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{{Infobox ET}}
{{Infobox ET}}
'''[[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.
{{ED intro}}


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.
41edt is related to [[26edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 6.12 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 {{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
| num = 3
| num = 3
| denom = 1
| denom = 1
| intervals = integer
| intervals = prime
}}
}}
{{Harmonics in equal
{{Harmonics in equal
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| start = 12
| start = 12
| collapsed = 1
| collapsed = 1
| intervals = integer
| intervals = prime
}}
}}


=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> = 46.3927
POTE generator: ~{{monzo| 140 -59 -20 }} = 46.3927


Mapping: [<1 0 7|, <0 41 -121|]
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: [<1 0 7 3|, <0 41 -121 -5|]
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: [<1 0 7 3 4|, <0 41 -121 -5 -14|]
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: [<1 0 7 3 4 2|, <0 41 -121 -5 -14 44|]
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: [<1 0 7 3 4 2 2|, <0 41 -121 -5 -14 44 54|]
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> = 46.3883
POTE generator: ~{{monzo| -16 57 -32 }} = 46.3883


Mapping: [<2 0 -1|, <0 41 73|]
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}}
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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: [<2 0 -1 6|, <0 41 73 -5|]
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: [<2 0 -1 6 8|, <0 41 73 -5 -14|]
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}}
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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: [<2 0 -1 6 8 4|, <0 41 73 -5 -14 44|]
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}}
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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: [<2 0 -1 6 8 4 4|, <0 41 73 -5 -14 44 54|]
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}}
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Badness: 0.0326
Badness: 0.0326


[[Category:Edt]]
[[Category:Edonoi]]
{{todo|expand}}
{{todo|expand}}
Retrieved from "https://en.xen.wiki/w/41edt"