2016edo: Difference between revisions

{{Infobox ET}}

{{ED intro}}

 

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.

 

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]].

 

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 {{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.  

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 {{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}.

{{Harmonics in equal|2016}}

=== 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]], 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.

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).

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br />8ve stretch (¢)

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

| 2.3.5

| {{monzo| -83 26 18 }}, {{monzo| 30 47 -45 }}

| {{mapping| 2016 3195 4681 }}

| +0.036

| 0.050

| 8.4

| 2.3.5.7

| 5250987/5242880, 40353607/40310784, {{monzo| 14 11 -22 7 }}

| {{mapping| 2016 3195 4681 5659 }} (2016d)

| +0.060

| 0.060

| 10.1

| 2.3.5.11.13

| 196625/196608, 53144100/53094899, {{monzo| 14 8 -10 -1 0 }}, {{monzo| -13 9 5 -8 4 }}

| {{mapping| 2016 3195 4681 6974 7460 }}

| +0.032

| 0.040

| 6.7

| 2.5.11.13.19.41.47

| 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865

| {{mapping| 2016 4681 6974 7460 8564 10801 11198 }}

| +0.002

| 0.019

| 3.2

=== Rank-2 temperaments ===

| 24

| 29\2016

| 17.262

| 72