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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|2016}}
{{ED intro}}


== Theory ==
== Theory ==
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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. One such temperament is 311 & 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.  
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 & 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.  


=== Fractional-octave temperaments ===
=== 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 171 & 306, period 9 and 270 & 936bd, period 18.
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 & 306}}, period 9 and {{nowrap|270 & 936bd}}, period 18.


In the 2016dijk val, which is tuned better than the patent val, it supports the [[32nd-octave temperaments|dike temperament]], defined as 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 1944 & 2016 and named after an eponymous song. It has a comma basis 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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== 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|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| {{monzo| -83 26 18 }}, {{monzo| 30 47 -45 }}
| {{monzo| -83 26 18 }}, {{monzo| 30 47 -45 }}
| {{mapping| 2016 3195 4681 }}
| {{mapping| 2016 3195 4681 }}
| 0.036
| +0.036
| 0.050
| 0.050
| 8.4
| 8.4
Line 47: Line 48:
| 250047/250000, {{monzo| 7 18 -2 -11 }}, {{monzo| 43 -1 -13 -4 }}
| 250047/250000, {{monzo| 7 18 -2 -11 }}, {{monzo| 43 -1 -13 -4 }}
| {{mapping| 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;"
| style="border-top: double;" | 2.3.5.7
| 2.3.5.7
| style="border-top: double;" | 5250987/5242880, 40353607/40310784, {{monzo| 14 11 -22 7 }}
| 5250987/5242880, 40353607/40310784, {{monzo| 14 11 -22 7 }}
| style="border-top: double;" | {{mapping| 2016 3195 4681 5659 }} (2016d)
| {{mapping| 2016 3195 4681 5659 }} (2016d)
| style="border-top: double;" | 0.060
| +0.060
| style="border-top: double;" | 0.060
| 0.060
| style="border-top: double;" | 10.1
| 10.1
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5.11
| 2.3.5.11
| style="border-top: double;" | {{monzo| 14  8 -10 -1 }}, {{monzo| -26 15 -5  4 }}, {{monzo| -29 27 3 -6 }}
| {{monzo| 14  8 -10 -1 }}, {{monzo| -26 15 -5  4 }}, {{monzo| -29 27 3 -6 }}
| style="border-top: double;" | {{mapping| 2016 3195 4681 6974}}
| {{mapping| 2016 3195 4681 6974}}
| style="border-top: double;" | 0.036
| +0.036
| style="border-top: double;" | 0.043
| 0.043
| style="border-top: double;" | 7.3
| 7.3
|-
|-
| 2.3.5.11.13
| 2.3.5.11.13
| 196625/196608, 53144100/53094899, {{monzo| 14 8 -10 -1 0 }}, {{monzo| -13 9 5 -8 4 }}
| 196625/196608, 53144100/53094899, {{monzo| 14 8 -10 -1 0 }}, {{monzo| -13 9 5 -8 4 }}
| {{mapping| 2016 3195 4681 6974 7460 }}
| {{mapping| 2016 3195 4681 6974 7460 }}
| 0.032
| +0.032
| 0.040
| 0.040
| 6.7
| 6.7
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| 2601/2600, 120285/120224, 140625/140608, 161109/161051, 196625/196608
| 2601/2600, 120285/120224, 140625/140608, 161109/161051, 196625/196608
| {{mapping| 2016 3195 4681 6974 7460 8240}}]
| {{mapping| 2016 3195 4681 6974 7460 8240}}]
| 0.034
| +0.034
| 0.036
| 0.036
| 6.2
| 6.2
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.5.11.13.19.41.47
| 2.5.11.13.19.41.47
| style="border-top: double;" | 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865
| 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865
| style="border-top: double;" | {{mapping| 2016 4681 6974 7460 8564 10801 11198 }}
| {{mapping| 2016 4681 6974 7460 8564 10801 11198 }}
| style="border-top: double;" | 0.002
| +0.002
| style="border-top: double;" | 0.019
| 0.019
| style="border-top: double;" | 3.2
| 3.2
|}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
| 21
| 21
| 983\2016<br>(23\2016)
| 983\2016<br />(23\2016)
| 585.119<br>(13.690)
| 585.119<br />(13.690)
| 91875/65536<br>(126/125)
| 91875/65536<br />(126/125)
| [[Akjayland]]
| [[Akjayland]]
|-
|-
| 24
| 24
| 979\2016<br>(55\2016)
| 979\2016<br />(55\2016)
| 582.738<br>(32.738)
| 582.738<br />(32.738)
| 7/5<br>(?)
| 7/5<br />(?)
| [[Chromium]]
| [[Chromium]]
|-
|-
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|-
|-
| 72
| 72
| 925\2016<br>(1\2016)
| 925\2016<br />(1\2016)
| 550.595<br>(0.595)
| 550.595<br />(0.595)
| 73205/53248<br>(?)
| 73205/53248<br />(?)
| [[Jamala]]
| [[Jamala]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Music ==
== Music ==
; [[Mercury Amalgam]]
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=ILMS8XT1bPs ''Hemoclysm Totem''] (2022)
* [http://www.youtube.com/watch?v=ILMS8XT1bPs ''Hemoclysm Totem''] (2022)


[[Category:Akjayland]]
[[Category:Akjayland]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/2016edo"