37th-octave temperaments

Fractional-octave temperaments

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← 36th-octave • 37th-octave • 38th-octave →

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⟩

Subgroup-val mapping: [⟨37 28]]

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, ...