Misty family: Difference between revisions

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{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}
{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}
: mapping generators: ~512/405, ~3
: mapping generators: ~512/405, ~3


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~512/405 = 1\3, ~3/2 = 703.2481
* [[CTE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.2481{{c}}
* [[POTE]]: ~512/405 = 1\3, ~3/2 = 703.111
* [[POTE]]: ~512/405 = 400.000{{c}}, ~3/2 = 703.111{{c}}


{{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 }}


[[Badness]]: 0.106540
[[Badness]] (Smith): 0.106540


=== Overview to extensions ===
=== Overview to extensions ===
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{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}


: mapping generators: ~63/50, ~3
[[Optimal tuning]] ([[CTE]]): ~63/50 = 400.0000{{c}}, ~3/2 = 703.1448{{c}}
 
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 }}


[[Badness]]: 0.036802
[[Badness]] (Smith): 0.036802


=== Undecimal misty ===
=== Undecimal misty ===
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Optimal tunings:
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3127
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3127{{c}}
* POTE: ~44/35 = 1\3, ~3/2 = 703.210
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.210{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}
{{Optimal ET sequence|legend=0| 12, …, 87, 99e, 186e }}


Badness: 0.035932
Badness (Smith): 0.035932


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3859
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3859{{c}}
* POTE: ~44/35 = 1\3, ~3/2 = 703.303
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.303{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}
{{Optimal ET sequence|legend=0| 12f, …, 87, 186ef }}


Badness: 0.027820
Badness (Smith): 0.027820


===== 17-limit =====
===== 17-limit =====
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Optimal tunings:
Optimal tunings:
* CTE: ~34/27 = 1\3, ~3/2 = 703.3686
* CTE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.3686{{c}}
* POTE: ~34/27 = 1\3, ~3/2 = 703.233
* POTE: ~34/27 = 400.000{{c}}, ~3/2 = 703.233{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efg }}
{{Optimal ET sequence|legend=0| 12f, 87, 99ef, 186efg }}


Badness: 0.023908
Badness (Smith): 0.023908


===== 19-limit =====
===== 19-limit =====
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Optimal tunings:
Optimal tunings:
* CTE: ~24/19 = 1\3, ~3/2 = 703.3626
* CTE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.3626{{c}}
* POTE: ~34/27 = 1\3, ~3/2 = 703.253
* POTE: ~34/27 = 400.000{{c}}, ~3/2 = 703.253{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efgh }}
{{Optimal ET sequence|legend=0| 12f, 87, 99ef, 186efgh }}


Badness: 0.019878
Badness (Smith): 0.019878


==== Mystic ====
==== Mystic ====
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Optimal tunings:
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3285
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3285{{c}}
* POTE: ~44/35 = 1\3, ~3/2 = 703.311
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.311{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, 87, 186e, 273e }}
{{Optimal ET sequence|legend=0| 12, 87, 186e, 273e }}


Badness: 0.036872
Badness (Smith): 0.036872


=== Semimist ===
=== Semimist ===
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Mapping: {{mapping| 6 0 52 112 173 | 0 1 -4 -10 -16 }}
Mapping: {{mapping| 6 0 52 112 173 | 0 1 -4 -10 -16 }}
: mapping generators: ~55/49, ~3
: mapping generators: ~55/49, ~3


Optimal tunings:
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 703.0864
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0864{{c}}
* POTE: ~55/49 = 1\6, ~3/2 = 702.9764
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.9764{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, 174e, 186, 198, 408cde, 606bcde }}
{{Optimal ET sequence|legend=0| 12, 174e, 186, 198, 408cde, 606bcde }}


Badness: 0.068509
Badness (Smith): 0.068509


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 703.1045
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.1045{{c}}
* POTE: ~55/49 = 1\6, ~3/2 = 703.0087
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0087{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, 174ef, 186, 198 }}
{{Optimal ET sequence|legend=0| 12f, 174ef, 186, 198 }}


Badness: 0.044073
Badness (Smith): 0.044073


=== Hemimist ===
=== Hemimist ===
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Mapping: {{mapping| 3 0 26 56 8 | 0 2 -8 -20 1 }}
Mapping: {{mapping| 3 0 26 56 8 | 0 2 -8 -20 1 }}
: mapping generators: ~63/50, ~400/231
: mapping generators: ~63/50, ~400/231


Optimal tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.5721
Optimal tuning (POTE): ~63/50 = 400.0000{{c}}, ~400/231 = 951.5721{{c}}


Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}
{{Optimal ET sequence|legend=0| 24d, 87, 111, 198 }}


Badness: 0.047439
Badness (Smith): 0.047439


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 3 0 26 56 8 23 | 0 2 -8 -20 1 -5 }}
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 tuning (POTE): ~63/50 = 400.0000{{c}}, ~26/15 = 951.5810{{c}}


Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}
{{Optimal ET sequence|legend=0| 24d, 87, 111, 198 }}


Badness: 0.022923
Badness (Smith): 0.022923


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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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 }}
: mapping generators: ~80/63, ~3
: mapping generators: ~80/63, ~3


Optimal tunings:
Optimal tunings:
* CTE: ~80/63 = 1\3, ~3/2 = 704.3694
* CTE: ~80/63 = 400.0000{{c}}, ~3/2 = 704.3694{{c}}
* POTE: ~80/63 = 1\3, ~3/2 = 704.118
* POTE: ~80/63 = 400.000{{c}}, ~3/2 = 704.118{{c}}


{{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: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}
Mapping: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}
: mapping generators: ~44/35, ~3


Optimal tunings:
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 704.6253
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 704.6253{{c}}
* POTE: ~44/35 = 1\3, ~3/2 = 704.411
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 704.411{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, 39ce, 51, 63, 138de }}
{{Optimal ET sequence|legend=0| 12, 39ce, 51, 63, 138de }}


Badness: 0.051348
Badness (Smith): 0.051348


== Subgroup extensions ==
== Subgroup extensions ==
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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]])}.  
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
[[Subgroup]]: 2.3.5.7.17


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


Subgroup-val mapping: {{mapping| 3 0 26 56 -2 | 0 1 -4 -10 3 }}
{{Mapping|legend=2| 3 0 26 56 -2 | 0 1 -4 -10 3 }}
: mapping generators: ~34/27, ~3
: mapping generators: ~34/27, ~3


Optimal tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.0963
[[Optimal tuning]] ([[CTE]]): ~34/27 = 400.0000{{c}}, ~3/2 = 703.0963{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}
{{Optimal ET sequence|legend=1| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}


Badness: 0.0186
[[Badness]] (Smith): 0.0186


==== 2.3.5.7.17.19 subgroup ====
==== 2.3.5.7.17.19 subgroup ====
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Subgroup-val mapping: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}
Subgroup-val mapping: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}


Optimal tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.0778  
Optimal tuning (CTE): ~24/19 = 400.0000{{c}}, ~3/2 = 703.0778{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, 87, 99, 210gh, 309cdghh }}
{{Optimal ET sequence|legend=0| 12, 87, 99, 210gh, 309cdghh }}


Badness: 0.0132
Badness (Smith): 0.0132


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Misty family| ]] <!-- main article -->
[[Category:Misty family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 13:40, 3 January 2026

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 (monzo: [26 -12 -3⟩, ratio: 67108864/66430125).

Misty

Main article: Misty

Misty features a third-octave period, which is represented by 512/405 and acts as the interval separating ~128/81 from ~5/4. It also equates the Pythagorean comma with a stack of three schismas, making it a member of the schismic–Pythagorean equivalence continuum with n = 3.

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 = 400.0000 ¢, ~3/2 = 703.2481 ¢
  • POTE: ~512/405 = 400.000 ¢, ~3/2 = 703.111 ¢

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

Badness (Smith): 0.106540

Overview to extensions

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.

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]]

Optimal tuning (CTE): ~63/50 = 400.0000 ¢, ~3/2 = 703.1448 ¢

Optimal ET sequence: 12, …, 87, 99

Badness (Smith): 0.036802

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 = 400.0000 ¢, ~3/2 = 703.3127 ¢
  • POTE: ~44/35 = 400.000 ¢, ~3/2 = 703.210 ¢

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

Badness (Smith): 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 = 400.0000 ¢, ~3/2 = 703.3859 ¢
  • POTE: ~44/35 = 400.000 ¢, ~3/2 = 703.303 ¢

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

Badness (Smith): 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 = 400.0000 ¢, ~3/2 = 703.3686 ¢
  • POTE: ~34/27 = 400.000 ¢, ~3/2 = 703.233 ¢

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

Badness (Smith): 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 = 400.0000 ¢, ~3/2 = 703.3626 ¢
  • POTE: ~34/27 = 400.000 ¢, ~3/2 = 703.253 ¢

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

Badness (Smith): 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 = 400.0000 ¢, ~3/2 = 703.3285 ¢
  • POTE: ~44/35 = 400.000 ¢, ~3/2 = 703.311 ¢

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

Badness (Smith): 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 = 200.0000 ¢, ~3/2 = 703.0864 ¢
  • POTE: ~55/49 = 200.0000 ¢, ~3/2 = 702.9764 ¢

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

Badness (Smith): 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 = 200.0000 ¢, ~3/2 = 703.1045 ¢
  • POTE: ~55/49 = 200.0000 ¢, ~3/2 = 703.0087 ¢

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

Badness (Smith): 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 = 400.0000 ¢, ~400/231 = 951.5721 ¢

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

Badness (Smith): 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 = 400.0000 ¢, ~26/15 = 951.5810 ¢

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

Badness (Smith): 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 = 400.0000 ¢, ~3/2 = 704.3694 ¢
  • POTE: ~80/63 = 400.000 ¢, ~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]]

Optimal tunings:

  • CTE: ~44/35 = 400.0000 ¢, ~3/2 = 704.6253 ¢
  • POTE: ~44/35 = 400.000 ¢, ~3/2 = 704.411 ¢

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

Badness (Smith): 0.051348

Subgroup extensions

Misty (2.3.5.7.17)

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

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

mapping generators: ~34/27, ~3

Optimal tuning (CTE): ~34/27 = 400.0000 ¢, ~3/2 = 703.0963 ¢

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

Badness (Smith): 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

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

Optimal tuning (CTE): ~24/19 = 400.0000 ¢, ~3/2 = 703.0778 ¢

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

Badness (Smith): 0.0132

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