Misty: Difference between revisions
From XenReference
| Line 8: | Line 8: | ||
== Theory == | == Theory == | ||
=== Intervals === | === Intervals === | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
| Line 16: | Line 15: | ||
|- | |- | ||
!# gens | !# gens | ||
!Cents | !Cents* | ||
!JI | !JI | ||
!# gens | !# gens | ||
!Cents | !Cents* | ||
!JI | !JI | ||
!# gens | !# gens | ||
!Cents | !Cents* | ||
!JI | !JI | ||
|- | |- | ||
| Line 145: | Line 144: | ||
|50/36 | |50/36 | ||
|} | |} | ||
(* exact-2/1, exact-7/4 tuning) | |||
=== Derivation of 1/3-octave period === | === Derivation of 1/3-octave period === | ||
Revision as of 16:07, 2 April 2026
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
| +0 periods | +1 periods | +2 periods | ||||||
|---|---|---|---|---|---|---|---|---|
| # gens | Cents* | JI | # gens | Cents* | JI | # gens | Cents* | JI |
| -1 | 1103.1 | 189/100 | -1 | 303.1 | 25/21 | -1 | 703.1 | 3/2 |
| 0 | 0 | 1/1 | 0 | 400 | 63/50 | 0 | 800 | 100/63 |
| 1 | 96.9 | 200/189 | 1 | 496.9 | 4/3 | 1 | 896.9 | 42/25 |
| 2 | 193.8 | 28/25 | 2 | 593.8 | 2 | 993.8 | 16/9 | |
| 3 | 290.6 | 32/27 | 3 | 690.6 | 3 | 1090.6 | 15/8 | |
| 4 | 387.5 | 5/4 | 4 | 787.5 | 63/40, 128/81 | 4 | 1187.5 | 125/126, 224/225 |
| 5 | 484.4 | 5 | 884.4 | 5/3 | 5 | 84.4 | 21/20 | |
| 6 | 581.3 | 7/5 | 6 | 981.3 | 6 | 181.3 | 10/9 | |
| 7 | 678.2 | 7 | 1078.2 | 28/15 | 7 | 278.2 | 75/64 | |
| 8 | 775.1 | 25/16 | 8 | 1175.1 | 63/64, 80/81 | 8 | 375.1 | |
| 9 | 871.9 | 9 | 71.9 | 25/24 | 9 | 471.9 | 21/16 | |
| 10 | 968.8 | 7/4 | 10 | 168.8 | 10 | 568.8 | 50/36 | |
(* exact-2/1, exact-7/4 tuning)
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
