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At its most basic level, '''skwares''' can be thought of as a [[2.3.7 subgroup]] temperament, 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 243/121~99/49, in a manner similar to [[sensi]]'s relation to [[BPS]].
{{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 = 14c | Edo join 2 = 17c
| 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]]
| 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]].


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.


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. Meanwhile squad tempers out [[351/343]] (which is the same as 3.7.11.13 [[Mintaka#Minalzidar|Minalzidar]]'s tempering of that prime) so that 13 is equated with (7/3)<sup>3</sup>, and found 15 generators down. Finally, agora tempers out [[105/104]] to equate [[8/7]] with [[15/13]], finding the 13th harmonic 29 generators down.
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]].


See [[Meantone family #Squares]] and [[No-fives subgroup temperaments #Skwares]] for more technical data.  
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.


== Interval chain ==
== Interval chain ==
In the following table, prime harmonics are labeled 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"
|-
! rowspan="3" | #
! rowspan="3" | #
! rowspan="3" | Cents {{asterisk}}
! 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
Line 24: Line 39:
| 0
| 0
| 0.0
| 0.0
| 1/1
| '''1/1'''
|
|
|
|
Line 30: Line 45:
|-
|-
| 1
| 1
| 425.6
| 425.8
| 9/7, 14/11
| 9/7, 14/11
|
|
Line 37: Line 52:
|-
|-
| 2
| 2
| 851.2
| 851.7
| 18/11, 33/20, 44/27
| 18/11, 33/20, 44/27
| '''13/8'''
| '''13/8'''
Line 44: Line 59:
|-
|-
| 3
| 3
| 76.8
| 77.5
| 21/20, 28/27
| 21/20, 28/27
|
|
Line 51: Line 66:
|-
|-
| 4
| 4
| 502.4
| 503.4
| '''4/3'''
| '''4/3'''
|
|
Line 58: Line 73:
|-
|-
| 5
| 5
| 928.0
| 929.2
| 12/7
| 12/7
|
|
Line 65: Line 80:
|-
|-
| 6
| 6
| 153.6
| 155.1
| 11/10, 12/11
| 11/10, 12/11
| 13/12
| 13/12
Line 72: Line 87:
|-
|-
| 7
| 7
| 579.2
| 580.9
| 7/5
| 7/5
|
|
Line 79: Line 94:
|-
|-
| 8
| 8
| 1004.8
| 1006.8
| 9/5, 16/9
| 9/5, '''16/9'''
|
|
|
|
Line 86: Line 101:
|-
|-
| 9
| 9
| 230.4
| 232.6
| '''8/7'''
| '''8/7'''
|
|
Line 93: Line 108:
|-
|-
| 10
| 10
| 656.0
| 658.4
| '''16/11''', 22/15
| '''16/11''', 22/15
| 13/9
| 13/9
Line 100: Line 115:
|-
|-
| 11
| 11
| 1081.6
| 1084.3
| 28/15
| 28/15
| 13/7
| 13/7
Line 107: Line 122:
|-
|-
| 12
| 12
| 307.2
| 310.1
| 6/5
| 6/5
| 13/11
| 13/11
Line 114: Line 129:
|-
|-
| 13
| 13
| 732.8
| 736.0
| 32/21
| 32/21
|
|
Line 121: Line 136:
|-
|-
| 14
| 14
| 1158.4
| 1161.8
| 49/25, 64/33, 96/49
| 49/25, 64/33, 96/49
| 52/27
| 52/27
Line 128: Line 143:
|-
|-
| 15
| 15
| 384.0
| 387.7
| 56/45
| 56/45
| 26/21
| 26/21
Line 135: Line 150:
|-
|-
| 16
| 16
| 809.6
| 813.5
| '''8/5'''
| '''8/5'''
|
|
Line 142: Line 157:
|-
|-
| 17
| 17
| 35.2
| 39.3
| 36/35, 64/63
| 36/35, 64/63
|
|
Line 148: Line 163:
|
|
|}
|}
{{asterisk}} in 11-limit CTE tuning
<nowiki/>* In 11-limit CWE tuning


== Scales ==
== Scales ==
Line 154: Line 169:
* [[Skwares11]]
* [[Skwares11]]
* [[Skwares14]]
* [[Skwares14]]
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~14/9 = 774.3052{{c}}
| CWE: ~14/9 = 774.1560{{c}}
| POTE: ~14/9 = 774.0585{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~11/7 = 774.4005{{c}}
| CWE: ~11/7 = 774.1754{{c}}
| POTE: ~11/7 = 774.0427{{c}}
|}


== 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:Temperaments]]
[[Category:Squares| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Meantone family]]
[[Category:Meantone family]]
[[Category:Nuwell temperaments]]
[[Category:Nuwell temperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Squares"