Misty family: Difference between revisions

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Line 14: Line 14:
: mapping generators: ~512/405, ~3
: mapping generators: ~512/405, ~3


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[CTE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.2481{{c}}
* [[WE]]: ~512/405 = 399.8595{{c}}, ~3/2 = 702.8645{{c}}
* [[POTE]]: ~512/405 = 400.000{{c}}, ~3/2 = 703.111{{c}}
: [[error map]]: {{val| -0.421 +0.488 +0.261 }}
* [[CWE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.1489{{c}}
: error map: {{val| 0.000 +1.194 +1.091 }}
<!-- * [[CTE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.2481{{c}}
* [[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, 867bc }}


[[Badness]] (Smith): 0.106540
[[Badness]] (Sintel): 2.50


=== Overview to extensions ===
=== Overview to extensions ===
Line 36: Line 40:
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}


[[Optimal tuning]] ([[CTE]]): ~63/50 = 400.0000{{c}}, ~3/2 = 703.1448{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.8579{{c}}, ~3/2 = 702.7714{{c}}
: [[error map]]: {{val| -0.426 +0.390 +0.611 -0.236 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 703.0551{{c}}
: error map: {{val| 0.000 +1.100 +1.466 +0.623 }}
<!-- * [[CTE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 703.1448{{c}} -->


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


[[Badness]] (Smith): 0.036802
[[Badness]] (Sintel): 0.931


=== Undecimal misty ===
=== 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, 896/891, 3136/3125


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 tunings:
Optimal tunings:  
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3127{{c}}
* WE: ~44/35 = 399.8727{{c}}, ~3/2 = 702.9867{{c}}
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.210{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.2385{{c}}
<!-- * CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3127{{c}}
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.210{{c}} -->


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


Badness (Smith): 0.035932
Badness (Sintel): 1.19


==== 13-limit ====
==== 13-limit ====
Line 64: Line 75:
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 tunings:
Optimal tunings:  
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3859{{c}}
* WE: ~44/35 = 399.8888{{c}}, ~3/2 = 703.1080{{c}}
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.303{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3258{{c}}
<!-- * CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3859{{c}}
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.303{{c}} -->


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


Badness (Smith): 0.027820
Badness (Sintel): 1.15


===== 17-limit =====
===== 17-limit =====
Line 79: Line 92:
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 tunings:
Optimal tunings:  
* CTE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.3686{{c}}
* WE: ~34/27 = 399.8046{{c}}, ~3/2 = 702.8894{{c}}
* POTE: ~34/27 = 400.000{{c}}, ~3/2 = 703.233{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.2597{{c}}
<!-- * CTE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.3686{{c}}
* POTE: ~34/27 = 400.000{{c}}, ~3/2 = 703.233{{c}} -->


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


Badness (Smith): 0.023908
Badness (Sintel): 1.22


===== 19-limit =====
===== 19-limit =====
Line 94: Line 109:
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 tunings:
Optimal tunings:  
* CTE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.3626{{c}}
* WE: ~24/19 = 399.7530{{c}}, ~3/2 = 702.7601{{c}}
* POTE: ~34/27 = 400.000{{c}}, ~3/2 = 703.253{{c}}
* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.2216{{c}}
<!-- * CTE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.3626{{c}}
* POTE: ~24/19 = 400.000{{c}}, ~3/2 = 703.253{{c}} -->


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


Badness (Smith): 0.019878
Badness (Sintel): 1.21


==== Mystic ====
==== Mystic ====
Line 109: Line 126:
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 tunings:
Optimal tunings:  
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3285{{c}}
* WE: ~44/35 = 399.9543{{c}}, ~3/2 = 702.2310{{c}}
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.311{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3140{{c}}
<!-- * CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3285{{c}}
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.311{{c}} -->


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


Badness (Smith): 0.036872
Badness (Sintel): 1.52


=== Semimist ===
=== Semimist ===
Line 125: Line 144:
: mapping generators: ~55/49, ~3
: mapping generators: ~55/49, ~3


Optimal tunings:
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0864{{c}}
* WE: ~55/49 = 199.9260{{c}}, ~3/2 = 702.7162{{c}}
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.9764{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0062{{c}}
<!-- * CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0864{{c}}
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.9764{{c}} -->


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


Badness (Smith): 0.068509
Badness (Sintel): 2.26


==== 13-limit ====
==== 13-limit ====
Line 140: Line 161:
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 tunings:
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.1045{{c}}
* WE: ~55/49 = 199.9293{{c}}, ~3/2 = 702.7602{{c}}
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0087{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0343{{c}}
<!-- * CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.1045{{c}}
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0087{{c}} -->


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


Badness (Smith): 0.044073
Badness (Sintel): 1.82


=== Hemimist ===
=== Hemimist ===
Line 156: Line 179:
: mapping generators: ~63/50, ~400/231
: mapping generators: ~63/50, ~400/231


Optimal tuning (POTE): ~63/50 = 400.0000{{c}}, ~400/231 = 951.5721{{c}}
Optimal tunings:
* WE: ~63/50 = 399.8922{{c}}, ~400/231 = 951.2704{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~400/231 = 951.5363{{c}}
<!-- * POTE: ~63/50 = 400.0000{{c}}, ~400/231 = 951.5721{{c}} -->


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


Badness (Smith): 0.047439
Badness (Sintel): 1.57


==== 13-limit ====
==== 13-limit ====
Line 169: Line 195:
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 = 400.0000{{c}}, ~26/15 = 951.5810{{c}}
Optimal tunings:
* WE: ~63/50 = 399.8818{{c}}, ~26/15 = 951.2442{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.5358{{c}}
<!-- * POTE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.5810{{c}} -->


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


Badness (Smith): 0.022923
Badness (Sintel): 0.947


== Fog ==
== Fog ==
Line 183: Line 212:


{{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


Optimal tunings:
[[Optimal tuning]]s:  
* CTE: ~80/63 = 400.0000{{c}}, ~3/2 = 704.3694{{c}}
* [[WE]]: ~80/63 = 399.7918{{c}}, ~3/2 = 703.7511{{c}}
* POTE: ~80/63 = 400.000{{c}}, ~3/2 = 704.118{{c}}
: [[error map]]: {{val| -0.624 +1.172 -4.232 +4.712 }}
* [[CWE]]: ~80/63 = 400.0000{{c}}, ~3/2 = 704.1865{{c}}
: error map: {{val| 0.000 +2.232 -3.060 +6.055 }}
<!-- * CTE: ~80/63 = 400.0000{{c}}, ~3/2 = 704.3694{{c}}
* 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 }}


[[Badness]]: 0.086866
[[Badness]] (Sintel): 2.20


=== 11-limit ===
=== 11-limit ===
Line 200: Line 232:
Mapping: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}
Mapping: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 704.6253{{c}}
* WE: ~44/35 = 399.8078{{c}}, ~3/2 = 704.0728{{c}}
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 704.411{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 704.4697{{c}}
<!-- * CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 704.6253{{c}}
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 704.411{{c}} -->


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


Badness (Smith): 0.051348
Badness (Sintel): 1.70


== Subgroup extensions ==
== Subgroup extensions ==
Line 212: Line 246:
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


{{Mapping|legend=2| 3 0 26 56 -2 | 0 1 -4 -10 3 }}
Subgroup-val mapping: {{mapping| 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 = 400.0000{{c}}, ~3/2 = 703.0963{{c}}
Optimal tunings:
* WE: ~34/27 = 399.8023{{c}}, ~3/2 = 702.6087{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 702.9837{{c}}
<!-- * CTE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.0963{{c}} -->


{{Optimal ET sequence|legend=1| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}
{{Optimal ET sequence|legend=0| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}


[[Badness]] (Smith): 0.0186
Badness (Sintel): 0.688


==== 2.3.5.7.17.19 subgroup ====
==== 2.3.5.7.17.19 subgroup ====
Line 232: Line 269:
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 = 400.0000{{c}}, ~3/2 = 703.0778{{c}}
Optimal tunings:
* WE: ~24/19 = 399.7556{{c}}, ~3/2 = 702.4862{{c}}
* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 702.9418{{c}}
<!-- * CTE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.0778{{c}} -->


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


Badness (Smith): 0.0132
Badness (Sintel): 0.637


[[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 15:52, 4 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 represents the ptolemaic diminished fourth (512/405). Four perfect fourths (i.e. a minor sixth, ~128/81) minus a period give the interval class of 5. Misty has a ploidacot of triploid monocot, and 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:

  • WE: ~512/405 = 399.8595 ¢, ~3/2 = 702.8645 ¢
error map: ⟨-0.421 +0.488 +0.261]
  • CWE: ~512/405 = 400.0000 ¢, ~3/2 = 703.1489 ¢
error map: ⟨0.000 +1.194 +1.091]

Optimal ET sequence: 12, 51, 63, 75, 87, 99, 285, 384, 483c, 867bc

Badness (Sintel): 2.50

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 tunings:

  • WE: ~63/50 = 399.8579 ¢, ~3/2 = 702.7714 ¢
error map: ⟨-0.426 +0.390 +0.611 -0.236]
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 703.0551 ¢
error map: ⟨0.000 +1.100 +1.466 +0.623]

Optimal ET sequence: 12, …, 87, 99

Badness (Sintel): 0.931

Undecimal misty

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 3136/3125

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

Optimal tunings:

  • WE: ~44/35 = 399.8727 ¢, ~3/2 = 702.9867 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 703.2385 ¢

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

Badness (Sintel): 1.19

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:

  • WE: ~44/35 = 399.8888 ¢, ~3/2 = 703.1080 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 703.3258 ¢

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

Badness (Sintel): 1.15

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:

  • WE: ~34/27 = 399.8046 ¢, ~3/2 = 702.8894 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~3/2 = 703.2597 ¢

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

Badness (Sintel): 1.22

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:

  • WE: ~24/19 = 399.7530 ¢, ~3/2 = 702.7601 ¢
  • CWE: ~24/19 = 400.0000 ¢, ~3/2 = 703.2216 ¢

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

Badness (Sintel): 1.21

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:

  • WE: ~44/35 = 399.9543 ¢, ~3/2 = 702.2310 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 703.3140 ¢

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

Badness (Sintel): 1.52

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:

  • WE: ~55/49 = 199.9260 ¢, ~3/2 = 702.7162 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 703.0062 ¢

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

Badness (Sintel): 2.26

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:

  • WE: ~55/49 = 199.9293 ¢, ~3/2 = 702.7602 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 703.0343 ¢

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

Badness (Sintel): 1.82

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 tunings:

  • WE: ~63/50 = 399.8922 ¢, ~400/231 = 951.2704 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~400/231 = 951.5363 ¢

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

Badness (Sintel): 1.57

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 tunings:

  • WE: ~63/50 = 399.8818 ¢, ~26/15 = 951.2442 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~26/15 = 951.5358 ¢

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

Badness (Sintel): 0.947

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]]

Optimal tunings:

  • WE: ~80/63 = 399.7918 ¢, ~3/2 = 703.7511 ¢
error map: ⟨-0.624 +1.172 -4.232 +4.712]
  • CWE: ~80/63 = 400.0000 ¢, ~3/2 = 704.1865 ¢
error map: ⟨0.000 +2.232 -3.060 +6.055]

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

Badness (Sintel): 2.20

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:

  • WE: ~44/35 = 399.8078 ¢, ~3/2 = 704.0728 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 704.4697 ¢

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

Badness (Sintel): 1.70

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 tunings:

  • WE: ~34/27 = 399.8023 ¢, ~3/2 = 702.6087 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~3/2 = 702.9837 ¢

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

Badness (Sintel): 0.688

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 tunings:

  • WE: ~24/19 = 399.7556 ¢, ~3/2 = 702.4862 ¢
  • CWE: ~24/19 = 400.0000 ¢, ~3/2 = 702.9418 ¢

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

Badness (Sintel): 0.637

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