20567edo: Difference between revisions
I doubled checked that these are all the inconsistencies but ffs don't delete the entire thing if it has one mistake; build upon it instead |
79-limit vs 81-odd-limit |
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20567edo is a remarkable very high-limit system, distinctly (and almost purely, as all odd harmonics 57 and below, except 49, are within 25% relative error) [[consistent]] through the [[57-odd-limit]], with a lower [[relative error]] than any previous equal temperaments in the 43-limit. It tempers out 33814/33813, 35344/35343, 37180/37179, 42484/42483, 42688/42687, 47125/47124, 48504/48503, 67915/67914, 70500/70499, 91885/91884, 126225/126224, 156520/156519, 194580/194579, 206800/206793, and 561925/561924 in the 53-limit. | 20567edo is a remarkable very high-limit system, distinctly (and almost purely, as all odd harmonics 57 and below, except 49, are within 25% relative error) [[consistent]] through the [[57-odd-limit]], with a lower [[relative error]] than any previous equal temperaments in the 43-limit. It tempers out 33814/33813, 35344/35343, 37180/37179, 42484/42483, 42688/42687, 47125/47124, 48504/48503, 67915/67914, 70500/70499, 91885/91884, 126225/126224, 156520/156519, 194580/194579, 206800/206793, and 561925/561924 in the 53-limit. | ||
Although [[prime interval|prime]] [[61/1|61]] is poorly approximated, the next four primes are fairly accurate, so it can be used all the way to the no-61 [[79-limit]], with the only inconsistent intervals in the no-61 81-odd-limit being 81/47, 81/55, 81/59, 81/67, 63/47, 63/55, 63/59, 67/63, 67/49, 59/41, 59/42, 59/49, 59/51, 59/54, and their [[octave complement]]s. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|20567|columns=11}} | {{Harmonics in equal|20567|columns=11}} | ||
{{Harmonics in equal|20567|columns=11|start=12|collapsed=1|title=Approximation of prime harmonics in 20567edo (continued)}} | {{Harmonics in equal|20567|columns=11|start=12|collapsed=1|title=Approximation of prime harmonics in 20567edo (continued)}} | ||
Latest revision as of 19:01, 2 February 2026
| ← 20566edo | 20567edo | 20568edo → |
20567 equal divisions of the octave (abbreviated 20567edo or 20567ed2), also called 20567-tone equal temperament (20567tet) or 20567 equal temperament (20567et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 20567 equal parts of about 0.0583 ¢ each. Each step represents a frequency ratio of 21/20567, or the 20567th root of 2.
20567edo is a remarkable very high-limit system, distinctly (and almost purely, as all odd harmonics 57 and below, except 49, are within 25% relative error) consistent through the 57-odd-limit, with a lower relative error than any previous equal temperaments in the 43-limit. It tempers out 33814/33813, 35344/35343, 37180/37179, 42484/42483, 42688/42687, 47125/47124, 48504/48503, 67915/67914, 70500/70499, 91885/91884, 126225/126224, 156520/156519, 194580/194579, 206800/206793, and 561925/561924 in the 53-limit.
Although prime 61 is poorly approximated, the next four primes are fairly accurate, so it can be used all the way to the no-61 79-limit, with the only inconsistent intervals in the no-61 81-odd-limit being 81/47, 81/55, 81/59, 81/67, 63/47, 63/55, 63/59, 67/63, 67/49, 59/41, 59/42, 59/49, 59/51, 59/54, and their octave complements.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0044 | -0.0056 | +0.0077 | -0.0076 | +0.0033 | +0.0089 | -0.0073 | -0.0058 | -0.0056 | +0.0026 |
| Relative (%) | +0.0 | +7.6 | -9.5 | +13.1 | -13.0 | +5.6 | +15.2 | -12.5 | -9.9 | -9.5 | +4.4 | |
| Steps (reduced) |
20567 (0) |
32598 (12031) |
47755 (6621) |
57739 (16605) |
71150 (9449) |
76107 (14406) |
84067 (1799) |
87367 (5099) |
93036 (10768) |
99914 (17646) |
101893 (19625) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0101 | +0.0133 | +0.0007 | -0.0134 | -0.0082 | -0.0186 | -0.0277 | -0.0150 | +0.0088 | -0.0071 | +0.0082 |
| Relative (%) | +17.3 | +22.8 | +1.3 | -22.9 | -14.0 | -32.0 | -47.5 | -25.6 | +15.1 | -12.2 | +14.1 | |
| Steps (reduced) |
107143 (4308) |
110189 (7354) |
111602 (8767) |
114241 (11406) |
117806 (14971) |
120988 (18153) |
121977 (19142) |
124761 (1359) |
126482 (3080) |
127306 (3904) |
129650 (6248) | |