Squares: Difference between revisions
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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 | 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]] | 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 [[ | 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. | ||
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{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
|- | |- | ||
! rowspan="3" | | ! rowspan="3" | # | ||
! rowspan="3" | Cents* | ! rowspan="3" | Cents* | ||
! colspan="4" | Approximate | ! colspan="4" | Approximate ratios | ||
|- | |- | ||
! rowspan="2" | 11-limit | ! rowspan="2" | 11-limit | ||
! colspan="3" | 13-limit | ! colspan="3" | 13-limit extensions | ||
|- | |- | ||
! Squares | ! Squares | ||
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== Music == | == Music == | ||
* [ | ; [[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 --> | ||
Latest revision as of 12:44, 24 August 2025
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, prime harmonics and subharmonics are labelled in bold.
| # | Cents* | Approximate ratios | |||
|---|---|---|---|---|---|
| 11-limit | 13-limit extensions | ||||
| Squares | Squad | Agora | |||
| 0 | 0.0 | 1/1 | |||
| 1 | 425.6 | 9/7, 14/11 | 13/10 | ||
| 2 | 851.2 | 18/11, 33/20, 44/27 | 13/8 | 21/13 | |
| 3 | 76.8 | 21/20, 28/27 | 27/26 | ||
| 4 | 502.4 | 4/3 | |||
| 5 | 928.0 | 12/7 | 22/13, 26/15 | ||
| 6 | 153.6 | 11/10, 12/11 | 13/12 | 14/13 | |
| 7 | 579.2 | 7/5 | 18/13 | ||
| 8 | 1004.8 | 9/5, 16/9 | |||
| 9 | 230.4 | 8/7 | 15/13 | ||
| 10 | 656.0 | 16/11, 22/15 | 13/9 | ||
| 11 | 1081.6 | 28/15 | 13/7 | 24/13 | |
| 12 | 307.2 | 6/5 | 13/11 | ||
| 13 | 732.8 | 32/21 | 20/13 | ||
| 14 | 1158.4 | 49/25, 64/33, 96/49 | 52/27 | ||
| 15 | 384.0 | 56/45 | 26/21 | 16/13 | |
| 16 | 809.6 | 8/5 | 21/13 | ||
| 17 | 35.2 | 36/35, 64/63 | |||
* In 11-limit CTE 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 ¢ |