41edt: Difference between revisions
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'''[[Edt|Division of the third harmonic]] into 41 equal parts''' (41edt) is related to [[26edo|26 edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 6.1178 cents stretched and the step size is about 46.3891 cents. Unlike 26edo, it is only consistent up to the 10-[[integer-limit]], with discrepancy for the 11th harmonic. | '''[[Edt|Division of the third harmonic]] into 41 equal parts''' (41edt) is related to [[26edo|26 edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 6.1178 cents stretched and the step size is about 46.3891 cents. Unlike 26edo, it is only consistent up to the 10-[[integer-limit]], with discrepancy for the 11th harmonic. | ||
41edt is related to the regular temperament which tempers out 823543/820125 and 2199023255552/2197176384375 in the 7-limit, which is supported by | 41edt is related to the regular temperament which tempers out 823543/820125 and 2199023255552/2197176384375 in the 7-limit, which is supported by {{EDOs| 181, 207, 388, 569, and 595 }} EDOs. | ||
== Intervals == | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
==Harmonics== | == Harmonics == | ||
{{Harmonics in equal | {{Harmonics in equal | ||
| steps = 41 | | steps = 41 | ||
| Line 25: | Line 25: | ||
}} | }} | ||
=Related regular temperaments= | = Related regular temperaments = | ||
==181&207 temperament== | == 181 & 207 temperament == | ||
===5-limit=== | === 5-limit === | ||
Comma: |287 -121 -41 | Comma: {{monzo| 287 -121 -41 }} | ||
POTE generator: ~|140 -59 -20 | POTE generator: ~{{monzo| 140 -59 -20 }} = 46.3927 | ||
Mapping: [ | Mapping: [{{map| 1 0 7 }}, {{map| 0 41 -121 }}] | ||
EDOs: {{EDOs|181, 207, 388, 569, 595, 957, 1345}} | EDOs: {{EDOs|181, 207, 388, 569, 595, 957, 1345}} | ||
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Badness: 17.5651 | Badness: 17.5651 | ||
===7-limit=== | === 7-limit === | ||
Commas: 823543/820125, 2199023255552/2197176384375 | Commas: 823543/820125, 2199023255552/2197176384375 | ||
POTE generator: ~131072/127575 = 46.3932 | POTE generator: ~131072/127575 = 46.3932 | ||
Mapping: [ | Mapping: [{{map| 1 0 7 3 }}, {{map| 0 41 -121 -5}}] | ||
EDOs: {{EDOs|181, 207, 388, 569, 595}} | EDOs: {{EDOs|181, 207, 388, 569, 595}} | ||
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Badness: 0.6461 | Badness: 0.6461 | ||
===11-limit=== | === 11-limit === | ||
Commas: 42592/42525, 43923/43904, 184877/184320 | Commas: 42592/42525, 43923/43904, 184877/184320 | ||
POTE generator: ~352/343 = 46.3934 | POTE generator: ~352/343 = 46.3934 | ||
Mapping: [ | Mapping: [{{map| 1 0 7 3 4 }}, {{map| 0 41 -121 -5 -14}}] | ||
EDOs: {{EDOs|181, 207, 388, 569, 595}} | EDOs: {{EDOs|181, 207, 388, 569, 595}} | ||
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Badness: 0.1362 | Badness: 0.1362 | ||
===13-limit=== | === 13-limit === | ||
Commas: 847/845, 4096/4095, 4459/4455, 17303/17280 | Commas: 847/845, 4096/4095, 4459/4455, 17303/17280 | ||
POTE generator: ~352/343 = 46.3921 | POTE generator: ~352/343 = 46.3921 | ||
Mapping: [ | Mapping: [{{map| 1 0 7 3 4 2 }}, {{map| 0 41 -121 -5 -14 44 }}] | ||
EDOs: {{EDOs|181, 207, 388, 569, 595}} | EDOs: {{EDOs|181, 207, 388, 569, 595}} | ||
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Badness: 0.0707 | Badness: 0.0707 | ||
===17-limit=== | === 17-limit === | ||
Commas: 833/832, 847/845, 1089/1088, 2058/2057, 2431/2430 | Commas: 833/832, 847/845, 1089/1088, 2058/2057, 2431/2430 | ||
POTE generator: ~187/182 = 46.3918 | POTE generator: ~187/182 = 46.3918 | ||
Mapping: [ | Mapping: [{{map| 1 0 7 3 4 2 2 }}, {{map| 0 41 -121 -5 -14 44 54 }}] | ||
EDOs: {{EDOs|181, 207, 388, 569, 595}} | EDOs: {{EDOs|181, 207, 388, 569, 595}} | ||
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Badness: 0.0411 | Badness: 0.0411 | ||
==26&388 temperament== | == 26 & 388 temperament == | ||
===5-limit=== | === 5-limit === | ||
Comma: |-41 146 -82 | Comma: {{monzo| -41 146 -82 }} | ||
POTE generator: ~|-16 57 -32 | POTE generator: ~{{monzo| -16 57 -32 }} = 46.3883 | ||
Mapping: [ | Mapping: [{{map| 2 0 -1 }}, {{map| 0 41 73 }}] | ||
EDOs: {{EDOs|26, 388, 414, 802, 1190, 1578, 1966, 2354}} | EDOs: {{EDOs|26, 388, 414, 802, 1190, 1578, 1966, 2354}} | ||
| Line 94: | Line 94: | ||
Badness: 3.9285 | Badness: 3.9285 | ||
===7-limit=== | === 7-limit === | ||
Commas: 4375/4374, |-62 -1 2 21 | Commas: 4375/4374, {{monzo| -62 -1 2 21 }} | ||
POTE generator: ~17294403/16777216 = 46.3835 | POTE generator: ~17294403/16777216 = 46.3835 | ||
Mapping: [ | Mapping: [{{map| 2 0 -1 6 }}, {{map| 0 41 73 -5 }}] | ||
EDOs: {{EDOs|26, 362, 388, 414, 802}} | EDOs: {{EDOs|26, 362, 388, 414, 802}} | ||
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Badness: 0.4543 | Badness: 0.4543 | ||
===11-limit=== | === 11-limit === | ||
Commas: 3025/3024, 4375/4374, 5931980229/5905580032 | Commas: 3025/3024, 4375/4374, 5931980229/5905580032 | ||
POTE generator: ~352/343 = 46.3827 | POTE generator: ~352/343 = 46.3827 | ||
Mapping: [ | Mapping: [{{map| 2 0 -1 6 8 }}, {{map| 0 41 73 -5 -14 }}] | ||
EDOs: {{EDOs|26, 362, 388, 414, 802}} | EDOs: {{EDOs|26, 362, 388, 414, 802}} | ||
| Line 116: | Line 116: | ||
Badness: 0.1020 | Badness: 0.1020 | ||
===13-limit=== | === 13-limit === | ||
Commas: 2200/2197, 3025/3024, 4375/4374, 50421/50336 | Commas: 2200/2197, 3025/3024, 4375/4374, 50421/50336 | ||
POTE generator: ~352/343 = 46.3825 | POTE generator: ~352/343 = 46.3825 | ||
Mapping: [ | Mapping: [{{map| 2 0 -1 6 8 4 }}, {{map| 0 41 73 -5 -14 44 }}] | ||
EDOs: {{EDOs|26, 362, 388, 414, 802}} | EDOs: {{EDOs|26, 362, 388, 414, 802}} | ||
| Line 127: | Line 127: | ||
Badness: 0.0595 | Badness: 0.0595 | ||
===17-limit=== | === 17-limit === | ||
Commas: 833/832, 1089/1088, 1225/1224, 1701/1700, 2200/2197 | Commas: 833/832, 1089/1088, 1225/1224, 1701/1700, 2200/2197 | ||
POTE generator: ~187/182 = 46.3824 | POTE generator: ~187/182 = 46.3824 | ||
Mapping: [ | Mapping: [{{map| 2 0 -1 6 8 4 4 }}, {{map| 0 41 73 -5 -14 44 54 }}] | ||
EDOs: {{EDOs|26, 362, 388, 414, 802}} | EDOs: {{EDOs|26, 362, 388, 414, 802}} | ||