Squares: Difference between revisions

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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 = 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]].
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 ==
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== 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" |  

Latest revision as of 09:44, 9 February 2026

Skwares; Squares
Subgroups 2.3.7, 2.3.7.11, 2.3.5.7.11
Comma basis 19683/19208 (2.3.7);
99/98, 243/242 (2.3.7.11);
81/80, 99/98, 121/120 (11-limit)
Reduced mapping ⟨1; -4 -16 -9 -10]
ET join 14c & 17c
Generators (CWE) ~9/7 = 426.0 ¢
MOS scales 3L 2s, 3L 5s, 3L 8s, 3L 11s, 14L 3s
Ploidacot beta-tetracot
Pergen (P8, P11/4)
Minimax error 2.3.7 7-odd-limit: 4.7 ¢;
11-odd-limit: 10.8 ¢
Target scale size 2.3.7 7-odd-limit: 11 notes;
11-odd-limit: 17 notes

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 3.7.11-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. 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.

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 minalzidar's tempering of that prime) so that 13 is equated with (7/3)3, and found 15 generators down.

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

Interval chain

In the following table, harmonics and subharmonics 1–13 are labelled in bold.

# Cents* Approximate ratios
11-limit 13-limit extensions
Squares Squad Agora
0 0.0 1/1
1 425.8 9/7, 14/11 13/10
2 851.7 18/11, 33/20, 44/27 13/8 21/13
3 77.5 21/20, 28/27 27/26
4 503.4 4/3
5 929.2 12/7 22/13, 26/15
6 155.1 11/10, 12/11 13/12 14/13
7 580.9 7/5 18/13
8 1006.8 9/5, 16/9
9 232.6 8/7 15/13
10 658.4 16/11, 22/15 13/9
11 1084.3 28/15 13/7 24/13
12 310.1 6/5 13/11
13 736.0 32/21 20/13
14 1161.8 49/25, 64/33, 96/49 52/27
15 387.7 56/45 26/21 16/13
16 813.5 8/5 21/13
17 39.3 36/35, 64/63

* In 11-limit CWE tuning

Scales

Tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~14/9 = 774.3052 ¢ CWE: ~14/9 = 774.1560 ¢ POTE: ~14/9 = 774.0585 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~11/7 = 774.4005 ¢ CWE: ~11/7 = 774.1754 ¢ POTE: ~11/7 = 774.0427 ¢

Music

Joel Kivelä
Chris Vaisvil
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