2016edo: Difference between revisions
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== Theory == | == Theory == | ||
2016edo shares the mapping for 3 with [[224edo]], albeit with a 28 relative | 2016edo shares the mapping for 3 with [[224edo]], albeit with a 28% relative error. First 7 prime harmonics with less than 25% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47. | ||
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's patent val 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's patent val corresponds to [[63/50]]. | ||
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In the 2016dijk val, which is tuned better than the patent val, it supports the [[32nd-octave temperaments|dike temperament]], defined as {{nowrap|1600 & 2016dijk}} in the 37-limit with period 32. | In the 2016dijk val, which is tuned better than the patent val, it supports the [[32nd-octave temperaments|dike temperament]], defined as {{nowrap|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 {{nowrap|1944 & 2016}} and named after an eponymous song. It has a comma basis of {47012251/47000000, 2502280/2501369, 2680291328/2679296875, 410041489/410000000, 52448351813/52428800000 | In the 2.5.11.13.19.41.47, 2016edo supports the period 72 Jamala temperament, defined as {{nowrap|1944 & 2016}} and named after an eponymous song. It has a comma basis of {47012251/47000000, 2502280/2501369, 2680291328/2679296875, 410041489/410000000, 52448351813/52428800000}. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
2016 is a significantly composite number, with its subset edos 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. Some of its divisors have found applied use. 72edo has been used in [[Wikipedia:Byzantine music|Byzantine chanting]], has been theoreticized by [[Wikipedia:Alois Hába|Alois Haba]] and [[Ivan Wyschnegradsky]] | 2016 is a significantly composite number, with its subset edos 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. Some of its divisors have found applied use. 72edo has been used in [[Wikipedia:Byzantine music|Byzantine chanting]], has been theoreticized by [[Wikipedia:Alois Hába|Alois Haba]] and used by [[Ivan Wyschnegradsky]] and jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]], and 224edo is a member of [[The Riemann zeta function and tuning|zeta]] edos. | ||
2016 is a divisor of some [[highly composite edo]]s, such as [[10080edo]], [[20160edo]], etc. As a subset of 20160edo, one step of 2016edo is exactly 10 pians (10\20160). | 2016 is a divisor of some [[highly composite edo]]s, such as [[10080edo]], [[20160edo]], etc. As a subset of 20160edo, one step of 2016edo is exactly 10 pians (10\20160). | ||
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| [[Jamala]] | | [[Jamala]] | ||
|} | |} | ||
<nowiki />* [[Normal | <nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
== Music == | == Music == | ||