thought of as an octavization of the 3.7.11-subgroup [[mintaka]] temperament by identifying [[2/1]] with a false octave corresponding to 99/49~243/121, in a manner similar to [[sensi]]'s relation to [[BPS]].

FloraC: Text replacement - "prime-optimized" to "norm-based"

2025-09-12T14:05:43Z

Text replacement - "prime-optimized" to "norm-based"

← Older revision		Revision as of 14:05, 12 September 2025
Line 160:		Line 160:
	== Tunings ==		== Tunings ==
	{\| class="wikitable mw-collapsible mw-collapsed"		{\| class="wikitable mw-collapsible mw-collapsed"
	\|+ style="font-size: 105%; white-space: nowrap;" \| 7-limit ~~prime~~-~~optimized~~ tunings		\|+ style="font-size: 105%; white-space: nowrap;" \| 7-limit norm-based tunings
	\|-		\|-
	! rowspan="2" \|		! rowspan="2" \|
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	\|}		\|}
	{\| class="wikitable mw-collapsible mw-collapsed"		{\| class="wikitable mw-collapsible mw-collapsed"
	\|+ style="font-size: 105%; white-space: nowrap;" \| 11-limit ~~prime~~-~~optimized~~ tunings		\|+ style="font-size: 105%; white-space: nowrap;" \| 11-limit norm-based tunings
	\|-		\|-
	! rowspan="2" \|		! rowspan="2" \|

FloraC: Fix a dead link & + an alt link

2025-08-24T12:44:19Z

Fix a dead link & + an alt link

← Older revision		Revision as of 12:44, 24 August 2025
Line 191:		Line 191:

	== Music ==		== Music ==
	* [~~http~~://~~clones~~.~~soonlabel~~.com/~~public/micro/tuning-survey~~/~~daily20100603~~-~~squares8piano.mp3 Square 8] by [[Chris Vaisvil~~]]		; [[Joel Kivelä]]
	* [https://www.youtube.com/watch?v=NUJOVrLqtdk ~~Optimum Rains~~] by [[~~Joel Kivelä~~]]		* ''Optimum Rains'' (2023) – [https://joelkivela.bandcamp.com/album/optimum-rains Bandcamp] \| [https://www.youtube.com/watch?v=NUJOVrLqtdk YouTube]

			; [[Chris Vaisvil]]
			* [https://web.archive.org/web/20201127015038/http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3 ''Square 8'']

	[[Category:Squares\| ]] <!-- main article -->		[[Category:Squares\| ]] <!-- main article -->

FloraC: Linking & style

2025-08-24T12:40:50Z

Linking & style

← Older revision		Revision as of 12:40, 24 August 2025
Line 1:		Line 1:
	At its most basic level, '''squares''' can be thought of as a [[2.3.7 subgroup]] temperament (sometimes called ''skwares''), generated by a flat [[~]][[9/7]] such that four of them stack to the perfect eleventh, [[8/3]], therefore tempering out the comma [[19683/19208]]. However, it is more natural to think of the temperament first as [[2.3.7.11 subgroup]], tempering out [[99/98]] so as to identify the generator with [[14/11]] in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, [[18/11]], two of which are then identified with 8/3 due to tempering out [[243/242]]. This can also be thought of as an octavization of the ~~[[3/1\|tritave]]-equivalent~~ 3.7.11 [[~~Mintaka~~]] temperament by identifying 2/1 with a false octave corresponding to ~~{{nowrap\|~~243/121 ~~~ 99/49}}~~, in a manner similar to [[sensi]]'s relation to [[BPS]].		At its most basic level, '''squares''' can be thought of as a [[2.3.7 subgroup\|2.3.7-subgroup]] temperament (sometimes called ''skwares''), generated by a flat [[~]][[9/7]] such that four of them stack to the perfect eleventh, [[8/3]], therefore [[tempering out]] the comma [[19683/19208]]. However, it is more natural to think of the temperament first as [[2.3.7.11 subgroup]], tempering out [[99/98]] so as to identify the generator with [[14/11]] in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, [[18/11]], two of which are then identified with 8/3 due to tempering out [[243/242]]. This can also be thought of as an octavization of the 3.7.11-subgroup [[mintaka]] temperament by identifying [[2/1]] with a false octave corresponding to 99/49~243/121, in a manner similar to [[sensi]]'s relation to [[BPS]].

	However, since the fifth in skwares is tuned flat, it is very natural to combine the temperament with [[meantone]] to create full [[11-limit]] ~~'''~~squares~~'''~~, which additionally can be restricted to the [[7-limit]] as the temperament with comma basis [[81/80]] and [[2401/2400]]. ~~The~~ 11-limit temperament is considered below.		However, since the fifth in skwares is tuned flat, it is very natural to combine the temperament with [[meantone]] to create full [[11-limit]] squares, which additionally can be restricted to the [[7-limit]] as the temperament with comma basis [[81/80]] and [[2401/2400]]. This 11-limit temperament is considered below.

	There is also a natural extension adding prime 23 by equating the generator to [[23/18]], and so finding 23 itself seven generators down, tempering out [[162/161]].		There is also a natural extension adding [[prime interval\|prime]] [[23/1\|23]] by equating the generator to [[23/18]], and so finding 23 itself seven generators down, tempering out [[162/161]].

	As for prime 13, the way to map it is less clear. The canonical squares mapping tempers out [[144/143]] in order to equate the tridecimal neutral sixth, [[13/8]], with 18/11, finding 13 two generators up, while '''agora''' tempers out [[105/104]] to equate [[8/7]] with [[15/13]], finding the 13th harmonic 29 generators down. These two mappings are enharmonically equivalent in [[31edo]]. Finally, '''squad''' tempers out [[351/343]] (which is the same as 3.7.11.13 [[~~Mintaka#Extensions of Mintaka\|Minalzidar~~]]'s tempering of that prime) so that 13 is equated with (7/3)<sup>3</sup>, and found 15 generators down.		As for prime [[13/1\|13]], the way to map it is less clear. The canonical squares mapping tempers out [[144/143]] in order to equate the tridecimal neutral sixth, [[13/8]], with 18/11, finding 13 two generators up, while '''agora''' tempers out [[105/104]] to equate [[8/7]] with [[15/13]], finding the 13th harmonic 29 generators down. These two mappings are enharmonically equivalent in [[31edo]]. Finally, '''squad''' tempers out [[351/343]] (which is the same as 3.7.11.13 [[minalzidar]]'s tempering of that prime) so that 13 is equated with (7/3)<sup>3</sup>, and found 15 generators down.

	See [[Meantone family #Squares]] and [[No-fives subgroup temperaments #Skwares]] for more technical data.		See [[Meantone family #Squares]] and [[No-fives subgroup temperaments #Skwares]] for more technical data.
Line 14:		Line 14:
	{\| class="wikitable center-1 right-2"		{\| class="wikitable center-1 right-2"
	\|-		\|-
	! rowspan="3" \| &#~~35;~~		! rowspan="3" \| #
	! rowspan="3" \| Cents*		! rowspan="3" \| Cents*
	! colspan="4" \| Approximate ~~Ratios~~		! colspan="4" \| Approximate ratios
	\|-		\|-
	! rowspan="2" \| 11-limit		! rowspan="2" \| 11-limit
	! colspan="3" \| 13-limit ~~Extension~~		! colspan="3" \| 13-limit extensions
	\|-		\|-
	! Squares		! Squares

FloraC: + tunings

2025-08-24T12:34:51Z

+ tunings

@@ Line 157: / Line 157: @@
 * [[Skwares11]]
 * [[Skwares14]]
 == Music ==

Lériendil at 02:16, 22 June 2025

2025-06-22T02:16:11Z

← Older revision		Revision as of 02:16, 22 June 2025
Line 3:		Line 3:
	However, since the fifth in skwares is tuned flat, it is very natural to combine the temperament with [[meantone]] to create full [[11-limit]] '''squares''', which additionally can be restricted to the [[7-limit]] as the temperament with comma basis [[81/80]] and [[2401/2400]]. The 11-limit temperament is considered below.		However, since the fifth in skwares is tuned flat, it is very natural to combine the temperament with [[meantone]] to create full [[11-limit]] '''squares''', which additionally can be restricted to the [[7-limit]] as the temperament with comma basis [[81/80]] and [[2401/2400]]. The 11-limit temperament is considered below.

	There is also a natural extension adding prime 23 by equating the generator to [[23/18]], and so finding 23 itself seven generators down, tempering out [[161~~/160~~]].		There is also a natural extension adding prime 23 by equating the generator to [[23/18]], and so finding 23 itself seven generators down, tempering out [[162/161]].

	As for prime 13, the way to map it is less clear. The canonical squares mapping tempers out [[144/143]] in order to equate the tridecimal neutral sixth, [[13/8]], with 18/11, finding 13 two generators up, while '''agora''' tempers out [[105/104]] to equate [[8/7]] with [[15/13]], finding the 13th harmonic 29 generators down. These two mappings are enharmonically equivalent in [[31edo]]. Finally, '''squad''' tempers out [[351/343]] (which is the same as 3.7.11.13 [[Mintaka#Extensions of Mintaka\|Minalzidar]]'s tempering of that prime) so that 13 is equated with (7/3)<sup>3</sup>, and found 15 generators down.		As for prime 13, the way to map it is less clear. The canonical squares mapping tempers out [[144/143]] in order to equate the tridecimal neutral sixth, [[13/8]], with 18/11, finding 13 two generators up, while '''agora''' tempers out [[105/104]] to equate [[8/7]] with [[15/13]], finding the 13th harmonic 29 generators down. These two mappings are enharmonically equivalent in [[31edo]]. Finally, '''squad''' tempers out [[351/343]] (which is the same as 3.7.11.13 [[Mintaka#Extensions of Mintaka\|Minalzidar]]'s tempering of that prime) so that 13 is equated with (7/3)<sup>3</sup>, and found 15 generators down.