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'''Misty''', 87 & 99, is a 7-limit temperament with
'''Misty''', 12 & 87, is a 7-limit temperament with
* generator 3/2 (period-reduced: 25/21)
* generator 3/2 (period-reduced: 25/21)
* period 1/3-octave which represents 63/50, the difference between 56/25 and 16/9.
* 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:
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.
* 5120/5103, the aberschisma, which equates 64/63 and 81/80
 
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
* 3136/3125, the [[Didacus]] comma


Notable Misty edos include [[87edo]], [[99edo]], and [[111edo]].
== Theory ==
== Theory ==
=== Intervals ===
=== Intervals ===
Misty has the structural property of dividing the comma 81/80~64/63 into two equal kleismas 126/125~225/224.
{| class="wikitable"
{| class="wikitable"
|+
|+
! colspan="3" |+0 periods
!| # periods
! colspan="3" |+1 periods
! colspan="2" |-1 (mod 3)
! colspan="3" |+2 periods
! colspan="2" |0 (mod 3)
! colspan="2" |+1 (mod 3)
|-
|-
!# gens
!# gens
!Cents*
!Cents*
!JI
!JI
!# gens
!Cents*
!Cents*
!JI
!JI
!# gens
!Cents*
!Cents*
!JI
!JI
|-
|-
| -1
! -1
|'''703.1'''
|'''3/2'''
|1103.1
|1103.1
|189/100
|189/100
| -1
|303.1
|303.1
|25/21
|25/21
| -1
|'''703.1'''
|'''3/2'''
|-
|-
|class="thl"|0
! class="thl" |0
|class="thl"|'''0'''
|class="thl"|800
|class="thl"|100/63
| class="thl" |'''0'''
|class="thl"|'''1/1'''
|class="thl"|'''1/1'''
|class="thl"|0
|class="thl"|400
|class="thl"|400
|class="thl"|63/50
|class="thl"|63/50
|class="thl"|0
|class="thl"|800
|class="thl"|100/63
|-
|-
|1
!1
|896.9
|42/25
|96.9
|96.9
|200/189
|135/128, 200/189
|1
|496.9
|496.9
|4/3
|4/3
|1
|896.9
|42/25
|-
|-
|2
!2
|993.8
|16/9
|193.8
|193.8
|28/25
|28/25
|2
|593.8
|593.8
|
|45/32
|2
|993.8
|16/9
|-
|-
|3
!3
|1090.6
|15/8
|290.6
|290.6
|32/27
|32/27
|3
|690.6
|690.6
|
|
|3
|1090.6
|15/8
|-
|-
|4
!4
|1187.5
|125/126, 224/225
|'''387.5'''
|'''387.5'''
|'''5/4'''
|'''5/4'''
|4
|787.5
|787.5
|63/40, 128/81
|63/40, 128/81
|4
|1187.5
|125/126, 224/225
|-
|-
|5
!5
|84.4
|21/20
|484.4
|484.4
|
|
|5
|884.4
|884.4
|5/3
|5/3
|5
|84.4
|21/20
|-
|-
|6
!6
|181.3
|10/9
|581.3
|581.3
|7/5
|7/5
|6
|981.3
|981.3
|
|
|6
|181.3
|10/9
|-
|-
|7
!7
|278.2
|75/64
|678.2
|678.2
|
|40/27
|7
|1078.2
|1078.2
|28/15
|28/15
|7
|278.2
|75/64
|-
|-
|8
!8
|375.1
|56/45
|775.1
|775.1
|25/16
|25/16
|8
|1175.1
|1175.1
|63/64, 80/81
|63/64, 80/81
|8
|375.1
|
|-
|-
|9
!9
|471.9
|21/16
|871.9
|871.9
|
|
|9
|71.9
|71.9
|25/24
|25/24
|9
|471.9
|21/16
|-
|-
|10
!10
|568.8
|50/36
|'''968.8'''
|'''968.8'''
|'''7/4'''
|'''7/4'''
|10
|168.8
|168.8
|
|
|10
|-
|568.8
!11
|50/36
|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)
(* exact-2/1, exact-7/4 tuning; octave-reduced)


=== Derivation of 1/3-octave period ===
=== Derivation of 1/3-octave period ===
Line 154: Line 165:
# {{adv|Hence, 2/1 {{=}} 5/4 * 5/4 * 5/4 * 128/125 ~{{=}} (5/4 * 126/125)<sup>3</sup> {{=}} (63/50)<sup>3</sup>}}
# {{adv|Hence, 2/1 {{=}} 5/4 * 5/4 * 5/4 * 128/125 ~{{=}} (5/4 * 126/125)<sup>3</sup> {{=}} (63/50)<sup>3</sup>}}


== Patent vals ==
The following patent vals support 5-limit Misty, which tempers out {{monzo|26 -12 -3}}. Vals that are contorted in the 5-limit are not included.
{| class="wikitable sortable"
!|Edo
!|7-limit extension
!|Fifth
|-
||12||12 & 87||700.000
|-
||123||12 & 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||87 & 360||703.333
|-
||87||12 & 87, 87 & 75||703.448
|-
||336||87 & 75||703.571
|-
||249||87 & 75||703.614
|-
||162||87 & 75||703.704
|-
||237||237 & 312||703.797
|-
||312||237 & 312||703.846
|-
||75||12 & 51, 87 & 75||704.000
|-
||138|| ||704.348
|-
||63||12 & 51||704.762
|-
||51||12 & 51||705.882
|}
{{Navbox regtemp}}
{{Navbox regtemp}}
{{Cat|Temperaments}}
{{Cat|Temperaments}}

Latest revision as of 00:39, 4 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:

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

  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

Patent vals

The following patent vals support 5-limit Misty, which tempers out [26 -12 -3⟩. Vals that are contorted in the 5-limit are not included.

Edo 7-limit extension Fifth
12 12 & 87 700.000
123 12 & 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 87 & 360 703.333
87 12 & 87, 87 & 75 703.448
336 87 & 75 703.571
249 87 & 75 703.614
162 87 & 75 703.704
237 237 & 312 703.797
312 237 & 312 703.846
75 12 & 51, 87 & 75 704.000
138 704.348
63 12 & 51 704.762
51 12 & 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
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