58edf: Difference between revisions
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Created page with "'''Division of the just perfect fifth into 58 equal parts''' (58EDF) is related to 99 edo, but with the 3/2 rather than the 2/1 being just. The octave is abo..." Tags: Mobile edit Mobile web edit |
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== Theory == | |||
58edf corresponds to 99.1517…edo. It is related to [[99edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1|octave]] being just. The octave is [[stretched and compressed tuning|compressed]] by about 1.84 cents. 58edf is [[consistent]] to the [[integer limit|12-integer-limit]]. In comparison, 99edo is only consistent up to the 10-integer-limit. 58edf has a flat tendency, with [[prime harmonic]]s 2, [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]] all tuned flat of just. | |||
[[ | === Harmonics === | ||
[[ | {{Harmonics in equal|58|3|2|intervals=integer|columns=11}} | ||
{{Harmonics in equal|58|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edf (continued)}} | |||
=== Subsets and supersets === | |||
Since 58 factors into primes as {{nowrap| 2 × 29 }}, 58edf contains [[2edf]] and [[29edf]] as subset edts. | |||
== See also == | |||
* [[99edo]] – relative edo | |||
* [[157edt]] – relative edt | |||
* [[256ed6]] – relative ed6 | |||
Latest revision as of 13:20, 18 April 2025
| ← 57edf | 58edf | 59edf → |
58 equal divisions of the perfect fifth (abbreviated 58edf or 58ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 58 equal parts of about 12.1 ¢ each. Each step represents a frequency ratio of (3/2)1/58, or the 58th root of 3/2.
Theory
58edf corresponds to 99.1517…edo. It is related to 99edo, but with the perfect fifth rather than the octave being just. The octave is compressed by about 1.84 cents. 58edf is consistent to the 12-integer-limit. In comparison, 99edo is only consistent up to the 10-integer-limit. 58edf has a flat tendency, with prime harmonics 2, 3, 5, 7, and 11 all tuned flat of just.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.84 | -1.84 | -3.67 | -2.70 | -3.67 | -4.28 | -5.51 | -3.67 | -4.53 | -0.10 | -5.51 |
| Relative (%) | -15.2 | -15.2 | -30.3 | -22.3 | -30.3 | -35.4 | -45.5 | -30.3 | -37.5 | -0.8 | -45.5 | |
| Steps (reduced) |
99 (41) |
157 (41) |
198 (24) |
230 (56) |
256 (24) |
278 (46) |
297 (7) |
314 (24) |
329 (39) |
343 (53) |
355 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.15 | +5.98 | -4.53 | +4.76 | -3.37 | -5.51 | -2.29 | +5.73 | +5.98 | -1.94 | +5.83 | +4.76 |
| Relative (%) | +9.5 | +49.4 | -37.5 | +39.3 | -27.9 | -45.5 | -18.9 | +47.4 | +49.4 | -16.0 | +48.1 | +39.3 | |
| Steps (reduced) |
367 (19) |
378 (30) |
387 (39) |
397 (49) |
405 (57) |
413 (7) |
421 (15) |
429 (23) |
436 (30) |
442 (36) |
449 (43) |
455 (49) | |
Subsets and supersets
Since 58 factors into primes as 2 × 29, 58edf contains 2edf and 29edf as subset edts.