37th-octave temperaments
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)
Subgroup: 2.11
Comma list: {{monzo|128 -37
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, ...