Misty
From XenReference
Misty, 12 & 87, is a 7-limit temperament with
- generator 3/2 (period-reduced: 25/21)
- period 1/3-octave which represents 63/50, the difference between 56/25 and 16/9.
In Misty, the diesis 128/125 is split into three 126/125s (which also represent 225/224). As a result, the octave is also split into three, because by definition a 5/4 and a third of 128/125 reach 1\3. The generator is 135/128, which stacks four times to 5/4 and raises by a 400c period to 4/3. Because it is a weak extension of Didacus, 5/4 is split into two parts that stack 5 times to 7/4.
Misty results from tempering out the following 2 commas:
- 5120/5103, the aberschisma, which equates 64/63 and 81/80
- 3136/3125, the Didacus comma
Theory
Intervals
Misty has the structural property of dividing the comma 81/80~64/63 into two equal kleismas 126/125~225/224.
| # periods | +0 (mod 3) | +1 (mod 3) | +2 (mod 3) | |||
|---|---|---|---|---|---|---|
| # gens | Cents* | JI | Cents* | JI | Cents* | JI |
| -1 | 1103.1 | 189/100 | 303.1 | 25/21 | 703.1 | 3/2 |
| 0 | 0 | 1/1 | 400 | 63/50 | 800 | 100/63 |
| 1 | 96.9 | 135/128, 200/189 | 496.9 | 4/3 | 896.9 | 42/25 |
| 2 | 193.8 | 28/25 | 593.8 | 45/32 | 993.8 | 16/9 |
| 3 | 290.6 | 32/27 | 690.6 | 1090.6 | 15/8 | |
| 4 | 387.5 | 5/4 | 787.5 | 63/40, 128/81 | 1187.5 | 125/126, 224/225 |
| 5 | 484.4 | 884.4 | 5/3 | 84.4 | 21/20 | |
| 6 | 581.3 | 7/5 | 981.3 | 181.3 | 10/9 | |
| 7 | 678.2 | 40/27 | 1078.2 | 28/15 | 278.2 | 75/64 |
| 8 | 775.1 | 25/16 | 1175.1 | 63/64, 80/81 | 375.1 | 56/45 |
| 9 | 871.9 | 71.9 | 25/24 | 471.9 | 21/16 | |
| 10 | 968.8 | 7/4 | 168.8 | 568.8 | 50/36 | |
(* exact-2/1, exact-7/4 tuning; octave-reduced)
Derivation of 1/3-octave period
- 128/125 = 126/125 * 64/63 is equated to 126/125 * 81/80 by the aberschisma
- 81/80 itself = 126/125 * 225/224
- Didacus equates 225/224 to 126/125
- So we have 128/125 ~= 126/125 * 81/80 = 126/125 * 126/125 * 225/224 ~= (126/125)3
- Hence, 2/1 = 5/4 * 5/4 * 5/4 * 128/125 ~= (5/4 * 126/125)3 = (63/50)3
