Misty family

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:

• 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