Misty family: Difference between revisions

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Move back subgroup extensions since they're still septimal misty extensions
- CTE & POTE tunings
Line 19: Line 19:
* [[CWE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.1489{{c}}
* [[CWE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.1489{{c}}
: error map: {{val| 0.000 +1.194 +1.091 }}
: 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, 867bc }}
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c, 867bc }}
Line 46: Line 44:
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 703.0551{{c}}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 703.0551{{c}}
: error map: {{val| 0.000 +1.100 +1.466 +0.623 }}
: 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 }}
Line 65: Line 62:
* WE: ~34/27 = 399.8023{{c}}, ~3/2 = 702.6087{{c}}
* WE: ~34/27 = 399.8023{{c}}, ~3/2 = 702.6087{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 702.9837{{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=0| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}
{{Optimal ET sequence|legend=0| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}
Line 81: Line 77:
* WE: ~24/19 = 399.7556{{c}}, ~3/2 = 702.4862{{c}}
* WE: ~24/19 = 399.7556{{c}}, ~3/2 = 702.4862{{c}}
* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 702.9418{{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 }}
Line 97: Line 92:
* WE: ~44/35 = 399.8727{{c}}, ~3/2 = 702.9867{{c}}
* WE: ~44/35 = 399.8727{{c}}, ~3/2 = 702.9867{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.2385{{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 }}
Line 114: Line 107:
* WE: ~44/35 = 399.8888{{c}}, ~3/2 = 703.1080{{c}}
* WE: ~44/35 = 399.8888{{c}}, ~3/2 = 703.1080{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3258{{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 }}
Line 131: Line 122:
* WE: ~34/27 = 399.8046{{c}}, ~3/2 = 702.8894{{c}}
* WE: ~34/27 = 399.8046{{c}}, ~3/2 = 702.8894{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.2597{{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 }}
Line 148: Line 137:
* WE: ~24/19 = 399.7530{{c}}, ~3/2 = 702.7601{{c}}
* WE: ~24/19 = 399.7530{{c}}, ~3/2 = 702.7601{{c}}
* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.2216{{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 }}
Line 165: Line 152:
* WE: ~44/35 = 399.9543{{c}}, ~3/2 = 702.2310{{c}}
* WE: ~44/35 = 399.9543{{c}}, ~3/2 = 702.2310{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3140{{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 }}
Line 183: Line 168:
* WE: ~55/49 = 199.9260{{c}}, ~3/2 = 702.7162{{c}}
* WE: ~55/49 = 199.9260{{c}}, ~3/2 = 702.7162{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0062{{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 }}
Line 200: Line 183:
* WE: ~55/49 = 199.9293{{c}}, ~3/2 = 702.7602{{c}}
* WE: ~55/49 = 199.9293{{c}}, ~3/2 = 702.7602{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0343{{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 }}
Line 218: Line 199:
* WE: ~63/50 = 399.8922{{c}}, ~400/231 = 951.2704{{c}}
* WE: ~63/50 = 399.8922{{c}}, ~400/231 = 951.2704{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~400/231 = 951.5363{{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 }}
Line 234: Line 214:
* WE: ~63/50 = 399.8818{{c}}, ~26/15 = 951.2442{{c}}
* WE: ~63/50 = 399.8818{{c}}, ~26/15 = 951.2442{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.5358{{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 }}
Line 254: Line 233:
* [[CWE]]: ~80/63 = 400.0000{{c}}, ~3/2 = 704.1865{{c}}
* [[CWE]]: ~80/63 = 400.0000{{c}}, ~3/2 = 704.1865{{c}}
: error map: {{val| 0.000 +2.232 -3.060 +6.055 }}
: 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 }}
Line 271: Line 248:
* WE: ~44/35 = 399.8078{{c}}, ~3/2 = 704.0728{{c}}
* WE: ~44/35 = 399.8078{{c}}, ~3/2 = 704.0728{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 704.4697{{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 }}

Revision as of 15:55, 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

7-limit

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

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

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

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

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