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 [[9-odd-limit|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 [[9-odd-limit|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 [[181edo|181]], [[207edo|207]], [[388edo|388]], 569, and 595 EDOs. | 41edt is related to the regular temperament which tempers out 823543/820125 and 2199023255552/2197176384375 in the 7-limit, which is supported by [[181edo|181]], [[207edo|207]], [[388edo|388]], [[569edo|569]], and [[595edo|595]] EDOs. | ||
=Related regular temperaments= | =Related regular temperaments= | ||
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Mapping: [<1 0 7|, <0 41 -121|] | Mapping: [<1 0 7|, <0 41 -121|] | ||
EDOs: 181, 207, 388, 569, 595, 957, 1345 | EDOs: {{EDOs|181, 207, 388, 569, 595, 957, 1345}} | ||
Badness: 17.5651 | Badness: 17.5651 | ||
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Mapping: [<1 0 7 3|, <0 41 -121 -5|] | Mapping: [<1 0 7 3|, <0 41 -121 -5|] | ||
EDOs: 181, 207, 388, 569, 595 | EDOs: {{EDOs|181, 207, 388, 569, 595}} | ||
Badness: 0.6461 | Badness: 0.6461 | ||
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Mapping: [<1 0 7 3 4|, <0 41 -121 -5 -14|] | Mapping: [<1 0 7 3 4|, <0 41 -121 -5 -14|] | ||
EDOs: 181, 207, 388, 569, 595 | EDOs: {{EDOs|181, 207, 388, 569, 595}} | ||
Badness: 0.1362 | Badness: 0.1362 | ||
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Mapping: [<1 0 7 3 4 2|, <0 41 -121 -5 -14 44|] | Mapping: [<1 0 7 3 4 2|, <0 41 -121 -5 -14 44|] | ||
EDOs: 181, 207, 388, 569, 595 | EDOs: {{EDOs|181, 207, 388, 569, 595}} | ||
Badness: 0.0707 | Badness: 0.0707 | ||
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Mapping: [<1 0 7 3 4 2 2|, <0 41 -121 -5 -14 44 54|] | Mapping: [<1 0 7 3 4 2 2|, <0 41 -121 -5 -14 44 54|] | ||
EDOs: 181, 207, 388, 569, 595 | EDOs: {{EDOs|181, 207, 388, 569, 595}} | ||
Badness: 0.0411 | Badness: 0.0411 | ||
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Mapping: [<2 0 -1|, <0 41 73|] | Mapping: [<2 0 -1|, <0 41 73|] | ||
EDOs: 26, 388, 414, 802, 1190, 1578, 1966, 2354 | EDOs: {{EDOs|26, 388, 414, 802, 1190, 1578, 1966, 2354}} | ||
Badness: 3.9285 | Badness: 3.9285 | ||
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Mapping: [<2 0 -1 6|, <0 41 73 -5|] | Mapping: [<2 0 -1 6|, <0 41 73 -5|] | ||
EDOs: 26, 362, 388, 414, 802 | EDOs: {{EDOs|26, 362, 388, 414, 802}} | ||
Badness: 0.4543 | Badness: 0.4543 | ||
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Mapping: [<2 0 -1 6 8|, <0 41 73 -5 -14|] | Mapping: [<2 0 -1 6 8|, <0 41 73 -5 -14|] | ||
EDOs: 26, 362, 388, 414, 802 | EDOs: {{EDOs|26, 362, 388, 414, 802}} | ||
Badness: 0.1020 | Badness: 0.1020 | ||
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Mapping: [<2 0 -1 6 8 4|, <0 41 73 -5 -14 44|] | Mapping: [<2 0 -1 6 8 4|, <0 41 73 -5 -14 44|] | ||
EDOs: 26, 362, 388, 414, 802 | EDOs: {{EDOs|26, 362, 388, 414, 802}} | ||
Badness: 0.0595 | Badness: 0.0595 | ||
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Mapping: [<2 0 -1 6 8 4 4|, <0 41 73 -5 -14 44 54|] | Mapping: [<2 0 -1 6 8 4 4|, <0 41 73 -5 -14 44 54|] | ||
EDOs: 26, 362, 388, 414, 802 | EDOs: {{EDOs|26, 362, 388, 414, 802}} | ||
Badness: 0.0326 | Badness: 0.0326 | ||
Revision as of 07:55, 19 April 2023
| ← 40edt | 41edt | 42edt → |
Division of the third harmonic into 41 equal parts (41edt) is related to 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 181, 207, 388, 569, and 595 EDOs.
Related regular temperaments
181&207 temperament
5-limit
Comma: |287 -121 -41>
POTE generator: ~|140 -59 -20> = 46.3927
Mapping: [<1 0 7|, <0 41 -121|]
EDOs: 181, 207, 388, 569, 595, 957, 1345
Badness: 17.5651
7-limit
Commas: 823543/820125, 2199023255552/2197176384375
POTE generator: ~131072/127575 = 46.3932
Mapping: [<1 0 7 3|, <0 41 -121 -5|]
Badness: 0.6461
11-limit
Commas: 42592/42525, 43923/43904, 184877/184320
POTE generator: ~352/343 = 46.3934
Mapping: [<1 0 7 3 4|, <0 41 -121 -5 -14|]
Badness: 0.1362
13-limit
Commas: 847/845, 4096/4095, 4459/4455, 17303/17280
POTE generator: ~352/343 = 46.3921
Mapping: [<1 0 7 3 4 2|, <0 41 -121 -5 -14 44|]
Badness: 0.0707
17-limit
Commas: 833/832, 847/845, 1089/1088, 2058/2057, 2431/2430
POTE generator: ~187/182 = 46.3918
Mapping: [<1 0 7 3 4 2 2|, <0 41 -121 -5 -14 44 54|]
Badness: 0.0411
26&388 temperament
5-limit
Comma: |-41 146 -82>
POTE generator: ~|-16 57 -32> = 46.3883
Mapping: [<2 0 -1|, <0 41 73|]
EDOs: 26, 388, 414, 802, 1190, 1578, 1966, 2354
Badness: 3.9285
7-limit
Commas: 4375/4374, |-62 -1 2 21>
POTE generator: ~17294403/16777216 = 46.3835
Mapping: [<2 0 -1 6|, <0 41 73 -5|]
Badness: 0.4543
11-limit
Commas: 3025/3024, 4375/4374, 5931980229/5905580032
POTE generator: ~352/343 = 46.3827
Mapping: [<2 0 -1 6 8|, <0 41 73 -5 -14|]
Badness: 0.1020
13-limit
Commas: 2200/2197, 3025/3024, 4375/4374, 50421/50336
POTE generator: ~352/343 = 46.3825
Mapping: [<2 0 -1 6 8 4|, <0 41 73 -5 -14 44|]
Badness: 0.0595
17-limit
Commas: 833/832, 1089/1088, 1225/1224, 1701/1700, 2200/2197
POTE generator: ~187/182 = 46.3824
Mapping: [<2 0 -1 6 8 4 4|, <0 41 73 -5 -14 44 54|]
Badness: 0.0326