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'''Misty''', 87 & 99, is a 7-limit temperament with generator 3/2 and 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:
'''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
* 5120/5103, the aberschisma, which equates 64/63 and 81/80
* 3136/3125, the [[Didacus]] comma
* 3136/3125, the [[Didacus]] comma
Line 5: Line 8:
== Theory ==
== Theory ==
=== Intervals ===
=== Intervals ===
(exact-7/4 tuning)
{| class="wikitable"
|+
! colspan="3" |Period 1
! colspan="3" |Period 2
! colspan="3" |Period 3
|-
!#
!Cents
!JI
!#
!Cents
!JI
!#
!Cents
!JI
|-
| -1
|1103.117
|189/100
| -1
|303.117
|25/21
| -1
|'''703.117'''
|'''3/2'''
|-
|class="thl"|0
|class="thl"|'''0'''
|class="thl"|'''1/1'''
|class="thl"|0
|class="thl"|400
|class="thl"|63/50
|class="thl"|0
|class="thl"|800
|class="thl"|100/63
|-
|1
|96.883
|200/189
|1
|496.883
|4/3
|1
|896.883
|42/25
|-
|2
|193.766
|28/25
|2
|593.766
|
|2
|993.766
|16/9
|-
|3
|290.649
|
|3
|690.649
|
|3
|1090.649
|15/8
|-
|4
|'''387.532'''
|'''5/4'''
|4
|787.532
|
|4
|1187.532
|
|-
|5
|484.415
|
|5
|885.415
|
|5
|85.415
|
|-
|6
|581.298
|7/5
|6
|981.298
|
|6
|181.298
|
|-
|7
|678.177
|
|7
|1078.177
|
|7
|278.177
|
|-
|8
|775.06
|25/16
|8
|1175.06
|
|8
|375.06
|
|-
|9
|871.943
|
|9
|71.943
|
|9
|471.943
|
|-
|10
|'''968.826'''
|'''7/4'''
|10
|168.826
|
|10
|568.826
|
|}
=== Derivation of 1/3-octave period ===
=== Derivation of 1/3-octave period ===
# {{adv|128/125 {{=}} 126/125 * 64/63 is equated to 126/125 * 81/80 by the aberschisma}}
# {{adv|128/125 {{=}} 126/125 * 64/63 is equated to 126/125 * 81/80 by the aberschisma}}

Revision as of 15:55, 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

(exact-7/4 tuning)

Period 1 Period 2 Period 3
# Cents JI # Cents JI # Cents JI
-1 1103.117 189/100 -1 303.117 25/21 -1 703.117 3/2
0 0 1/1 0 400 63/50 0 800 100/63
1 96.883 200/189 1 496.883 4/3 1 896.883 42/25
2 193.766 28/25 2 593.766 2 993.766 16/9
3 290.649 3 690.649 3 1090.649 15/8
4 387.532 5/4 4 787.532 4 1187.532
5 484.415 5 885.415 5 85.415
6 581.298 7/5 6 981.298 6 181.298
7 678.177 7 1078.177 7 278.177
8 775.06 25/16 8 1175.06 8 375.06
9 871.943 9 71.943 9 471.943
10 968.826 7/4 10 168.826 10 568.826

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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