2016edo: Difference between revisions
→Rank two temperaments by generator: documenting chromium as it is named now |
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=== Fractional octave temperaments === | === Fractional octave temperaments === | ||
The patent val 7-limit in 2016edo gives rise to the to rank two temperaments of | 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. | ||
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 on the 2.3.5.49 subgroup. 236 & 600 is the temperament which best represents that. | 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 on the 2.3.5.49 subgroup. 236 & 600 is the temperament which best represents that. | ||
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|91875/65536<br>(126/125) | |91875/65536<br>(126/125) | ||
|[[Akjayland]] | |[[Akjayland]] | ||
|- | |||
|24 | |||
|979\2016<br>(55\2016) | |||
|582.738 | |||
(32.738) | |||
|7/5 | |||
(?) | |||
|[[Chromium]] | |||
|- | |- | ||
|32 | |32 | ||
Revision as of 17:53, 23 October 2022
| ← 2015edo | 2016edo | 2017edo → |
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 millicents each.
Theory
2016 is a significantly composite number, with its divisors being 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 Byzantine chanting, has been theoreticized by 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 zeta edos.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.169 | -0.004 | +0.222 | +0.257 | -0.127 | -0.051 | -0.173 | -0.194 | +0.106 | +0.052 | +0.297 |
| Relative (%) | -28.4 | -0.7 | +37.2 | +43.1 | -21.4 | -8.6 | -29.1 | -32.5 | +17.8 | +8.8 | +49.9 | |
| Steps (reduced) |
3195 (1179) |
4681 (649) |
5660 (1628) |
6391 (343) |
6974 (926) |
7460 (1412) |
7876 (1828) |
8240 (176) |
8564 (500) |
8855 (791) |
9120 (1056) | |
2016edo is consistent in the no-7s 17-limit. 2016 shares the mapping for 3 with 224edo, albeit with a 28 relative cent error. 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, ⟨2016 3195 4681 5659], tempers out 5250987/5242880, 40353607/40310784 (tritrizo), and [14 11 -22 7⟩. As such, its circle of the interval 7/6 is the same as in 9edo. The patent val, ⟨2016 3195 4681 5658] tempers out 250047/250000, along with [7 18 -2 -11⟩ and [43 -1 -13 -4⟩. This means that the symmetrical major third (400 cents, 1/3 of the octave) in 2016edo corresponds to 63/50.
In the 11-limit, 2016edo tempers out the [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.
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.
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 on the 2.3.5.49 subgroup. 236 & 600 is the temperament which best represents that.
In the 2016dijk val it supports the dike temperament, defined as 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.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [-83 26 18⟩, [30 47 -45⟩ | [⟨2016 3195 4681]] | 0.036 | 0.050 | 8.4 |
| 2.3.5.7 | 250047/250000, [7 18 -2 -11⟩, [43 -1 -13 -4⟩ | [⟨2016 3195 4681 5660]] | 0.007 | 0.066 | 11.1 |
| 2.3.5.7 | 5250987/5242880, 40353607/40310784, [14 11 -22 7⟩ | [⟨2016 3195 4681 5659]] (2016d) | 0.060 | 0.060 | 10.1 |
| 2.3.5.11 | [14 8 -10 -1⟩, [-26 15 -5 4⟩, [-29 27 3 -6⟩ | [⟨2016 3195 4681 6974]] | 0.036 | 0.043 | 7.3 |
| 2.3.5.11.13 | 196625/196608, 53144100/53094899, [14 8 -10 -1 0⟩, [-13 9 5 -8 4⟩ |
[⟨2016 3195 4681 6974 7460]] | 0.032 | 0.040 | 6.7 |
| 2.3.5.11.13.17 | 2601/2600, 120285/120224, 140625/140608, 161109/161051, 196625/196608 |
[⟨2016 3195 4681 6974 7460 8240]] | 0.034 | 0.036 | 6.2 |
| 2.5.11.13.19.41.47 | 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865 |
[⟨2016 4681 6974 7460 8564 10801 11198]] | 0.002 | 0.019 | 3.2 |
Rank two temperaments by generator
| Periods
per octave |
Generator
(reduced) |
Cents
(reduced) |
Associated
ratio |
Temperaments |
|---|---|---|---|---|
| 21 | 983\2016 (23\2016) |
585.119 (13.690) |
91875/65536 (126/125) |
Akjayland |
| 24 | 979\2016 (55\2016) |
582.738
(32.738) |
7/5
(?) |
Chromium |
| 32 | 29\2016 | 17.2619 | (?) | Dike (2016dijk) |
| 72 | (?) 1\2016 |
(?) 0.5953 |
73205/53248 (?) |
Jamala |
Music
- Hemoclysm Totem by Mercury Amalgam