2016edo: Difference between revisions
-rank-2 temps (until they're named; otherwise there would be infinitely many); misc cleanup |
Correct infobox, misc readability improvements, and +category |
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{{Infobox ET | {{Infobox ET | ||
| Prime factorization = 2<sup>5</sup> × 3<sup>2</sup> × 7 | | Prime factorization = 2<sup>5</sup> × 3<sup>2</sup> × 7 | ||
| Step size = 0. | | Step size = 0.595238¢ | ||
| Fifth = 1179\2016 (701. | | Fifth = 1179\2016 (701.786¢) (→[[224edo|131\224]]) | ||
| Semitones = | | Semitones = 189:153 (112.500¢ : 91.071¢) | ||
}} | |||
The '''2016 equal divisions of the octave''' ('''2016edo'''), or the '''2016-tone equal temperament''' ('''2016tet'''), '''2016 equal temperament''' ('''2016et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 2016 [[equal]] parts of about 595 milli[[cent]]s each. | The '''2016 equal divisions of the octave''' ('''2016edo'''), or the '''2016-tone equal temperament''' ('''2016tet'''), '''2016 equal temperament''' ('''2016et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 2016 [[equal]] parts of about 595 milli[[cent]]s each. | ||
== Theory == | == Theory == | ||
2016 is a significantly composite number, with its divisors being {{EDOs| 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008 }}. Its abundancy index is 2.25. | |||
2016 is a significantly composite number, with its divisors being {{EDOs|1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008}}. | |||
Prime harmonics (below 61) with less than 22% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47. With next error being 26% on the 37th harmonic, it is reasonable to make cutoff here. | Prime harmonics (below 61) with less than 22% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47. With next error being 26% on the 37th harmonic, it is reasonable to make cutoff here. | ||
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2016 shares the mapping for 3 with [[224edo]], albeit with a 28 relative cent error. | 2016 shares the mapping for 3 with [[224edo]], albeit with a 28 relative cent error. | ||
2016edo has two reasonable mappings for 7. The 2016d val, {{ | 2016edo has two reasonable mappings for 7. The 2016d val, {{val| 2016 3195 4681 5659 }}, tempers out 5250987/5242880, 40353607/40310784 (tritrizo), and {{monzo| 14 11 -22 7 }}. As such, its circle of the interval 7/6 is the same as in [[9edo]]. | ||
The patent val, {{val| 2016 3195 4681 5658 }} tempers out [[250047/250000]], along with {{monzo| 7 18 -2 -11 }} and {{monzo| 43 -1 -13 -4 }}. This means that the symmetrical major third (400 cents, 1/3 of the octave) in 2016edo corresponds to [[63/50]]. | |||
In the 11-limit, 2016edo tempers out the {{monzo| 0 0 -22 0 3 11 }} comma, which equates a stack of eleven [[25/13]]'s with three [[11/1]]'s. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|2016}} | |||
=== Fractional octave temperaments === | === Fractional octave temperaments === | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
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| 3.2 | | 3.2 | ||
|} | |} | ||
[[Category:Equal divisions of the octave]] | |||