Misty
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
Notable Misty edos include 87edo, 99edo, and 111edo.
Theory
Intervals
Misty has the structural property of dividing the comma 81/80~64/63 into two equal kleismas 126/125~225/224.
| # periods | -1 (mod 3) | 0 (mod 3) | +1 (mod 3) | |||
|---|---|---|---|---|---|---|
| # gens | Cents* | JI | Cents* | JI | Cents* | JI |
| -1 | 703.1 | 3/2 | 1103.1 | 189/100 | 303.1 | 25/21 |
| 0 | 800 | 100/63 | 0 | 1/1 | 400 | 63/50 |
| 1 | 896.9 | 42/25 | 96.9 | 135/128, 200/189 | 496.9 | 4/3 |
| 2 | 993.8 | 16/9 | 193.8 | 28/25 | 593.8 | 45/32 |
| 3 | 1090.6 | 15/8 | 290.6 | 32/27 | 690.6 | |
| 4 | 1187.5 | 125/126, 224/225 | 387.5 | 5/4 | 787.5 | 63/40, 128/81 |
| 5 | 84.4 | 21/20 | 484.4 | 884.4 | 5/3 | |
| 6 | 181.3 | 10/9 | 581.3 | 7/5 | 981.3 | |
| 7 | 278.2 | 75/64 | 678.2 | 40/27 | 1078.2 | 28/15 |
| 8 | 375.1 | 56/45 | 775.1 | 25/16 | 1175.1 | 63/64, 80/81 |
| 9 | 471.9 | 21/16 | 871.9 | 71.9 | 25/24 | |
| 10 | 568.8 | 50/36 | 968.8 | 7/4 | 168.8 | |
| 11 | 665.7 | 1065.7 | 50/27 | 265.7 | 7/6 | |
| 12 | 762.6 | 14/9 | 1162.6 | 49/50, 125/128 | 362.6 | 100/81 |
| 13 | 859.5 | 59.5 | 28/27 | 459.5 | ||
| 14 | 956.4 | 156.4 | 35/32 | 556.4 | ||
(* 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
Patent vals
The following patent vals support 5-limit Misty, which tempers out [26 -12 -3⟩.
| Edo | 7-limit extension | Fifth |
|---|---|---|
| 12 | 12 & 87 | 700.000 |
| 123 | 702.439 | |
| 111 | 12 & 87 | 702.703 |
| 210 | 12 & 87 | 702.857 |
| 99 | 12 & 87 | 703.030 |
| 384 | 12 & 87 | 703.125 |
| 285 | 12 & 87 | 703.158 |
| 471 | 12 & 87 | 703.185 |
| 186 | 12 & 87 | 703.226 |
| 273 | 12 & 87 | 703.297 |
| 360 | 703.333 | |
| 87 | 12 & 87 | 703.448 |
| 336 | 703.571 | |
| 249 | 703.614 | |
| 162 | 703.704 | |
| 237 | 703.797 | |
| 312 | 703.846 | |
| 75 | 704.000 | |
| 138 | 704.348 | |
| 63 | 704.762 | |
| 51 | 705.882 |
| View • Talk • EditRegular temperaments | |
|---|---|
| Rank-2 | |
| Acot | Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave) |
| Monocot | Meantone • Schismic • Leapday • Archy |
| Complexity 2 | Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot) |
| Complexity 3 | Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot) |
| Complexity 4 | Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot) |
| Complexity 5-6 | Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot) |
| Higher complexity | Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (epsilon-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot) |
| Straddle-3 | A-Team (alter-tricot) • Machine (alter-monocot) |
| No-3 | Trismegistus (alpha-triseph) • Orgone (trimech) • Didacus (diseph) |
| No-octaves | Sensamagic (monogem) |
| Exotemperament | Dicot • Mavila • Father |
| Higher-rank | |
| Rank-3 | Hemifamity • Marvel • Parapyth |
