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The '''misty family''' tempers out the [[misty comma]], 67108864/66430125 = {{monzo| 26 -12 -3 }}.
{{Technical data page}}
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]].
 
The second comma of the [[Normal lists #Normal interval list|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 ==
== Misty ==
Subgroup: 2.3.5
{{Main| Misty }}
 
[[Subgroup]]: 2.3.5


[[Comma list]]: 67108864/66430125
[[Comma list]]: 67108864/66430125


[[Mapping]]: [{{val| 3 0 26 }}, {{val| 0 1 -4 }}]
{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}


[[POTE generator]]: ~3/2 = 703.111
: mapping generators: ~512/405, ~3


{{Val list|legend=1| 12, 51, 63, 75, 87, 99, 285, 384 }}
[[Optimal tuning]]s:
* [[CTE]]: ~512/405 = 1\3, ~3/2 = 703.2481
* [[POTE]]: ~512/405 = 1\3, ~3/2 = 703.111
 
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c }}


[[Badness]]: 0.106540
[[Badness]]: 0.106540
=== Extensions ===
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.
* Adding 3136/3125 (hemimean) leads septimal misty.
* Adding 225/224 (marvel) leads fog.


== Septimal misty ==
== Septimal misty ==
{{see also|Hemimean clan #Misty}}
{{Main| Misty }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 5120/5103
[[Comma list]]: 3136/3125, 5120/5103


[[Mapping]]: [{{val| 3 0 26 56 }}, {{val| 0 1 -4 -10 }}]
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}


{{Multival|legend=1| 3 -12 -30 -26 -56 -36 }}
: mapping generators: ~63/50, ~3


[[POTE generator]]: ~3/2 = 703.021
Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448


{{Val list|legend=1| 12, 87, 99 }}
{{Optimal ET sequence|legend=1| 12, …, 87, 99 }}


[[Badness]]: 0.036802
[[Badness]]: 0.036802


=== 11-limit ===
=== 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]])}.
 
Subgroup: 2.3.5.7.17
 
Comma list: 256/255, 1701/1700, 2025/2023
 
Sval mapping: {{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: {{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: {{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: {{Optimal ET sequence| 12, 87, 99, 210gh, 309cdghh }}
 
Badness: 0.0132
 
=== Undecimal misty ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 3 0 26 56 77 }}, {{val| 0 1 -4 -10 -14 }}]
Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}


POTE generator: ~3/2 = 703.210
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3127
* POTE: ~44/35 = 1\3, ~3/2 = 703.210


Vals: {{Val list| 12, 87, 99e, 186e }}
Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}


Badness: 0.035932
Badness: 0.035932
Line 54: Line 94:
Comma list: 196/195, 352/351, 364/363, 625/624
Comma list: 196/195, 352/351, 364/363, 625/624


Mapping: [{{val| 3 0 26 56 77 92 }}, {{val| 0 1 -4 -10 -14 -17 }}]
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}


POTE generator: ~3/2 = 703.303
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3859
* POTE: ~44/35 = 1\3, ~3/2 = 703.303


Vals: {{Val list| 12f, 87, 99ef, 186ef, 273ef }}
Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}


Badness: 0.027820
Badness: 0.027820
Line 67: Line 109:
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594


Mapping: [{{val| 3 0 26 56 77 92 -2 }}, {{val| 0 1 -4 -10 -14 -17 3 }}]
Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}


POTE generator: ~3/2 = 703.233
Optimal tunings:
* CTE: ~34/27 = 1\3, ~3/2 = 703.3686
* POTE: ~34/27 = 1\3, ~3/2 = 703.233


Vals: {{Val list| 12f, 87, 99ef, 186efg }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efg }}


Badness: 0.023908
Badness: 0.023908
Line 80: Line 124:
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594


Mapping: [{{val| 3 0 26 56 77 92 -2 8 }}, {{val| 0 1 -4 -10 -14 -17 3 1 }}]
Mapping: {{mapping| 3 0 26 56 77 92 -2 8 | 0 1 -4 -10 -14 -17 3 1 }}


POTE generator: ~3/2 = 703.253
Optimal tunings:
* CTE: ~24/19 = 1\3, ~3/2 = 703.3626
* POTE: ~34/27 = 1\3, ~3/2 = 703.253


Vals: {{Val list| 12f, 87, 99ef, 186efgh }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efgh }}


Badness: 0.019878
Badness: 0.019878
Line 93: Line 139:
Comma list: 325/324, 441/440, 896/891, 1001/1000
Comma list: 325/324, 441/440, 896/891, 1001/1000


Mapping: [{{val| 3 0 26 56 77 -46 }}, {{val| 0 1 -4 -10 -14 12 }}]
Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}


POTE generator: ~3/2 = 703.311
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3285
* POTE: ~44/35 = 1\3, ~3/2 = 703.311


Vals: {{Val list| 12, 87, 186e, 273e }}
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 186e, 273e }}


Badness: 0.036872
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 ===
=== Hemimist ===
Line 106: Line 186:
Comma list: 3025/3024, 3136/3125, 5120/5103
Comma list: 3025/3024, 3136/3125, 5120/5103


Mapping: [{{val| 3 0 26 56 8 }}, {{val| 0 2 -8 -20 1 }}]
Mapping: {{mapping| 3 0 26 56 8 | 0 2 -8 -20 1 }}


POTE generator: ~11/8 = 551.527
: mapping generators: ~63/50, ~400/231


Vals: {{Val list| 24d, 87, 111, 198 }}
Optimal tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.5721
 
Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}


Badness: 0.047439
Badness: 0.047439
Line 119: Line 201:
Comma list: 352/351, 676/675, 847/845, 3025/3024
Comma list: 352/351, 676/675, 847/845, 3025/3024


Mapping: [{{val| 3 0 26 56 8 23 }}, {{val| 0 2 -8 -20 1 -5 }}]
Mapping: {{mapping| 3 0 26 56 8 23 | 0 2 -8 -20 1 -5 }}


POTE generator: ~11/8 = 551.525
Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 951.5810


Vals: {{Val list| 24d, 87, 111, 198 }}
Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}


Badness: 0.022923
Badness: 0.022923


== Fog ==
== Fog ==
The ''fog'' temperament (12&63) tempers out the [[225/224|marvel comma]] 225/224.
The fog temperament (12 & 63) tempers out the marvel comma, [[225/224]].


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 156250/151263
[[Comma list]]: 225/224, 156250/151263


[[Mapping]]: [{{val|3 5 6 7}}, {{val|0 -1 4 6}}]
{{Mapping|legend=1| 3 0 26 37 | 0 1 -4 -6 }}


{{Multival|legend=1| 3 -12 -18 -26 -37 -8 }}
: mapping generators: ~80/63, ~3


[[POTE generator]]: ~3/2 = 704.118
Optimal tunings:
* CTE: ~80/63 = 1\3, ~3/2 = 704.3694
* POTE: ~80/63 = 1\3, ~3/2 = 704.118


{{Val list|legend=1| 12, 51, 63, 75 }}
{{Optimal ET sequence|legend=1| 12, 39c, 51, 63, 75 }}


[[Badness]]: 0.086866
[[Badness]]: 0.086866
Line 149: Line 233:
Comma list: 100/99, 225/224, 2420/2401
Comma list: 100/99, 225/224, 2420/2401


Mapping: [{{val|3 5 6 7 8}}, {{val|0 -1 4 6 10}}]
Mapping: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}
 
: mapping generators: ~44/35, ~3


POTE generator: ~3/2 = 704.411
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 704.6253
* POTE: ~44/35 = 1\3, ~3/2 = 704.411


Vals: {{Val list| 12, 51, 63, 138de }}
Optimal ET sequence: {{Optimal ET sequence| 12, 39ce, 51, 63, 138de }}


Badness: 0.051348
Badness: 0.051348


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Misty]]
[[Category:Misty family| ]] <!-- main article -->
[[Category:Rank 2]]
[[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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