Misty: Difference between revisions
From XenReference
m The cent values in the table are supposed to be octave-reduced, not period-reduced; more intuitive that way |
The cent values in the table are supposed to be period-reduced, not octave-reduced; more intuitive that way |
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!JI | !JI | ||
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| -1 | | -1 | ||
|303.1 | |303.1 | ||
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|'''703.1''' | |'''703.1''' | ||
|'''3/2''' | |'''3/2''' | ||
| -1 | |||
|1103.1 | |||
|189/100 | |||
|- | |- | ||
|class="thl"|0 | |class="thl"|0 | ||
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|125/126, 224/225 | |125/126, 224/225 | ||
|- | |- | ||
|5 | |||
|84.4 | |||
|21/20 | |||
|5 | |5 | ||
|484.4 | |484.4 | ||
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|884.4 | |884.4 | ||
|5/3 | |5/3 | ||
|- | |- | ||
|6 | |||
|181.3 | |||
|10/9 | |||
|6 | |6 | ||
|581.3 | |581.3 | ||
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|981.3 | |981.3 | ||
| | | | ||
|- | |- | ||
|7 | |||
|278.2 | |||
|75/64 | |||
|7 | |7 | ||
|678.2 | |678.2 | ||
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|1078.2 | |1078.2 | ||
|28/15 | |28/15 | ||
|- | |- | ||
|8 | |||
|375.1 | |||
|56/45 | |||
|8 | |8 | ||
|775.1 | |775.1 | ||
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|1175.1 | |1175.1 | ||
|63/64, 80/81 | |63/64, 80/81 | ||
|- | |- | ||
|9 | |9 | ||
|71.9 | |71.9 | ||
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|471.9 | |471.9 | ||
|21/16 | |21/16 | ||
|9 | |||
|871.9 | |||
| | |||
|- | |- | ||
|10 | |10 | ||
|168.8 | |168.8 | ||
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|568.8 | |568.8 | ||
|50/36 | |50/36 | ||
|10 | |||
|'''968.8''' | |||
|'''7/4''' | |||
|} | |} | ||
Revision as of 01:42, 3 April 2026
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 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.
| +0 periods | +1 periods | +2 periods | ||||||
|---|---|---|---|---|---|---|---|---|
| # gens | Cents* | JI | # gens | Cents* | JI | # gens | Cents* | JI |
| -1 | 303.1 | 25/21 | -1 | 703.1 | 3/2 | -1 | 1103.1 | 189/100 |
| 0 | 0 | 1/1 | 0 | 400 | 63/50 | 0 | 800 | 100/63 |
| 1 | 96.9 | 135/128, 200/189 | 1 | 496.9 | 4/3 | 1 | 896.9 | 42/25 |
| 2 | 193.8 | 28/25 | 2 | 593.8 | 45/32 | 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 | 84.4 | 21/20 | 5 | 484.4 | 5 | 884.4 | 5/3 | |
| 6 | 181.3 | 10/9 | 6 | 581.3 | 7/5 | 6 | 981.3 | |
| 7 | 278.2 | 75/64 | 7 | 678.2 | 40/27 | 7 | 1078.2 | 28/15 |
| 8 | 375.1 | 56/45 | 8 | 775.1 | 25/16 | 8 | 1175.1 | 63/64, 80/81 |
| 9 | 71.9 | 25/24 | 9 | 471.9 | 21/16 | 9 | 871.9 | |
| 10 | 168.8 | 10 | 568.8 | 50/36 | 10 | 968.8 | 7/4 | |
(* 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
