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__FORCETOC__
{{Technical data page}}
The misty family tempers out the [[misty_comma|misty comma]], 67108864/66430125 = |26 -12 -3>.
The '''misty family''' tempers out the [[misty comma]], 67108864/66430125 = {{monzo| 26 -12 -3 }}, equating the [[Pythagorean comma]] with a stack of three [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]]. It features a third-octave period, which is represented by [[512/405]] and acts as the interval separating ~[[128/81]] from ~[[5/4]].  


=Misty=
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.
Comma: 67108864/66430125 = |26 -12 -3>
* Adding 3136/3125 (hemimean) leads to septimal misty.
* Adding 225/224 (marvel) leads to fog.


POTE generator: ~3/2 = 703.111
== Misty ==
{{Main| Misty }}


Map: [<3 0 26|, <0 1 -4|]
[[Subgroup]]: 2.3.5


EDOs: 12, 51, 63, 75, 87, 99, 285, 384
[[Comma list]]: 67108864/66430125


Badness: 0.1065
{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}


==7-limit==
: mapping generators: ~512/405, ~3
Commas: 3136/3125, 5120/5103


[[POTE_tuning|POTE generator]]: 703.021
[[Optimal tuning]]s:
* [[CTE]]: ~512/405 = 1\3, ~3/2 = 703.2481
* [[POTE]]: ~512/405 = 1\3, ~3/2 = 703.111


Map: [<3 0 26 56|, <0 1 -4 -10|]
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c }}


Wedgie: <<3 -12 -30 -26 -56 -36||
[[Badness]]: 0.106540


EDOs: [[12edo|12]], [[87edo|87]], [[99edo|99]]
== Septimal misty ==
{{Main| Misty }}


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 441/440, 3136/3125, 4375/4356


[[POTE_tuning|POTE generator]]: 703.210
[[Comma list]]: 3136/3125, 5120/5103


Map: [<3 0 26 56 77|, <0 1 -4 -10 -14|]
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}


EDOs: 12, 87, 99, 186
: mapping generators: ~63/50, ~3


==13-limit==
Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448
Commas: 196/195, 352/351, 364/363, 635/624


[[POTE_tuning|POTE generator]]: 703.303
{{Optimal ET sequence|legend=1| 12, …, 87, 99 }}


Map: [<3 0 26 56 77 92|, <0 1 -4 -10 -14 -17|]
[[Badness]]: 0.036802


EDOs: 12, 87, 273
=== 2.3.5.7.17 subgroup ===
The [[S-expression]]-based comma list for this extension is {([[5120/5103|S8/S9]],) [[2025/2023|S15/S17]], [[256/255|S16]], [[1701/1700|S18/S20]](, [[2500/2499|S50]])}, and for the 2.3.5.7.17.19 subgroup extension, {([[5120/5103|S8/S9]],) [[256/255|S16]], [[324/323|S18]], [[400/399|S20]], [[1445/1444|S17/S19]](, [[2500/2499|S50]])}.


=Mystic=
Subgroup: 2.3.5.7.17
Commas: 325/324, 441/440, 896/891, 1001/1000


[[POTE_tuning|POTE generator]]: 703.311
Comma list: 256/255, 1701/1700, 2025/2023


Map: [<3 0 26 56 77 -46|, <0 1 -4 -10 -14 12|]
Sval mapping: {{mapping| 3 0 26 56 -2 | 0 1 -4 -10 3 }}


EDOs: 12, 87, 273
: sval mapping generators: ~34/27, ~3


==17-limit==
Optimal tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.0963
Commas: 196/195, 352/351, 364/363, 635/624


[[POTE_tuning|POTE generator]]: 703.233
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}


Map: [<3 0 26 56 77 92 -2|, <0 1 -4 -10 -14 -17 3|]
Badness: 0.0186


EDOs: 12, 87, 186
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19


==19-limit==
Comma list: 256/255, 324/323, 400/399, 476/475
Commas: 154/153, 196/195, 245/243, 273/272, 286/285, 325/324


[[POTE_tuning|POTE generator]]: 703.253
Sval mapping: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}


Map: [<3 0 26 56 77 92 -2 8|, <0 1 -4 -10 -14 -17 3 1|]
: sval mapping generators: ~24/19, ~3


EDOs: 12, 87, 186
Optimal tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.0778


=Hemimist=
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 99, 210gh, 309cdghh }}
Commas: 3136/3125, 5120/5103, 3025/3024


POTE generator: ~11/8 = 551.527
Badness: 0.0132


Map: [<3 0 26 56 8|, <0 2 -8 -20 1|]
=== Undecimal misty ===
Subgroup: 2.3.5.7.11


EDOs: 15, 24, 87, 111, 198
Comma list: 441/440, 3136/3125, 4375/4356


Badness: 0.0474
Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}


==13-limit==
Optimal tunings:
Commas: 352/351, 676/675, 847/845, 3025/3024
* CTE: ~44/35 = 1\3, ~3/2 = 703.3127
* POTE: ~44/35 = 1\3, ~3/2 = 703.210


POTE generator: ~11/8 = 551.525
Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}


Map: [<3 0 26 56 8 23|, <0 2 -8 -20 1 -5|]
Badness: 0.035932


EDOs: 15, 24, 87, 111, 198
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0229
Comma list: 196/195, 352/351, 364/363, 625/624
 
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}
 
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3859
* POTE: ~44/35 = 1\3, ~3/2 = 703.303
 
Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}
 
Badness: 0.027820
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594
 
Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}
 
Optimal tunings:
* CTE: ~34/27 = 1\3, ~3/2 = 703.3686
* POTE: ~34/27 = 1\3, ~3/2 = 703.233
 
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efg }}
 
Badness: 0.023908
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594
 
Mapping: {{mapping| 3 0 26 56 77 92 -2 8 | 0 1 -4 -10 -14 -17 3 1 }}
 
Optimal tunings:
* CTE: ~24/19 = 1\3, ~3/2 = 703.3626
* POTE: ~34/27 = 1\3, ~3/2 = 703.253
 
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efgh }}
 
Badness: 0.019878
 
==== Mystic ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 441/440, 896/891, 1001/1000
 
Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}
 
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3285
* POTE: ~44/35 = 1\3, ~3/2 = 703.311
 
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 186e, 273e }}
 
Badness: 0.036872
 
=== Semimist ===
Subgroup: 2.3.5.7.11
 
Comma list: 3136/3125, 5120/5103, 9801/9800
 
Mapping: {{mapping| 6 0 52 112 173 | 0 1 -4 -10 -16 }}
 
: mapping generators: ~55/49, ~3
 
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 703.0864
* POTE: ~55/49 = 1\6, ~3/2 = 702.9764
 
Optimal ET sequence: {{Optimal ET sequence| 12, 174e, 186, 198, 408cde, 606bcde }}
 
Badness: 0.068509
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 1716/1715, 3136/3125
 
Mapping: {{mapping| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}
 
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 703.1045
* POTE: ~55/49 = 1\6, ~3/2 = 703.0087
 
Optimal ET sequence: {{Optimal ET sequence| 12f, 174ef, 186, 198 }}
 
Badness: 0.044073
 
=== Hemimist ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 3136/3125, 5120/5103
 
Mapping: {{mapping| 3 0 26 56 8 | 0 2 -8 -20 1 }}
 
: mapping generators: ~63/50, ~400/231
 
Optimal tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.5721
 
Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}
 
Badness: 0.047439
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 676/675, 847/845, 3025/3024
 
Mapping: {{mapping| 3 0 26 56 8 23 | 0 2 -8 -20 1 -5 }}
 
Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 951.5810
 
Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}
 
Badness: 0.022923
 
== Fog ==
The fog temperament (12 & 63) tempers out the marvel comma, [[225/224]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 156250/151263
 
{{Mapping|legend=1| 3 0 26 37 | 0 1 -4 -6 }}
 
: mapping generators: ~80/63, ~3
 
Optimal tunings:
* CTE: ~80/63 = 1\3, ~3/2 = 704.3694
* POTE: ~80/63 = 1\3, ~3/2 = 704.118
 
{{Optimal ET sequence|legend=1| 12, 39c, 51, 63, 75 }}
 
[[Badness]]: 0.086866
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 225/224, 2420/2401
 
Mapping: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}
 
: mapping generators: ~44/35, ~3
 
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 704.6253
* POTE: ~44/35 = 1\3, ~3/2 = 704.411
 
Optimal ET sequence: {{Optimal ET sequence| 12, 39ce, 51, 63, 138de }}
 
Badness: 0.051348
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Misty family| ]] <!-- main article -->
[[Category:Rank 2]]

Latest revision as of 00:37, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The misty family tempers out the misty comma, 67108864/66430125 = [26 -12 -3⟩, equating the Pythagorean comma with a stack of three schismas, making it a member of the schismic–Pythagorean equivalence continuum. It features a third-octave period, which is represented by 512/405 and acts as the interval separating ~128/81 from ~5/4.

The second comma of the normal comma list defines which 7-limit family member we are looking at.

  • Adding 3136/3125 (hemimean) leads to septimal misty.
  • Adding 225/224 (marvel) leads to fog.

Misty

Main article: Misty

Subgroup: 2.3.5

Comma list: 67108864/66430125

Mapping: [⟨3 0 26], ⟨0 1 -4]]

mapping generators: ~512/405, ~3

Optimal tunings:

  • CTE: ~512/405 = 1\3, ~3/2 = 703.2481
  • POTE: ~512/405 = 1\3, ~3/2 = 703.111

Optimal ET sequence: 12, 51, 63, 75, 87, 99, 285, 384, 483c

Badness: 0.106540

Septimal misty

Main article: Misty

Subgroup: 2.3.5.7

Comma list: 3136/3125, 5120/5103

Mapping: [⟨3 0 26 56], ⟨0 1 -4 -10]]

mapping generators: ~63/50, ~3

Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448

Optimal ET sequence: 12, …, 87, 99

Badness: 0.036802

2.3.5.7.17 subgroup

The S-expression-based comma list for this extension is {(S8/S9,) S15/S17, S16, S18/S20(, S50)}, and for the 2.3.5.7.17.19 subgroup extension, {(S8/S9,) S16, S18, S20, S17/S19(, S50)}.

Subgroup: 2.3.5.7.17

Comma list: 256/255, 1701/1700, 2025/2023

Sval mapping: [⟨3 0 26 56 -2], ⟨0 1 -4 -10 3]]

sval mapping generators: ~34/27, ~3

Optimal tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.0963

Optimal ET sequence: 12, 87, 99, 309cdg, 408cdgg, 507cdggg

Badness: 0.0186

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 256/255, 324/323, 400/399, 476/475

Sval mapping: [⟨3 0 26 56 -2 8], ⟨0 1 -4 -10 3 1]]

sval mapping generators: ~24/19, ~3

Optimal tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.0778

Optimal ET sequence: 12, 87, 99, 210gh, 309cdghh

Badness: 0.0132

Undecimal misty

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 4375/4356

Mapping: [⟨3 0 26 56 77], ⟨0 1 -4 -10 -14]]

Optimal tunings:

  • CTE: ~44/35 = 1\3, ~3/2 = 703.3127
  • POTE: ~44/35 = 1\3, ~3/2 = 703.210

Optimal ET sequence: 12, …, 87, 99e, 186e

Badness: 0.035932

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 625/624

Mapping: [⟨3 0 26 56 77 92], ⟨0 1 -4 -10 -14 -17]]

Optimal tunings:

  • CTE: ~44/35 = 1\3, ~3/2 = 703.3859
  • POTE: ~44/35 = 1\3, ~3/2 = 703.303

Optimal ET sequence: 12f, …, 87, 186ef

Badness: 0.027820

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 352/351, 364/363, 375/374, 595/594

Mapping: [⟨3 0 26 56 77 92 -2], ⟨0 1 -4 -10 -14 -17 3]]

Optimal tunings:

  • CTE: ~34/27 = 1\3, ~3/2 = 703.3686
  • POTE: ~34/27 = 1\3, ~3/2 = 703.233

Optimal ET sequence: 12f, 87, 99ef, 186efg

Badness: 0.023908

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594

Mapping: [⟨3 0 26 56 77 92 -2 8], ⟨0 1 -4 -10 -14 -17 3 1]]

Optimal tunings:

  • CTE: ~24/19 = 1\3, ~3/2 = 703.3626
  • POTE: ~34/27 = 1\3, ~3/2 = 703.253

Optimal ET sequence: 12f, 87, 99ef, 186efgh

Badness: 0.019878

Mystic

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 441/440, 896/891, 1001/1000

Mapping: [⟨3 0 26 56 77 -46], ⟨0 1 -4 -10 -14 12]]

Optimal tunings:

  • CTE: ~44/35 = 1\3, ~3/2 = 703.3285
  • POTE: ~44/35 = 1\3, ~3/2 = 703.311

Optimal ET sequence: 12, 87, 186e, 273e

Badness: 0.036872

Semimist

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 5120/5103, 9801/9800

Mapping: [⟨6 0 52 112 173], ⟨0 1 -4 -10 -16]]

mapping generators: ~55/49, ~3

Optimal tunings:

  • CTE: ~55/49 = 1\6, ~3/2 = 703.0864
  • POTE: ~55/49 = 1\6, ~3/2 = 702.9764

Optimal ET sequence: 12, 174e, 186, 198, 408cde, 606bcde

Badness: 0.068509

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1716/1715, 3136/3125

Mapping: [⟨6 0 52 112 173 203], ⟨0 1 -4 -10 -16 -19]]

Optimal tunings:

  • CTE: ~55/49 = 1\6, ~3/2 = 703.1045
  • POTE: ~55/49 = 1\6, ~3/2 = 703.0087

Optimal ET sequence: 12f, 174ef, 186, 198

Badness: 0.044073

Hemimist

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 5120/5103

Mapping: [⟨3 0 26 56 8], ⟨0 2 -8 -20 1]]

mapping generators: ~63/50, ~400/231

Optimal tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.5721

Optimal ET sequence: 24d, 87, 111, 198

Badness: 0.047439

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845, 3025/3024

Mapping: [⟨3 0 26 56 8 23], ⟨0 2 -8 -20 1 -5]]

Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 951.5810

Optimal ET sequence: 24d, 87, 111, 198

Badness: 0.022923

Fog

The fog temperament (12 & 63) tempers out the marvel comma, 225/224.

Subgroup: 2.3.5.7

Comma list: 225/224, 156250/151263

Mapping: [⟨3 0 26 37], ⟨0 1 -4 -6]]

mapping generators: ~80/63, ~3

Optimal tunings:

  • CTE: ~80/63 = 1\3, ~3/2 = 704.3694
  • POTE: ~80/63 = 1\3, ~3/2 = 704.118

Optimal ET sequence: 12, 39c, 51, 63, 75

Badness: 0.086866

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 2420/2401

Mapping: [⟨3 0 26 37 58], ⟨0 1 -4 -6 -10]]

mapping generators: ~44/35, ~3

Optimal tunings:

  • CTE: ~44/35 = 1\3, ~3/2 = 704.6253
  • POTE: ~44/35 = 1\3, ~3/2 = 704.411

Optimal ET sequence: 12, 39ce, 51, 63, 138de

Badness: 0.051348

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