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 (see 37-11-comma, 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.

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 tunings:

• WE: ~50/49 = 32.408, ~3/2 = 702.859

error map: ⟨-0.906 -0.002 0.770 1.601]

• CWE: ~50/49 = 1\37, ~3/2 = 703.031

error map: ⟨0.000 1.076 2.875 4.147]

Optimal ET sequence: 37, 74, 111

Badness: 7.898

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]]

mapping generator: ~56/55, ~3/2

Optimal tunings:

• WE: ~56/55 = 32.408, ~3/2 = 702.635

error map: ⟨-0.683 -0.002 1.288 2.228 -2.329]

• CWE: ~56/55 = 1\37, ~3/2 = 702.819

error map: ⟨-0.000 0.864 2.875 4.147 0.033]

Optimal ET sequence: 37, 74, 111

Badness: 3.339

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 tunings:

• WE: ~65/64 = 32.414, ~3/2 = 702.646

error map: ⟨-0.693 -0.002 1.264 2.198 -2.365 0.149]

• CWE: ~65/64 = 1\37, ~3/2 = 702.869

error map: ⟨0.000 0.914 2.875 4.147 0.033 2.716]

Optimal ET sequence: 37, 74, 111

Badness: 2.014

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

Ludizemlic

Ludizemlic [lʲudi'zɛmlɪk] is notable because it is made by edo-joining two notable edos consistent in the 15-limit that are divisible by 37 - 296edo and 1665edo, and it is named after the Cyrillic numerals representation of the number 37 (Л = ludi = 30, З = zemlya = 7). Also notably, The 3rd harmonic is also is divided into 7 equal parts.

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⟩, ~[96 -17 3 -27⟩

• WE: ~[103 -18 3 -29⟩ = 32.433, ~[96 -17 3 -27⟩ = 271.711

error map: ⟨0.029, 0.021, -0.036, -0.074]

• CWE: ~[103 -18 3 -29⟩ = 1\37, ~[96 -17 3 -27⟩ = 271.706

error map: ⟨0.000 -0.014 -0.084 -0.159]

Optimal ET sequence: 296, 1073c, 1369, 1665, 1961, 3626

Badness: 33.301

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, ~11979/10240

Optimal tunings:

• WE: ~3993/3920 = 32.433 ¢, ~11979/10240 = 271.710 ¢

error map: ⟨0.018, 0.017, -0.037, -0.111, 0.095]

• CWE: ~3993/3920 = 1\37, ~11979/10240 = 271.707 ¢

error map: ⟨0.000, -0.006, -0.069, -0.163, 0.033]

Optimal ET sequence: 296, 1073c, 1369, 1665, 3626

Badness: 6.891

13-limit

This is the best and most usable tuning for this temperament. 14 periods map to 13/10 (a property inherited from 37edo), thus equating a stack of three 11/8 with one 13/10 and making ludizemlic 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, ~117/100

Optimal tunings:

• WE: ~1248/1225 = 32.433 ¢, ~117/100 = 271.713 ¢

error map: ⟨0.022 0.036 -0.005 -0.105 0.109 -0.135]

• CWE: ~1248/1225 = 1\37, ~117/100 = 271.709 ¢

error map: ⟨-0.000 0.008 -0.043 -0.169 0.033 -0.203]

Optimal ET sequence: 296, 1073c, 1369, 1665

Badness: 3.574