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37th-octave temperaments: Difference between revisions

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{{Infobox fractional-octave|37}}
{{Infobox fractional-octave|37}}
[[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.
[[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.  
== 37-11-commatic (rank-1) ==
{{Main|37-11-comma}}
 
Subgroup: 2.11
 
Comma list: {{monzo|128 -37}}
 
{{mapping|legend=2| 37 28 }}
 
: mapping generators: ~35184372088832/34522712143931 = 1\37
 
[[Support]]ing [[ET]]s: 37N, N = 1 to 485


== Rubidium ==
== Rubidium ==
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[[Mapping]]: [{{val| 37 0 86 104 }}, {{val| 0 1 0 0 }}]
[[Mapping]]: [{{val| 37 0 86 104 }}, {{val| 0 1 0 0 }}]
: mapping generators: ~50/49, ~3


Mapping generators: ~50/49, ~3
Optimal tunings:
 
* [[WE]]: ~50/49 = 32.408, ~3/2 = 702.859
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.3903
: [[error map]]: {{val| -0.906 -0.002 0.770 1.601 }}
* [[CWE]]: ~50/49 = 1\37, ~3/2 = 703.031
: [[error map]]: {{val| 0.000 1.076 2.875 4.147 }}


{{Optimal ET sequence|legend=1| 37, 74, 111 }}
{{Optimal ET sequence|legend=1| 37, 74, 111 }}


[[Badness]]: 0.312105
[[Badness]]: 7.898


=== 11-limit ===
=== 11-limit ===
Line 37: Line 28:


Mapping: [{{val| 37 0 86 104 128 }}, {{val| 0 1 0 0 0 }}]
Mapping: [{{val| 37 0 86 104 128 }}, {{val| 0 1 0 0 0 }}]
: mapping generator: ~56/55, ~3/2


Optimal tuning (POTE): ~3/2 = 703.0355
Optimal tunings:
* [[WE]]: ~56/55 = 32.408, ~3/2 = 702.635
: [[error map]]: {{val| -0.683 -0.002 1.288 2.228 -2.329 }}
* [[CWE]]: ~56/55 = 1\37, ~3/2 = 702.819
: [[error map]]: {{val| -0.000 0.864 2.875 4.147 0.033 }}


{{Optimal ET sequence|legend=1| 37, 74, 111 }}
{{Optimal ET sequence|legend=1| 37, 74, 111 }}


Badness: 0.101001
Badness: 3.339


=== 13-limit ===
=== 13-limit ===
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Mapping: [{{val| 37 0 86 104 128 137 }}, {{val| 0 1 0 0 0 0 }}]
Mapping: [{{val| 37 0 86 104 128 137 }}, {{val| 0 1 0 0 0 0 }}]


Optimal tuning (POTE): ~3/2 = 703.0520
Optimal tunings:
* [[WE]]: ~65/64 = 32.414, ~3/2 = 702.646
: [[error map]]: {{val| -0.693 -0.002 1.264 2.198 -2.365 0.149 }}
* [[CWE]]: ~65/64 = 1\37, ~3/2 = 702.869
: [[error map]]: {{val| 0.000 0.914 2.875 4.147 0.033 2.716 }}


{{Optimal ET sequence|legend=1| 37, 74, 111 }}
{{Optimal ET sequence|legend=1| 37, 74, 111 }}


Badness: 0.048732
Badness: 2.014


== Triacontaheptoid ==
== Triacontaheptoid ==
Line 112: Line 112:


==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
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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}}
* [[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 }}


Mapping generators: ~{{monzo|103 -18 3 -29}} = 1\37, ~{{monzo|96 -17 3 -27}} = 271.709
{{Optimal ET sequence|legend=1|296, 1073c, 1369, 1665, 1961, 3626}}


Optimal tuning (CTE): ~{{monzo|96 -17 3 -27}} = 271.709
Badness: 33.301


===11-limit===
===11-limit===
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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 ludidzelic 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
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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}}
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