Misty: Difference between revisions

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|+
|+
!| # periods
!| # periods
! colspan="2" |+0 (mod 3)
! colspan="2" |-1 (mod 3)
! colspan="2" |0 (mod 3)
! colspan="2" |+1 (mod 3)
! colspan="2" |+1 (mod 3)
! colspan="2" |+2 (mod 3)
|-
|-
!# gens
!# gens
Line 27: Line 27:
|-
|-
! -1
! -1
|'''703.1'''
|'''3/2'''
|1103.1
|1103.1
|189/100
|189/100
|303.1
|303.1
|25/21
|25/21
|'''703.1'''
|'''3/2'''
|-
|-
! class="thl" |0
! class="thl" |0
|class="thl"|800
|class="thl"|100/63
| class="thl" |'''0'''
| class="thl" |'''0'''
|class="thl"|'''1/1'''
|class="thl"|'''1/1'''
|class="thl"|400
|class="thl"|400
|class="thl"|63/50
|class="thl"|63/50
|class="thl"|800
|class="thl"|100/63
|-
|-
!1
!1
|896.9
|42/25
|96.9
|96.9
|135/128, 200/189
|135/128, 200/189
|496.9
|496.9
|4/3
|4/3
|896.9
|42/25
|-
|-
!2
!2
|993.8
|16/9
|193.8
|193.8
|28/25
|28/25
|593.8
|593.8
|45/32
|45/32
|993.8
|16/9
|-
|-
!3
!3
|1090.6
|15/8
|290.6
|290.6
|32/27
|32/27
|690.6
|690.6
|
|
|1090.6
|15/8
|-
|-
!4
!4
|1187.5
|125/126, 224/225
|'''387.5'''
|'''387.5'''
|'''5/4'''
|'''5/4'''
|787.5
|787.5
|63/40, 128/81
|63/40, 128/81
|1187.5
|125/126, 224/225
|-
|-
!5
!5
|84.4
|21/20
|484.4
|484.4
|
|
|884.4
|884.4
|5/3
|5/3
|84.4
|21/20
|-
|-
!6
!6
|181.3
|10/9
|581.3
|581.3
|7/5
|7/5
|981.3
|981.3
|
|
|181.3
|10/9
|-
|-
!7
!7
|278.2
|75/64
|678.2
|678.2
|40/27
|40/27
|1078.2
|1078.2
|28/15
|28/15
|278.2
|75/64
|-
|-
!8
!8
|375.1
|56/45
|775.1
|775.1
|25/16
|25/16
|1175.1
|1175.1
|63/64, 80/81
|63/64, 80/81
|375.1
|56/45
|-
|-
!9
!9
|471.9
|21/16
|871.9
|871.9
|
|
|71.9
|71.9
|25/24
|25/24
|471.9
|21/16
|-
|-
!10
!10
|568.8
|50/36
|'''968.8'''
|'''968.8'''
|'''7/4'''
|'''7/4'''
|168.8
|168.8
|
|
|568.8
|50/36
|}
|}


Revision as of 02:23, 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 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

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

(* 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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