415edt: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
''' | '''415edt''' is the [[EDT|equal division of the third harmonic]] into 415 parts of 4.5830 [[cent|cents]] each, corresponding to 261.8358 [[edo]]. It is notable for its impressive [[consistency]] records in very high no-evens [[odd limit#Nonoctave equaves|throdd limit]]s: specifically, it is consistent to the entirety of the no-23s no-47s no-59s add-71 add-77 65-throdd limit, and all additional intervals if primes 59, 67, and 73 are added to this are within 60.2% of a step of their [[patent val]] approximation. This makes it a potential candidate for the tritave-based version of [[311edo]], although its performance is not quite as spectacular as that miracle edo. | ||
== Harmonics == | == Harmonics == | ||
{{Harmonics in equal|415|3|1|intervals=odd|columns=17}} | {{Harmonics in equal|415|3|1|intervals=odd|columns=17}} | ||
{{Harmonics in equal|415|3|1|intervals=odd|start=18|columns= | {{Harmonics in equal|415|3|1|intervals=odd|start=18|columns=22|collapsed=1|title=Approximation of odd harmonics in 415edt (continued)}} | ||
Latest revision as of 16:08, 29 October 2024
| ← 414edt | 415edt | 416edt → |
415edt is the equal division of the third harmonic into 415 parts of 4.5830 cents each, corresponding to 261.8358 edo. It is notable for its impressive consistency records in very high no-evens throdd limits: specifically, it is consistent to the entirety of the no-23s no-47s no-59s add-71 add-77 65-throdd limit, and all additional intervals if primes 59, 67, and 73 are added to this are within 60.2% of a step of their patent val approximation. This makes it a potential candidate for the tritave-based version of 311edo, although its performance is not quite as spectacular as that miracle edo.
Harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | 33 | 35 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.16 | -0.30 | +0.00 | +0.90 | +0.42 | +0.16 | -1.12 | -1.19 | -0.30 | -1.97 | +0.33 | +0.00 | +0.03 | -0.85 | +0.90 | -0.14 |
| Relative (%) | +0.0 | +3.6 | -6.6 | +0.0 | +19.7 | +9.2 | +3.6 | -24.4 | -26.0 | -6.6 | -43.1 | +7.2 | +0.0 | +0.6 | -18.6 | +19.7 | -3.0 | |
| Steps (reduced) |
415 (0) |
608 (193) |
735 (320) |
830 (0) |
906 (76) |
969 (139) |
1023 (193) |
1070 (240) |
1112 (282) |
1150 (320) |
1184 (354) |
1216 (386) |
1245 (0) |
1272 (27) |
1297 (52) |
1321 (76) |
1343 (98) | |
| Harmonic | 37 | 39 | 41 | 43 | 45 | 47 | 49 | 51 | 53 | 55 | 57 | 59 | 61 | 63 | 65 | 67 | 69 | 71 | 73 | 75 | 77 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.10 | +0.42 | +0.92 | +0.96 | +0.16 | -1.79 | -0.61 | -1.12 | +1.03 | +1.07 | -1.19 | -1.31 | +0.55 | -0.30 | +0.59 | -1.46 | -1.97 | -1.03 | +1.29 | +0.33 | +0.60 | +2.04 |
| Relative (%) | -2.2 | +9.2 | +20.1 | +20.9 | +3.6 | -39.0 | -13.2 | -24.4 | +22.5 | +23.3 | -26.0 | -28.7 | +12.0 | -6.6 | +12.8 | -32.0 | -43.1 | -22.4 | +28.2 | +7.2 | +13.1 | +44.4 | |
| Steps (reduced) |
1364 (119) |
1384 (139) |
1403 (158) |
1421 (176) |
1438 (193) |
1454 (209) |
1470 (225) |
1485 (240) |
1500 (255) |
1514 (269) |
1527 (282) |
1540 (295) |
1553 (308) |
1565 (320) |
1577 (332) |
1588 (343) |
1599 (354) |
1610 (365) |
1621 (376) |
1631 (386) |
1641 (396) |
1651 (406) | |