37th-octave temperaments
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37edo has an extremely precise mapping for the 11th harmonic, and it is a strong 2.5.7.13 tuning besides that, therefore various 37th-octave temperaments occur naturally between any two numbers whose greatest common divisor is 37.
37-11-commatic (rank-1)
- Main article: 37-11-comma
Subgroup: 2.11
Comma list: [128 -37⟩
- mapping generators: ~35184372088832/34522712143931 = 1\37
Supporting ETs: 37N, N = 1 to 485
Rubidium
The name of rubidium temperament comes from the 37th element. Developed by Xenllium, rubidium preserves the mappings for the 2.5.7.11.13 subgroup in 37edo and leaves the 3rd harmonic as a generator.
Subgroup: 2.3.5.7
Comma list: 3136/3125, 4194304/4117715
Mapping: [⟨37 0 86 104], ⟨0 1 0 0]]
Mapping generators: ~50/49, ~3
Optimal tuning (POTE): ~3/2 = 703.3903
Optimal ET sequence: 37, 74, 111
Badness: 0.312105
11-limit
Subgroup: 2.3.5.7.11
Comma list: 176/175, 1375/1372, 65536/65219
Mapping: [⟨37 0 86 104 128], ⟨0 1 0 0 0]]
Optimal tuning (POTE): ~3/2 = 703.0355
Optimal ET sequence: 37, 74, 111
Badness: 0.101001
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 640/637, 847/845, 1375/1372
Mapping: [⟨37 0 86 104 128 137], ⟨0 1 0 0 0 0]]
Optimal tuning (POTE): ~3/2 = 703.0520
Optimal ET sequence: 37, 74, 111
Badness: 0.048732
Triacontaheptoid
Subgroup: 2.3.5.7
Comma list: 244140625/242121642, 283115520/282475249
Mapping: [⟨37 2 67 85], ⟨0 3 1 1]]
Mapping generator: ~50/49, ~24000/16807
Optimal tuning (CTE): ~24000/16807 = 612.4003
Optimal ET sequence: 37, 222b, 259b, 296, 629
Badness: 0.784746
11-limit
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 4000/3993, 226492416/226474325
Mapping: [⟨37 2 67 85 128], ⟨0 3 1 1 0]]
Optimal tuning (CTE): ~768/359 = 612.4003
Optimal ET sequence: 37, 259b, 296, 629
Badness: 0.167327
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1375/1372, 4000/3993, 15379/15360
Mapping: [⟨37 2 67 85 128 118], ⟨0 3 1 1 0 1]]
Optimal tuning (CTE): ~462/325 = 612.4206
Optimal ET sequence: 37, 259b, 296, 629f
Badness: 0.076183
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 625/624, 715/714, 1225/1224, 4000/3993, 11271/11264
Mapping: [⟨37 2 67 85 128 118 189], ⟨0 3 1 1 0 1 -2]]
Optimal tuning (CTE): ~121/85 = 612.4187
Optimal ET sequence: 37, 259b, 296, 629f
Badness: 0.052475
Dzelic
Dzelic ['d͡zɛlɪk] is named after the Slavic letter dzelo, which represents the number 7, as it takes 7 generator grave minor thirds to reach the third harmonic.
Subgroup: 2.3.5.7
Comma list: 4375/4374, [-223 47 -11 62⟩
Mapping: [⟨37 0 -23 129], ⟨0 7 13 -3]]
Mapping generators: ~[103 -18 3 -29⟩ = 1\37, ~[96 -17 3 -27⟩ = 271.709
Optimal tuning (CTE): ~[96 -17 3 -27⟩ = 271.709
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 13423439799/13421772800, 113468578083/113379904000
Mapping: [⟨37 0 -23 129 128], ⟨0 7 13 -3 0]]
Mapping generators: ~3993/3920 = 1\37, ~11979/10240 = 271.709
Optimal tuning (CTE): ~11979/10240 = 271.709
13-limit
14 periods map to 13/10, thus equating a stack of three 11/8 with one 13/10 and making dzelic a jacobin temperament.
Subgroup: 2.3.5.7.11.13
Comma list: 4375/4374, 6656/6655, 405769/405504, 34034175/34027136
Mapping: [⟨37 0 -23 129 128 28], ⟨0 7 13 -3 0 13]]
Mapping generators: ~1248/1225 = 1\37, ~117/100 = 271.712
Optimal tuning (CTE): ~11979/10240 = 271.712
Optimal ET sequence: 296, 1369, 1665, ...