Luna family
The luna family tempers out the hemithirds comma aka luna comma, 274877906944/274658203125, which has a monzo of [38 -2 -15⟩.
Temperaments discussed elsewhere include:
- Subneutral (+2401/2400) → Breedsmic temperaments
Luna
Subgroup: 2.3.5
Comma list: 274877906944/274658203125
Mapping: [⟨1 4 2], ⟨0 -15 2]]
Optimal tuning (POTE): ~2 = 1\1, ~262144/234375 = 193.201
Optimal ET sequence: 25, 31, 56, 87, 118, 323, 441, 559, 1000, 1559
Badness: 0.020576
- Music
- "Moongazing" from Lessor Groove (2020) Bandcamp | YouTube – atmospheric-electro, luna[25] in 1000edo by Xotla
Hemithirds
- Main article: Hemithirds
- See also: Hemimean clan #Hemithirds
Subgroup: 2.3.5.7
Comma list: 1029/1024, 3136/3125
Mapping: [⟨1 4 2 2], ⟨0 -15 2 5]]
Wedgie: ⟨⟨15 -2 -5 -38 -50 -6]]
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.244
- 7-odd-limit: ~28/25 = [1/10 -1/20 0 1/20⟩
- [[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [11/5 -1/10 0 1/10⟩, [5/2 -1/4 0 1/4⟩]
- eigenmonzo (unchanged-interval) basis: 2.7/3
- 9-odd-limit: ~28/25 = [6/35 -2/35 0 1/35⟩
- [[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [82/35 -4/35 0 2/35⟩, [20/7 -2/7 0 1/7⟩]
- eigenmonzo (unchanged-interval) basis: 2.9/7
Optimal ET sequence: 25, 31, 87, 118
Badness: 0.044284
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 441/440, 3136/3125
Mapping: [⟨1 4 2 2 7], ⟨0 -15 2 5 -22]]
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227
Minimax tuning:
- 11-odd-limit: ~28/25 = [5/27 0 0 1/27 -1/27⟩
- [[1 0 0 0 0⟩, [11/9 0 0 -5/9 5/9⟩, [64/27 0 0 2/27 -2/27⟩, [79/27 0 0 5/27 -5/27⟩, [79/27 0 0 -22/27 22/27⟩]
- Eigenmonzos (unchanged-intervals): 2, 11/7
Optimal ET sequence: 25e, 31, 87, 118
Badness: 0.019003
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 385/384, 625/624
Mapping: [⟨1 4 2 2 7 0], ⟨0 -15 2 5 -22 23]]
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166
Optimal ET sequence: 31, 56, 87, 118, 205d
Badness: 0.021738
Lunatic
Subgroup: 2.3.5.7
Comma list: 4375/4374, 274877906944/274658203125
Mapping: [⟨1 4 2 21], ⟨0 -15 2 -113]]
Wedgie: ⟨⟨15 -2 113 -38 137 268]]
Optimal tuning (POTE): ~2 = 1\1, ~262144/234375 = 193.196
Optimal ET sequence: 87d, 118, 323, 441, 1205, 1646
Badness: 0.037966
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 12005/11979, 172032/171875
Mapping: [⟨1 4 2 21 14], ⟨0 -15 2 -113 -140]]
Optimal tuning (POTE): ~2 = 1\1, ~352/315 = 193.200
Optimal ET sequence: 118, 323e, 441, 559, 1000e
Badness: 0.058588
Hemiluna
Subgroup: 2.3.5.7
Comma list: 48828125/48771072, 67108864/66976875
Mapping: [⟨1 4 2 -5], ⟨0 -30 4 97]]
- mapping generators: ~2, ~200/189
Wedgie: ⟨⟨30 -4 -97 -76 -238 -214]]
Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 96.591
Optimal ET sequence: 87, 236, 323, 410, 733, 1056, 1789bd, 2845bdd
Badness: 0.179304
11-limit
Subgroup: 2.3.5.7.11
Comma list: 5632/5625, 14641/14580, 131072/130977
Mapping: [⟨1 4 2 -5 7], ⟨0 -30 4 97 -44]]
Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 96.587
Optimal ET sequence: 87, 236, 323, 410
Badness: 0.100077
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 4096/4095, 14641/14580
Mapping: [⟨1 4 2 -5 7 7], ⟨0 -30 4 97 -44 -41]]
Optimal tuning (POTE): ~2 = 1\1, ~143/135 = 96.586
Optimal ET sequence: 87, 236, 323, 410
Badness: 0.046151
Semiluna
Subgroup: 2.3.5.7
Comma list: 4802000/4782969, 95703125/95551488
Mapping: [⟨2 8 4 23], ⟨0 -15 2 -54]]
- mapping generators: ~10125/7168, ~2187/1960
Wedgie: ⟨⟨30 -4 108 -76 87 262]]
Optimal tuning (POTE): ~99/70 = 1\2, ~2187/1960 = 193.1725
Optimal ET sequence: 56d, 118, 292, 410
Badness: 0.192167
11-limit
Subgroup: 2.3.5.7.11
Comma list: 5632/5625, 9801/9800, 14641/14580
Mapping: [⟨2 8 4 23 14], ⟨0 -15 2 -54 -22]]
Optimal tuning (POTE): ~99/70 = 1\2, ~121/108 = 193.1732
Optimal ET sequence: 56d, 118, 292, 410
Badness: 0.067783