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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>5</sup> × 3<sup>2</sup> × 7
{{ED intro}}
| Step size = 0.595238¢
| Fifth = 1179\2016 (701.786¢) (&rarr;[[224edo|131\224]])
| Semitones = 189:153 (112.500¢ : 91.071¢)
| Consistency = 5
}}
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.
2016edo shares the mapping for 3 with [[224edo]], albeit with a 28 relative cent error. First 7 prime harmonics with less than 25% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47.


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 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]].  


2016 shares the mapping for 3 with [[224edo]], albeit with a 28 relative cent error.  
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]].


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]].  
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. One such temperament is {{nowrap|311 &amp; 2016}}, produced by stacking 1465\2016, and defined for the 2.5.11.13.19.41 subgroup with the comma basis 16777475/16777216, 1171280/1171001, 615288025/615120896, 1180029296875/1179517976576.  


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]].  
=== Fractional-octave temperaments ===
The patent val 7-limit in 2016edo gives rise to the to rank two temperaments of [[chromium]] with period 24 and the [[akjayland]], period 21. The 2016d val gives rise to {{nowrap|171 &amp; 306}}, period 9 and {{nowrap|270 &amp; 936bd}}, period 18.


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.
In the 2016dijk val, which is tuned better than the patent val, it supports the [[32nd-octave temperaments|dike temperament]], defined as {{nowrap|1600 &amp; 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 &amp; 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 ===
{{Harmonics in equal|2016}}
{{Harmonics in equal|2016}}


=== Fractional octave temperaments ===
=== Subsets and supersets ===
The patent val 7-limit in 2016edo gives rise to the to rank two temperaments of 72 & 624 with period 24 and the [[akjayland]], period 21. The 2016d val gives rise to 171 & 306, period 9 and 270 & 936bd, period 18.
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]], and used by 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.


If we assume that 2016edo is a dual-seventh system, where 5659th and 5660th steps represent 7- and 7+, two distinct dimensions amounting to 49/1, then this allows for mixing of these two temperaments. 236 & 600 is the temperament which best represents that.
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).


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 41: Line 39:
|-
|-
| 2.3.5
| 2.3.5
| {{Monzo| -83 26 18 }}, {{Monzo| 30 47 -45 }}
| {{monzo| -83 26 18 }}, {{monzo| 30 47 -45 }}
| [{{Val| 2016 3195 4681 }}]
| {{mapping| 2016 3195 4681 }}
| 0.036
| +0.036
| 0.050
| 0.050
| 8.4
| 8.4
Line 49: Line 47:
| 2.3.5.7
| 2.3.5.7
| 250047/250000, {{monzo| 7 18 -2 -11 }}, {{monzo| 43 -1 -13 -4 }}
| 250047/250000, {{monzo| 7 18 -2 -11 }}, {{monzo| 43 -1 -13 -4 }}
| [{{Val| 2016 3195 4681 5660 }}]
| {{mapping| 2016 3195 4681 5660 }}
| 0.007
| +0.007
| 0.066
| 0.066
| 11.1
| 11.1
|-
|- style="border-top: double;"
| 2.3.5.7
| 2.3.5.7
| 5250987/5242880, 40353607/40310784, {{monzo| 14 11 -22 7 }}
| 5250987/5242880, 40353607/40310784, {{monzo| 14 11 -22 7 }}
| [{{Val| 2016 3195 4681 5659 }}] (2016d)
| {{mapping| 2016 3195 4681 5659 }} (2016d)
| 0.060
| +0.060
| 0.060
| 0.060
| 10.1
| 10.1
|- style="border-top: double;"
| 2.3.5.11
| {{monzo| 14  8 -10 -1 }}, {{monzo| -26 15 -5  4 }}, {{monzo| -29 27 3 -6 }}
| {{mapping| 2016 3195 4681 6974}}
| +0.036
| 0.043
| 7.3
|-
|-
| 2.5.11.13
| 2.3.5.11.13
| {{monzo| 5 -6 9 6 }}, {{monzo| -38 12 4 -1 }}, {{monzo| 0 -22 3 11 }}
| 196625/196608, 53144100/53094899, {{monzo| 14 8 -10 -1 0 }}, {{monzo| -13 9 5 -8 4 }}
| [{{Val| 2016 4681 6974 7460 }}]
| {{mapping| 2016 3195 4681 6974 7460 }}
| 0.013
| +0.032
| 0.015
| 0.040
| 2.5
| 6.7
|-
|-
| 2.3.5.11.13.17
| 2601/2600, 120285/120224, 140625/140608, 161109/161051, 196625/196608
| {{mapping| 2016 3195 4681 6974 7460 8240}}]
| +0.034
| 0.036
| 6.2
|- style="border-top: double;"
| 2.5.11.13.19.41.47
| 2.5.11.13.19.41.47
| 7943/7942, 322465/322373, 415292/415207, 511225/511024, <br>5078491/5078125, 22151168/22150865
| 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865
| {{Val| 2016 4681 6974 7460 8564 10801 11198 }}
| {{mapping| 2016 4681 6974 7460 8564 10801 11198 }}
| 0.002
| +0.002
| 0.019
| 0.019
| 3.2
| 3.2
|}
|}


[[Category:Equal divisions of the octave]]
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 21
| 983\2016<br />(23\2016)
| 585.119<br />(13.690)
| 91875/65536<br />(126/125)
| [[Akjayland]]
|-
| 24
| 979\2016<br />(55\2016)
| 582.738<br />(32.738)
| 7/5<br />(?)
| [[Chromium]]
|-
| 32
| 29\2016
| 17.262
| (?)
| [[Dike]] (2016dijk)
|-
| 72
| 925\2016<br />(1\2016)
| 550.595<br />(0.595)
| 73205/53248<br />(?)
| [[Jamala]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Music ==
; [[Mercury Amalgam]]
* [http://www.youtube.com/watch?v=ILMS8XT1bPs ''Hemoclysm Totem''] (2022)
 
[[Category:Akjayland]]
[[Category:Akjayland]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/2016edo"