43edf: Difference between revisions

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'''43EDF''' is the [[EDF|equal division of the just perfect fifth]] into 43 parts of 16.3245 [[cent|cents]] each, corresponding to 73.5090 [[edo]] (similar to every second step of [[147edo]]). It is related to the [[microtempering|microtemperament]] which tempers out |-135 135 -86 43> (0.45970 cents) in the 7-limit, which is supported by 441, 4190, 4631, 5072, 7204, 11394, 11835, 12276, and 16466 EDOs.
{{Infobox ET}}
{{ED intro}}


==Related temperament==
== Theory ==
===7-limit 441&4631&12276===
43edf corresponds to 73.5090 [[edo]], similar to every second step of [[147edo]]. It is related to the [[microtemperament]] which [[tempering out|tempers out]] {{monzo| -135 135 -86 43 }} (0.45970 cents) in the [[7-limit]], which is supported by {{EDOs| 441-, 4190-, 4631-, 5072-, 7204-, 11394-, 11835-, 12276-, and 16466edo }}.
 
=== Harmonics ===
{{Harmonics in equal|43|3|2}}
{{Harmonics in equal|43|3|2|start=12|collapsed=true|title=Approximation of harmonics in 43edf (continued)}}
 
== Related temperament ==
===7-limit 441 & 4631 & 12276===
Comma: |-135 135 -86 43>
Comma: |-135 135 -86 43>


POTE generators: ~5/4 = 386.3143, ~|-22 22 -14 7> = 16.3245
POTE generators: ~5/4 = 386.3143, ~|-22 22 -14 7> = 16.3245


Map: [<1 1 0 0|, <0 43 0 -135|, <0 0 1 2|]
Mapping: [<1 1 0 0|, <0 43 0 -135|, <0 0 1 2|]


EDOs: 441, 3749, 4190, 4631, 5072, 5513, 6763, 7204, 11394, 11835, 12276, 12717, 16466, 23229, 24111
EDOs: {{EDOs|441, 3749, 4190, 4631, 5072, 5513, 6763, 7204, 11394, 11835, 12276, 12717, 16466, 23229, 24111}}


===11-limit 441&4631&12276===
===11-limit 441&4631&12276===
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POTE generators: ~5/4 = 386.3178, ~|-22 22 -14 7> = 16.3245
POTE generators: ~5/4 = 386.3178, ~|-22 22 -14 7> = 16.3245


Map: [<1 1 0 0 -2|, <0 43 0 -135 572|, <0 0 1 2 -1|]
Mapping: [<1 1 0 0 -2|, <0 43 0 -135 572|, <0 0 1 2 -1|]


EDOs: 441, 3455, 3749, 3896, 4190, 4631, 7645, 8086, 8821, 11835, 12276, 16466, 16907, 24111, 28742
EDOs: {{EDOs|441, 3455, 3749, 3896, 4190, 4631, 7645, 8086, 8821, 11835, 12276, 16466, 16907, 24111, 28742}}


===13-limit 441&4631&12276===
===13-limit 441 & 4631 & 12276===
Commas: 33792000/33787663, 703096443/703040000, 33319272448/33317578125
Commas: 33792000/33787663, 703096443/703040000, 33319272448/33317578125


POTE generators: ~5/4 = 386.3240, ~|-22 22 -14 7> = 16.3246
POTE generators: ~5/4 = 386.3240, ~|-22 22 -14 7> = 16.3246


Map: [<1 1 0 0 -2 2|, <0 43 0 -135 572 125|, <0 0 1 2 -1 0|]
Mapping: [<1 1 0 0 -2 2|, <0 43 0 -135 572 125|, <0 0 1 2 -1 0|]


EDOs: 441, 3455, 3749, 3896, 4190, 4631, 7645, 8086, 8821, 12276
EDOs: {{EDOs|441, 3455, 3749, 3896, 4190, 4631, 7645, 8086, 8821, 12276}}


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

Latest revision as of 20:57, 11 February 2025

← 42edf 43edf 44edf →
Prime factorization 43 (prime)
Step size 16.3245 ¢ 
Octave 74\43edf (1208.02 ¢)
Twelfth 117\43edf (1909.97 ¢)
Consistency limit 3
Distinct consistency limit 3

43 equal divisions of the perfect fifth (abbreviated 43edf or 43ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 43 equal parts of about 16.3 ¢ each. Each step represents a frequency ratio of (3/2)1/43, or the 43rd root of 3/2.

Theory

43edf corresponds to 73.5090 edo, similar to every second step of 147edo. It is related to the microtemperament which tempers out [-135 135 -86 43⟩ (0.45970 cents) in the 7-limit, which is supported by 441-, 4190-, 4631-, 5072-, 7204-, 11394-, 11835-, 12276-, and 16466edo.

Harmonics

Approximation of harmonics in 43edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +8.02 +8.02 -0.29 +5.18 -0.29 -5.97 +7.72 -0.29 -3.13 -4.89 +7.72
Relative (%) +49.1 +49.1 -1.8 +31.7 -1.8 -36.6 +47.3 -1.8 -19.2 -29.9 +47.3
Steps
(reduced) 		74
(31) 		117
(31) 		147
(18) 		171
(42) 		190
(18) 		206
(34) 		221
(6) 		233
(18) 		244
(29) 		254
(39) 		264
(6)
Approximation of harmonics in 43edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -0.25 +2.04 -3.13 -0.59 -7.59 +7.72 -4.26 +4.89 +2.04 +3.13 +7.80
Relative (%) -1.6 +12.5 -19.2 -3.6 -46.5 +47.3 -26.1 +29.9 +12.5 +19.2 +47.8
Steps
(reduced) 		272
(14) 		280
(22) 		287
(29) 		294
(36) 		300
(42) 		307
(6) 		312
(11) 		318
(17) 		323
(22) 		328
(27) 		333
(32)

Related temperament

7-limit 441 & 4631 & 12276

Comma: |-135 135 -86 43>

POTE generators: ~5/4 = 386.3143, ~|-22 22 -14 7> = 16.3245

Mapping: [<1 1 0 0|, <0 43 0 -135|, <0 0 1 2|]

EDOs: 441, 3749, 4190, 4631, 5072, 5513, 6763, 7204, 11394, 11835, 12276, 12717, 16466, 23229, 24111

11-limit 441&4631&12276

Commas: |-31 29 -11 5 -1>, |20 -10 -31 18 5>

POTE generators: ~5/4 = 386.3178, ~|-22 22 -14 7> = 16.3245

Mapping: [<1 1 0 0 -2|, <0 43 0 -135 572|, <0 0 1 2 -1|]

EDOs: 441, 3455, 3749, 3896, 4190, 4631, 7645, 8086, 8821, 11835, 12276, 16466, 16907, 24111, 28742

13-limit 441 & 4631 & 12276

Commas: 33792000/33787663, 703096443/703040000, 33319272448/33317578125

POTE generators: ~5/4 = 386.3240, ~|-22 22 -14 7> = 16.3246

Mapping: [<1 1 0 0 -2 2|, <0 43 0 -135 572 125|, <0 0 1 2 -1 0|]

EDOs: 441, 3455, 3749, 3896, 4190, 4631, 7645, 8086, 8821, 12276

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