Misty

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Revision as of 01:29, 3 April 2026 by Inthar (talk | contribs) (The cent values in the table are supposed to be octave-reduced, not period-reduced; more intuitive that way)

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 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 96.9 135/128, 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
5 484.4 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 40/27 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 56/45
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; octave-reduced)

Derivation of 1/3-octave period

  1. 128/125 = 126/125 * 64/63 is equated to 126/125 * 81/80 by the aberschisma
  2. 81/80 itself = 126/125 * 225/224
  3. Didacus equates 225/224 to 126/125
  4. So we have 128/125 ~= 126/125 * 81/80 = 126/125 * 126/125 * 225/224 ~= (126/125)3
  5. Hence, 2/1 = 5/4 * 5/4 * 5/4 * 128/125 ~= (5/4 * 126/125)3 = (63/50)3


View • Talk • EditRegular temperaments
Rank-2
Acot Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave)
Monocot Meantone • Schismic • Gentle-fifth temperaments • 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) • Miracle (hexacot) • Kleismic (alpha-hexacot) • Hemiseven (gamma-hexacot) • Hedgehog (diploid alpha-tricot)
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) • Paddle (alter-dicot)
No-3 Mabilic (alpha-triseph) • Orgone (trimech) • Didacus (diseph) • Sidewalk (epsilon-heptaseph)
No-2 Sensamagic (monogem)
Exotemperaments Dicot • Mavila • Father
Higher-rank
Rank-3 Aberschismic • Marvel • Parapyth • Pentagoth • Intergan
Straddle Rank-3 Hibernal • Monarch
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