851edo: Difference between revisions
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{{ | {{ED intro}} | ||
851edo is [[consistent]] to the [[15-odd-limit]] or the no-17 no-23 [[25-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] 2401/2400 ([[breedsma]]) and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[3025/3024]] and [[19712/19683]] in the 11-limit; and [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It provides the [[optimal patent val]] for 13-limit [[newt]] | 851edo is [[consistent]] to the [[15-odd-limit]] or the no-17 no-23 [[25-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the [[luna comma]] in the 5-limit; 2401/2400 ([[breedsma]]) and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[3025/3024]] and [[19712/19683]] in the 11-limit; and [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It provides the [[optimal patent val]] for 13-limit [[newt]], the 270 & 581 [[microtemperament]], as well as neonewt, its no-17 19-limit extension. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 851 factors into {{factorization|851}}, 851edo contains [[23edo]] and [[37edo]] as its subsets. | Since 851 factors into {{factorization|851}}, 851edo contains [[23edo]] and [[37edo]] as its subsets. [[1702edo]], which doubles it, provides a strong correction to the 17th and 23rd harmonics, making it notably good as a high-limit no-threes system. | ||
[[Category:Newt]] | [[Category:Newt]] | ||
Latest revision as of 00:07, 12 April 2025
| ← 850edo | 851edo | 852edo → |
851 equal divisions of the octave (abbreviated 851edo or 851ed2), also called 851-tone equal temperament (851tet) or 851 equal temperament (851et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 851 equal parts of about 1.41 ¢ each. Each step represents a frequency ratio of 21/851, or the 851st root of 2.
851edo is consistent to the 15-odd-limit or the no-17 no-23 25-odd-limit. As an equal temperament, it tempers out the luna comma in the 5-limit; 2401/2400 (breedsma) and 33554432/33480783 (garischisma) in the 7-limit; 3025/3024 and 19712/19683 in the 11-limit; and 2080/2079, 4096/4095, and 4225/4224 in the 13-limit. It provides the optimal patent val for 13-limit newt, the 270 & 581 microtemperament, as well as neonewt, its no-17 19-limit extension.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.278 | +0.055 | -0.083 | +0.033 | -0.105 | -0.608 | +0.019 | +0.633 | -0.200 | -0.030 |
| Relative (%) | +0.0 | +19.7 | +3.9 | -5.9 | +2.4 | -7.4 | -43.1 | +1.4 | +44.9 | -14.2 | -2.1 | |
| Steps (reduced) |
851 (0) |
1349 (498) |
1976 (274) |
2389 (687) |
2944 (391) |
3149 (596) |
3478 (74) |
3615 (211) |
3850 (446) |
4134 (730) |
4216 (812) | |
Subsets and supersets
Since 851 factors into 23 × 37, 851edo contains 23edo and 37edo as its subsets. 1702edo, which doubles it, provides a strong correction to the 17th and 23rd harmonics, making it notably good as a high-limit no-threes system.