2016edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro | {{EDO 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 | 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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| {{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 48: | 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 | ||
| Line 55: | Line 55: | ||
| 5250987/5242880, 40353607/40310784, {{monzo| 14 11 -22 7 }} | | 5250987/5242880, 40353607/40310784, {{monzo| 14 11 -22 7 }} | ||
| {{mapping| 2016 3195 4681 5659 }} (2016d) | | {{mapping| 2016 3195 4681 5659 }} (2016d) | ||
| 0.060 | | +0.060 | ||
| 0.060 | | 0.060 | ||
| 10.1 | | 10.1 | ||
| Line 62: | Line 62: | ||
| {{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 }} | ||
| {{mapping| 2016 3195 4681 6974}} | | {{mapping| 2016 3195 4681 6974}} | ||
| 0.036 | | +0.036 | ||
| 0.043 | | 0.043 | ||
| 7.3 | | 7.3 | ||
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| 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 | ||
| Line 76: | Line 76: | ||
| 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 | ||
| Line 83: | Line 83: | ||
| 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865 | | 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865 | ||
| {{mapping| 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 | ||
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| [[Jamala]] | | [[Jamala]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <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]] | ||
* [ | * [http://www.youtube.com/watch?v=ILMS8XT1bPs ''Hemoclysm Totem''] (2022) | ||
[[Category:Akjayland]] | [[Category:Akjayland]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||