Squares

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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 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

Scales

Music