57edf: Difference between revisions

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Created page with "'''57EDF''' is the equal division of the just perfect fifth into 57 parts of 12.3150 cents each, corresponding to 97.4421 edo. It is related to the regula..."
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'''57EDF''' is the [[EDF|equal division of the just perfect fifth]] into 57 parts of 12.3150 [[cent|cents]] each, corresponding to 97.4421 [[edo]]. It is related to the regular temperament which tempers out |-32 33 0 -6 -1> and |76 -8 0 -9 -11> in the 11-limit, which is supported by 877, 3313, 4190, 5067, 5944, 6821, 7698, and 11011 EDOs.
{{Infobox ET}}
{{ED intro}}


==Related regular temperaments==
== Theory ==
===2.3.7 subgroup 877&5067===
57edf corresponds to 97.4421[[edo]]. It is related to the [[regular temperament]] which [[tempering out|tempers out]] {{monzo| -32 33 0 -6 -1 }} and {{monzo| 76 -8 0 -9 -11 }} in the [[11-limit]], which is supported by {{EDOs| 877-, 3313-, 4190-, 5067-, 5944-, 6821-, 7698-, and 11011edo }}.
Commas: |-428 371 0 -57>
 
=== Harmonics ===
{{Harmonics in equal|57|3|2}}
{{Harmonics in equal|57|3|2|start=12|collapsed=true|title=Approximation of harmonics in 57edf (continued)}}
 
== Related regular temperaments ==
=== 2.3.7 subgroup 877&5067 ===
Commas: {{monzo| -428 371 0 -57 }}


POTE generator: ~1605632/1594323 = 12.3149
POTE generator: ~1605632/1594323 = 12.3149


Map: [<1 1 -1|, <0 57 371|]
Mapping: [{{val| 1 1 -1 }}, {{val| 0 57 371 }}]


EDOs: 877, 4190, 5067, 5944, 6821, 7698, 8575
EDOs: {{EDOs|877, 4190, 5067, 5944, 6821, 7698, 8575}}


===2.3.7.11 subgroup 877&5067===
=== 2.3.7.11 subgroup 877&5067 ===
Commas: |-32 33 0 -6 -1>, |76 -8 0 -9 -11>
Commas: {{monzo| -32 33 0 -6 -1 }}, {{monzo| 76 -8 0 -9 -11 }}


POTE generator: ~1605632/1594323 = 12.3150
POTE generator: ~1605632/1594323 = 12.3150


Map: [<1 1 -1 7|, <0 57 371 -345|]
Mapping: [{{val| 1 1 -1 7 }}, {{val| 0 57 371 -345 }}]


EDOs: 877, 3313, 4190, 5067, 5944, 6821, 7698, 11011
EDOs: {{EDOs|877, 3313, 4190, 5067, 5944, 6821, 7698, 11011}}


===2.3.7.11.13 subgroup 877&5067===
=== 2.3.7.11.13 subgroup 877&5067 ===
Commas: 257330216/257298363, 53722307808/53710650917, 1786706395136/1786568061663
Commas: 257330216/257298363, 53722307808/53710650917, 1786706395136/1786568061663


POTE generator: ~1605632/1594323 = 12.3150
POTE generator: ~1605632/1594323 = 12.3150


Map: [<1 1 -1 7 -10|, <0 57 371 -345 1335|]
Mapping: [{{val| 1 1 -1 7 -10 }}, {{val| 0 57 371 -345 1335 }}]
 
EDOs: {{EDOs|877, 3313, 4190, 5067, 5944, 9257}}


EDOs: 877, 3313, 4190, 5067, 5944, 9257
== Intervals ==
{{Interval table}}


[[Category:Edf]]
{{Todo|cleanup}}
[[Category:Edonoi]]

Latest revision as of 17:24, 17 January 2025

← 56edf 57edf 58edf →
Prime factorization 3 × 19
Step size 12.315 ¢ 
Octave 97\57edf (1194.56 ¢)
Twelfth 154\57edf (1896.51 ¢)
Consistency limit 3
Distinct consistency limit 3

57 equal divisions of the perfect fifth (abbreviated 57edf or 57ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 57 equal parts of about 12.3 ¢ each. Each step represents a frequency ratio of (3/2)1/57, or the 57th root of 3/2.

Theory

57edf corresponds to 97.4421edo. It is related to the regular temperament which tempers out [-32 33 0 -6 -1⟩ and [76 -8 0 -9 -11⟩ in the 11-limit, which is supported by 877-, 3313-, 4190-, 5067-, 5944-, 6821-, 7698-, and 11011edo.

Harmonics

Approximation of harmonics in 57edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.44 -5.44 +1.43 -3.12 +1.43 +5.48 -4.02 +1.43 +3.75 -1.16 -4.02
Relative (%) -44.2 -44.2 +11.6 -25.4 +11.6 +44.5 -32.6 +11.6 +30.4 -9.4 -32.6
Steps
(reduced) 		97
(40) 		154
(40) 		195
(24) 		226
(55) 		252
(24) 		274
(46) 		292
(7) 		309
(24) 		324
(39) 		337
(52) 		349
(7)
Approximation of harmonics in 57edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +5.19 +0.04 +3.75 +2.85 -3.59 -4.02 +0.90 -1.70 +0.04 +5.71 +2.64
Relative (%) +42.1 +0.3 +30.4 +23.1 -29.1 -32.6 +7.3 -13.8 +0.3 +46.3 +21.4
Steps
(reduced) 		361
(19) 		371
(29) 		381
(39) 		390
(48) 		398
(56) 		406
(7) 		414
(15) 		421
(22) 		428
(29) 		435
(36) 		441
(42)

Related regular temperaments

2.3.7 subgroup 877&5067

Commas: [-428 371 0 -57⟩

POTE generator: ~1605632/1594323 = 12.3149

Mapping: [⟨1 1 -1], ⟨0 57 371]]

EDOs: 877, 4190, 5067, 5944, 6821, 7698, 8575

2.3.7.11 subgroup 877&5067

Commas: [-32 33 0 -6 -1⟩, [76 -8 0 -9 -11⟩

POTE generator: ~1605632/1594323 = 12.3150

Mapping: [⟨1 1 -1 7], ⟨0 57 371 -345]]

EDOs: 877, 3313, 4190, 5067, 5944, 6821, 7698, 11011

2.3.7.11.13 subgroup 877&5067

Commas: 257330216/257298363, 53722307808/53710650917, 1786706395136/1786568061663

POTE generator: ~1605632/1594323 = 12.3150

Mapping: [⟨1 1 -1 7 -10], ⟨0 57 371 -345 1335]]

EDOs: 877, 3313, 4190, 5067, 5944, 9257

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 12.3
2 24.6
3 36.9
4 49.3 34/33
5 61.6 29/28
6 73.9
7 86.2
8 98.5
9 110.8
10 123.2
11 135.5
12 147.8
13 160.1 23/21
14 172.4 21/19
15 184.7 10/9, 29/26
16 197 19/17, 28/25
17 209.4 26/23
18 221.7 25/22, 33/29
19 234
20 246.3
21 258.6 29/25
22 270.9 34/29
23 283.2 33/28
24 295.6
25 307.9 31/26
26 320.2
27 332.5 17/14, 23/19
28 344.8
29 357.1
30 369.5 21/17, 26/21
31 381.8
32 394.1
33 406.4
34 418.7 14/11
35 431
36 443.3 22/17
37 455.7
38 468
39 480.3 29/22, 33/25
40 492.6
41 504.9
42 517.2 27/20, 31/23
43 529.5 19/14, 34/25
44 541.9 26/19
45 554.2 29/21
46 566.5
47 578.8
48 591.1
49 603.4
50 615.8
51 628.1
52 640.4
53 652.7 19/13
54 665 22/15, 25/17
55 677.3 31/21
56 689.6
57 702 3/2
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