37th-octave temperaments: Difference between revisions

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→Ludizemlic: switch to WE and CWE, explain why this exists, +ET sequences
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==Ludizemlic==
==Ludizemlic==
''Ludizemlic'' [lʲudi'zɛmlɪk] is named after the {{W|Cyrillic numeral}}s representation of the number 37 (Л = ludi = 30, З = zemlya = 7). Also notably, The 3rd harmonic is also is divided into 7 equal parts.
''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 {{W|Cyrillic numeral}}s 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
Subgroup: 2.3.5.7
Line 119: Line 119:


Mapping: [{{val|37 0 -23 129}}, {{val|0 7 13 -3}}]
Mapping: [{{val|37 0 -23 129}}, {{val|0 7 13 -3}}]
: mapping generators: ~{{monzo|103 -18 3 -29}}, ~{{monzo|96 -17 3 -27}}


Mapping generators: ~{{monzo|103 -18 3 -29}} = 1\37, ~{{monzo|96 -17 3 -27}} = 271.709
* [[WE]]: ~{{monzo|103 -18 3 -29}} = 32.433, ~{{monzo|96 -17 3 -27}} = 271.711
: [[error map]]: {{val| 0.029, 0.021, -0.036, -0.074 }}
* [[CWE]]: ~{{monzo|103 -18 3 -29}} = 1\37, ~{{monzo|96 -17 3 -27}} = 271.706
: [[error map]]: {{val| 0.000 -0.014 -0.084 -0.159 }}


Optimal tuning (CTE): ~{{monzo|96 -17 3 -27}} = 271.709
{{Optimal ET sequence|legend=1|296, 1073c, 1369, 1665, 1961, 3626}}
 
Badness: 33.301


===11-limit===
===11-limit===
Line 129: Line 135:
Comma list: 4375/4374, 13423439799/13421772800, 113468578083/113379904000
Comma list: 4375/4374, 13423439799/13421772800, 113468578083/113379904000


Mapping: [{{val|37 0 -23 129 128}}, {{val|0 7 13 -3 0}}]
{{Mapping|legend=1| 37 0 -23 129 128 | 0 7 13 -3 0 }}
: mapping generators: ~3993/3920, ~11979/10240
 
[[Optimal tuning]]s:
* [[WE]]: ~3993/3920 = 32.433{{c}}, ~11979/10240 = 271.710{{c}}
: [[error map]]: {{val| 0.018, 0.017, -0.037, -0.111, 0.095 }}
* [[CWE]]: ~3993/3920 = 1\37, ~11979/10240 = 271.707{{c}}
: error map: {{val| 0.000, -0.006, -0.069, -0.163, 0.033 }}


Mapping generators: ~3993/3920 = 1\37, ~11979/10240 = 271.709
{{Optimal ET sequence|legend=1|296, 1073c, 1369, 1665, 3626}}


Optimal tuning (CTE): ~11979/10240 = 271.709
Badness: 6.891


===13-limit===
===13-limit===
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 [[6656/6655|jacobin]] temperament.
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 [[6656/6655|jacobin]] temperament.


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 142: Line 155:
Comma list: 4375/4374, 6656/6655, 405769/405504, 34034175/34027136
Comma list: 4375/4374, 6656/6655, 405769/405504, 34034175/34027136


Mapping: [{{val|37 0 -23 129 128 28}}, {{val|0 7 13 -3 0 13}}]
{{Mapping|legend=1| 37 0 -23 129 128 28 | 0 7 13 -3 0 13}}
: mapping generators: ~1248/1225, ~117/100


Mapping generators: ~1248/1225 = 1\37, ~117/100 = 271.712
[[Optimal tuning]]s:
* [[WE]]: ~1248/1225 = 32.433{{c}}, ~117/100 = 271.713{{c}}
: [[error map]]: {{val| 0.022 0.036 -0.005 -0.105 0.109 -0.135 }}
* [[CWE]]: ~1248/1225 = 1\37, ~117/100 = 271.709{{c}}
: error map: {{val| -0.000 0.008 -0.043 -0.169 0.033 -0.203 }}


Optimal tuning (CTE): ~11979/10240 = 271.712
{{Optimal ET sequence|legend=1|296, 1073c, 1369, 1665}}


{{Optimal ET sequence|legend=1|296, 1369, 1665}}, ...
Badness: 3.574


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}

Revision as of 13:51, 13 October 2026

← 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

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

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