Misty family: Difference between revisions

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Normalization; POTE -> CTE; review optimal ET sequences
 
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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]].  
{{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.
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.
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: mapping generators: ~512/405, ~3
: mapping generators: ~512/405, ~3


[[Optimal tuning]] ([[CTE]]): ~512/405 = 1\3, ~3/2 = 703.2481
[[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 }}
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c }}
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: mapping generators: ~63/50, ~3
: mapping generators: ~63/50, ~3


{{Multival|legend=1| 3 -12 -30 -26 -56 -36 }}
Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448  
 
[[Optimal tuning]] ([[CTE]]): ~63/50 = 1\3, ~3/2 = 703.1448  


{{Optimal ET sequence|legend=1| 12, …, 87, 99 }}
{{Optimal ET sequence|legend=1| 12, …, 87, 99 }}
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Comma list: 256/255, 1701/1700, 2025/2023
Comma list: 256/255, 1701/1700, 2025/2023


Sval mapping: {{mapping| 3 0 ​26 ​56 -2 | 0 ​1 -4 -10​ 3​ }}
Sval mapping: {{mapping| 3 0 26 56 -2 | 0 1 -4 -10 3 }}


: sval mapping generators: ~34/27, ~3
: sval mapping generators: ~34/27, ~3
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Comma list: 256/255, 324/323, 400/399, 476/475
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: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}


: sval mapping generators: ~24/19, ~3
: sval mapping generators: ~24/19, ~3
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Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}
Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}


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


Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}
Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}
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Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}


Optimal tuning (CTE): ~44/35 = 1\3, ~3/2 = 703.3859
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}
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Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}
Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}


Optimal tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.3686
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efg }}
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Mapping: {{mapping| 3 0 26 56 77 92 -2 8 | 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 }}


Optimal tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.3626
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efgh }}
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Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}
Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}


Optimal tuning (CTE): ~44/35 = 1\3, ~3/2 = 703.3285
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 186e, 273e }}
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: mapping generators: ~55/49, ~3
: mapping generators: ~55/49, ~3


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 703.0864
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12, 174e, 186, 198, 408cde, 606bcde }}
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Mapping: {{mapping| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}
Mapping: {{mapping| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 703.1045
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 174ef, 186, 198 }}
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{{Mapping|legend=1| 3 0 26 37 | 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
: mapping generators: ~80/63, ~3


[[Optimal tuning]] ([[CTE]]): ~80/63 = 1\3, ~3/2 = 704.3694
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 }}
{{Optimal ET sequence|legend=1| 12, 39c, 51, 63, 75 }}
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: mapping generators: ~44/35, ~3
: mapping generators: ~44/35, ~3


Optimal tuning (CTE): ~44/35 = 1\3, ~3/2 = 704.6253
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12, 39ce, 51, 63, 138de }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Misty family| ]] <!-- main article -->
[[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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