2016edo: Difference between revisions
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== Theory == | == Theory == | ||
{{ | {{Harmonics in equal|2016}} | ||
2016 is a significantly composite number, with its divisors 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}}. It's abundancy index is 2.25. | 2016 is a significantly composite number, with its divisors 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}}. 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. Using the 2016f val gives the same mapping for 13 as [[224edo]], and unleashes the full power of 224edo's 13 limit chords. | 2016 shares the mapping for 3 with [[224edo]], albeit with a 28 relative cent error. Using the 2016f val gives the same mapping for 13 as [[224edo]], and unleashes the full power of 224edo's 13 limit chords. | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" |Subgroup | |||
! rowspan="2" |[[Comma list]] | |||
(zeroes skipped for clarity) | |||
! rowspan="2" |[[Mapping]] | |||
! rowspan="2" |Optimal | |||
8ve stretch (¢) | |||
! colspan="2" |Tuning error | |||
|- | |||
![[TE error|Absolute]] (¢) | |||
![[TE simple badness|Relative]] (%) | |||
|- | |||
|2.3.5 | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
|2.5.11.13 | |||
|{{monzo|5 -6 9 6}}, {{monzo|-38 12 4 -1}}, {{monzo|0 -22 3 11}} | |||
| | |||
|0.013 | |||
|0.015 | |||
|2.5 | |||
|} | |||
Revision as of 10:05, 11 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. Using the 2016f val gives the same mapping for 13 as 224edo, and unleashes the full power of 224edo's 13 limit chords.
Regular temperament properties
| Subgroup | Comma list
(zeroes skipped for clarity) |
Mapping | Optimal
8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | |||||
| 2.5.11.13 | [5 -6 9 6⟩, [-38 12 4 -1⟩, [0 -22 3 11⟩ | 0.013 | 0.015 | 2.5 | |