2016edo: Difference between revisions
→Rank two temperaments by generator: documenting chromium as it is named now |
→Odd harmonics: expanded, clarified |
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{{Harmonics in equal|2016}} | {{Harmonics in equal|2016}} | ||
2016edo is consistent in the no-7s 17-limit. 2016 shares the mapping for 3 with [[224edo]], albeit with a 28 relative cent error. 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. | 2016edo is naively consistent in the no-7s 17-limit. 2016 shares the mapping for 3 with [[224edo]], albeit with a 28 relative cent error. 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, and the fact that error below 25% guarantees consistency to distance 1, it is reasonable to make cutoff here. | ||
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]]. | 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. However, it does ''not'' temper out the [[jacobin comma]]. | 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. However, it does '''not''' temper out the [[jacobin comma]]. | ||
2016 has a total of 576 numbers coprime to it, which means this is how many generators can reach any point in the octave by being stacked. | 2016 has a total of 576 numbers coprime to it, which means this is how many generators can reach any point in the octave by being stacked. | ||
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In the 2016dijk val it supports the [[32nd-octave temperaments|dike temperament]], defined as 1600 & 2016dijk in the 37-limit with period 32. | In the 2016dijk val it supports the [[32nd-octave temperaments|dike temperament]], defined as 1600 & 2016dijk in the 37-limit with period 32. | ||
In the 2.5.11.13.19.41.47, 2016edo supports the period 72 Jamala temperament, defined as 1944 & 2016 and named after an eponymous song. | In the 2.5.11.13.19.41.47, 2016edo supports the period 72 Jamala temperament, defined as 1944 & 2016 and named after an eponymous song. It has a comma basis 47012251/47000000, 2502280/2501369, 2680291328/2679296875, 410041489/410000000, 52448351813/52428800000. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 118: | Line 118: | ||
|32 | |32 | ||
|29\2016 | |29\2016 | ||
|17. | |17.262 | ||
|(?) | |(?) | ||
|[[Dike]] (2016dijk) | |[[Dike]] (2016dijk) | ||
|- | |- | ||
|72 | |72 | ||
| | |925\2016<br>(1\2016) | ||
| | |550.595<br>(0.595) | ||
|73205/53248<br>(?) | |73205/53248<br>(?) | ||
|[[Jamala]] | |[[Jamala]] | ||