43edf: Difference between revisions
ArrowHead294 (talk | contribs) mNo edit summary |
|||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{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> | ||
| Line 21: | Line 28: | ||
EDOs: {{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 | ||
| Line 30: | Line 37: | ||
EDOs: {{EDOs|441, 3455, 3749, 3896, 4190, 4631, 7645, 8086, 8821, 12276}} | EDOs: {{EDOs|441, 3455, 3749, 3896, 4190, 4631, 7645, 8086, 8821, 12276}} | ||
{{Todo|cleanup|expand}} | |||
Latest revision as of 20:57, 11 February 2025
| ← 42edf | 43edf | 44edf → |
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
| 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) | |
| 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