Misty family

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 tuning (CTE): ~512/405 = 1\3, ~3/2 = 703.2481

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

Wedgie: ⟨⟨ 3 -12 -30 -26 -56 -36 ]]

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 tuning (CTE): ~44/35 = 1\3, ~3/2 = 703.3127

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 tuning (CTE): ~44/35 = 1\3, ~3/2 = 703.3859

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 tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.3686

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 tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.3626

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 tuning (CTE): ~44/35 = 1\3, ~3/2 = 703.3285

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 tuning (CTE): ~55/49 = 1\6, ~3/2 = 703.0864

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 tuning (CTE): ~55/49 = 1\6, ~3/2 = 703.1045

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]]

Wedgie: ⟨⟨ 3 -12 -18 -26 -37 -8 ]]

mapping generators: ~80/63, ~3

Optimal tuning (CTE): ~80/63 = 1\3, ~3/2 = 704.3694

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 tuning (CTE): ~44/35 = 1\3, ~3/2 = 704.6253

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

Badness: 0.051348