Misty
From XenReference
Misty, 87 & 99, 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.
It 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
(exact-7/4 tuning)
| Period 1 | Period 2 | Period 3 | ||||||
|---|---|---|---|---|---|---|---|---|
| # | Cents | JI | # | Cents | JI | # | Cents | JI |
| -1 | 1103.117 | 189/100 | -1 | 303.117 | 25/21 | -1 | 703.117 | 3/2 |
| 0 | 0 | 1/1 | 0 | 400 | 63/50 | 0 | 800 | 100/63 |
| 1 | 96.883 | 200/189 | 1 | 496.883 | 4/3 | 1 | 896.883 | 42/25 |
| 2 | 193.766 | 28/25 | 2 | 593.766 | 2 | 993.766 | 16/9 | |
| 3 | 290.649 | 3 | 690.649 | 3 | 1090.649 | 15/8 | ||
| 4 | 387.532 | 5/4 | 4 | 787.532 | 4 | 1187.532 | ||
| 5 | 484.415 | 5 | 885.415 | 5 | 85.415 | |||
| 6 | 581.298 | 7/5 | 6 | 981.298 | 6 | 181.298 | ||
| 7 | 678.177 | 7 | 1078.177 | 7 | 278.177 | |||
| 8 | 775.06 | 25/16 | 8 | 1175.06 | 8 | 375.06 | ||
| 9 | 871.943 | 9 | 71.943 | 9 | 471.943 | |||
| 10 | 968.826 | 7/4 | 10 | 168.826 | 10 | 568.826 | ||
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
