2016edo: Difference between revisions
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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 11 [[25/13]]<nowiki/>s with 3 [[11/1|hendecataves]]. | In the 11-limit, 2016edo tempers out the {{monzo|0 0 -22 0 3 11}} comma, which equates a stack of 11 [[25/13]]<nowiki/>s with 3 [[11/1|hendecataves]]. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
Revision as of 18:58, 22 March 2022
| ← 2015edo | 2016edo | 2017edo → |
2016 equal division divides the octave into steps of 595 millicents, or 25/42 cent each.
Theory
| 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) | |
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. It's abundancy index is 2.25.
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.
2016 shares the mapping for 3 with 224edo, albeit with a 28 relative cent error.
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 11 25/13s with 3 hendecataves.
Regular temperament properties
| Subgroup | Comma list
(zeroes skipped for clarity) |
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 5658]] | 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 |
| 2.5.11.13 | [5 -6 9 6⟩, [-38 12 4 -1⟩, [0 -22 3 11⟩ | [⟨2016 4681 6974 7460]] | 0.013 | 0.015 | 2.5 |
| 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 |