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Squares: Difference between revisions

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{{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 = 31 | Edo join 2 = 45
| 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]].


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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"
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| 0
| 0
| 0.0
| 0.0
| 1/1
| '''1/1'''
|
|
|
|
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|-
|-
| 1
| 1
| 425.6
| 425.8
| 9/7, 14/11
| 9/7, 14/11
|
|
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|-
|-
| 2
| 2
| 851.2
| 851.7
| 18/11, 33/20, 44/27
| 18/11, 33/20, 44/27
| '''13/8'''
| '''13/8'''
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|-
|-
| 3
| 3
| 76.8
| 77.5
| 21/20, 28/27
| 21/20, 28/27
|
|
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|-
|-
| 4
| 4
| 502.4
| 503.4
| '''4/3'''
| '''4/3'''
|
|
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|-
|-
| 5
| 5
| 928.0
| 929.2
| 12/7
| 12/7
|
|
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|-
|-
| 6
| 6
| 153.6
| 155.1
| 11/10, 12/11
| 11/10, 12/11
| 13/12
| 13/12
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|-
|-
| 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'''
|
|
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|-
|-
| 10
| 10
| 656.0
| 658.4
| '''16/11''', 22/15
| '''16/11''', 22/15
| 13/9
| 13/9
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|-
|-
| 11
| 11
| 1081.6
| 1084.3
| 28/15
| 28/15
| 13/7
| 13/7
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|-
|-
| 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
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|-
|-
| 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 ==
Retrieved from "https://en.xen.wiki/w/Squares"