Squares
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 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.
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.
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, 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 labeled in bold.
# | Cents* | Approximate Ratios | |||
---|---|---|---|---|---|
11-limit | 13-limit Extension | ||||
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