41edt: Difference between revisions

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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 [[9-odd-limit|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
Line 20: Line 22:
| 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}}
Line 36: Line 38:
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}}
Line 47: Line 49:
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}}
Line 58: Line 60:
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}}
Line 69: Line 71:
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}}
Line 80: Line 82:
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}}
Line 92: 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: [<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}}
Line 103: Line 105:
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}}
Line 114: 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: [<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}}
Line 125: 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: [<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}}
Line 136: Line 138:
Badness: 0.0326
Badness: 0.0326


[[Category:Edt]]
{{todo|expand}}
[[Category:Edonoi]]

Latest revision as of 19:22, 1 August 2025

← 40edt 41edt 42edt →
Prime factorization 41 (prime)
Step size 46.3891 ¢ 
Octave 26\41edt (1206.12 ¢)
Consistency limit 10
Distinct consistency limit 9

41 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 41edt or 41ed3), is a nonoctave tuning system that divides the interval of 3/1 into 41 equal parts of about 46.4 ¢ each. Each step represents a frequency ratio of 31/41, or the 41st root of 3.

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 181, 207, 388, 569, and 595 EDOs.

Intervals

Steps Cents Hekts Approximate ratios
0 0 0 1/1
1 46.4 31.7
2 92.8 63.4 18/17, 19/18, 20/19
3 139.2 95.1 13/12, 25/23, 27/25
4 185.6 126.8 10/9, 19/17, 29/26
5 231.9 158.5 8/7
6 278.3 190.2 20/17, 27/23
7 324.7 222 23/19, 29/24
8 371.1 253.7 21/17, 26/21
9 417.5 285.4 23/18
10 463.9 317.1 17/13, 21/16, 30/23
11 510.3 348.8
12 556.7 380.5 18/13, 29/21
13 603.1 412.2 17/12, 24/17, 27/19
14 649.4 443.9 29/20
15 695.8 475.6 3/2
16 742.2 507.3 20/13, 23/15, 26/17
17 788.6 539 19/12, 30/19
18 835 570.7 13/8, 21/13
19 881.4 602.4 5/3
20 927.8 634.1 12/7, 29/17
21 974.2 665.9 7/4
22 1020.6 697.6 9/5
23 1067 729.3 13/7, 24/13
24 1113.3 761 19/10
25 1159.7 792.7
26 1206.1 824.4 2/1
27 1252.5 856.1
28 1298.9 887.8 17/8, 19/9
29 1345.3 919.5 13/6
30 1391.7 951.2 29/13
31 1438.1 982.9 16/7, 23/10
32 1484.5 1014.6
33 1530.8 1046.3 17/7, 29/12
34 1577.2 1078
35 1623.6 1109.8 23/9
36 1670 1141.5 21/8
37 1716.4 1173.2 27/10
38 1762.8 1204.9 25/9
39 1809.2 1236.6 17/6
40 1855.6 1268.3
41 1902 1300 3/1

Harmonics

Approximation of prime harmonics in 41edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.1 +0.0 -3.0 +17.6 -22.7 +12.8 +12.3 +5.3 -0.7 +15.5 -7.2
Relative (%) +13.2 +0.0 -6.4 +37.9 -48.9 +27.7 +26.5 +11.4 -1.6 +33.3 -15.6
Steps
(reduced) 		26
(26) 		41
(0) 		60
(19) 		73
(32) 		89
(7) 		96
(14) 		106
(24) 		110
(28) 		117
(35) 		126
(3) 		128
(5)
Approximation of prime harmonics in 41edt
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +11.2 +19.0 -17.0 +14.5 -7.9 -8.0 -19.3 +3.8 -3.8 -5.5 -3.1
Relative (%) +24.1 +41.0 -36.7 +31.3 -17.1 -17.3 -41.7 +8.2 -8.2 -11.9 -6.7
Steps
(reduced) 		135
(12) 		139
(16) 		140
(17) 		144
(21) 		148
(25) 		152
(29) 		153
(30) 		157
(34) 		159
(36) 		160
(37) 		163
(40)

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]]

EDOs: 181, 207, 388, 569, 595

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]]

EDOs: 181, 207, 388, 569, 595

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]]

EDOs: 181, 207, 388, 569, 595

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]]

EDOs: 181, 207, 388, 569, 595

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]]

EDOs: 26, 362, 388, 414, 802

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]]

EDOs: 26, 362, 388, 414, 802

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]]

EDOs: 26, 362, 388, 414, 802

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]]

EDOs: 26, 362, 388, 414, 802

Badness: 0.0326

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