Squares: Difference between revisions
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{{Infobox | {{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 | ||
| | | 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]] | ||
| Pergen = (P8, P11/4) | | Pergen = (P8, P11/4) | ||
| Odd limit 1 = | | 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]]. | ||
Latest revision as of 09:44, 9 February 2026
| Skwares; Squares |
99/98, 243/242 (2.3.7.11);
81/80, 99/98, 121/120 (11-limit)
11-odd-limit: 10.8 ¢
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
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~14/9 = 774.3052 ¢ | CWE: ~14/9 = 774.1560 ¢ | POTE: ~14/9 = 774.0585 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~11/7 = 774.4005 ¢ | CWE: ~11/7 = 774.1754 ¢ | POTE: ~11/7 = 774.0427 ¢ |