Luna family

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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:

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 sequence25, 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

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

Minimax tuning:

[[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
[[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 sequence25, 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 sequence25e, 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 sequence31, 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 sequence87d, 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 sequence118, 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 sequence87, 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 sequence87, 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 sequence87, 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 sequence56d, 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 sequence56d, 118, 292, 410

Badness: 0.067783