User:Francium/5413edo: Difference between revisions
Created page with "{{Infobox ET}} {{ED intro}} == Theory == 5413edo is inconsistent to the 5-limit and its harmonic 3 is about halfway its steps. As a equal temperament, 5413edo is strong in the 2.9.15.11.19.23.31 subgroup, tempering out 178125/178112, 253952/253935, 173861875/173840256, 44921875/44914176, 14156615/14155776 and 484525998080/484404751413. === Odd harmonics === {{Harmonics in equal|5413}} === Subsets and supersets === 5413edo is the 714th pri..." |
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=== Subsets and supersets === | === Subsets and supersets === | ||
5413edo is the 714th [[prime edo]]. | 5413edo is the 714th [[prime edo]]. [[10826edo]], which doubles it, gives a good correction to its harmonics 3 and [[5/1|5]]. | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" |[[Subgroup]] | |||
! rowspan="2" |[[Comma list|Comma List]] | |||
! rowspan="2" |[[Mapping]] | |||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | |||
! colspan="2" |Tuning Error | |||
|- | |||
![[TE error|Absolute]] (¢) | |||
![[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.9 | |||
| {{monzo|17159 -5413}} | |||
| {{mapping|5413 17159}} | |||
| −0.0069 | |||
| 0.0069 | |||
| 3.11 | |||
|- | |||
| 2.9.15 | |||
| {{monzo|200 -52 -9}}, {{monzo|-49 -93 88}} | |||
| {{mapping|5413 17159 21148}} | |||
| −0.0046 | |||
| 0.0064 | |||
| 2.89 | |||
|} | |||
Revision as of 16:07, 11 February 2026
| ← 5412edo | 5413edo | 5414edo → |
5413 equal divisions of the octave (abbreviated 5413edo or 5413ed2), also called 5413-tone equal temperament (5413tet) or 5413 equal temperament (5413et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 5413 equal parts of about 0.222 ¢ each. Each step represents a frequency ratio of 21/5413, or the 5413th root of 2.
Theory
5413edo is inconsistent to the 5-limit and its harmonic 3 is about halfway its steps. As a equal temperament, 5413edo is strong in the 2.9.15.11.19.23.31 subgroup, tempering out 178125/178112, 253952/253935, 173861875/173840256, 44921875/44914176, 14156615/14155776 and 484525998080/484404751413.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.0891 | +0.0894 | -0.0470 | +0.0434 | +0.0214 | -0.1065 | +0.0003 | -0.0967 | -0.0070 | +0.0855 | -0.0091 |
| Relative (%) | -40.2 | +40.3 | -21.2 | +19.6 | +9.7 | -48.0 | +0.1 | -43.6 | -3.2 | +38.6 | -4.1 | |
| Steps (reduced) |
8579 (3166) |
12569 (1743) |
15196 (4370) |
17159 (920) |
18726 (2487) |
20030 (3791) |
21148 (4909) |
22125 (473) |
22994 (1342) |
23776 (2124) |
24486 (2834) | |
Subsets and supersets
5413edo is the 714th prime edo. 10826edo, which doubles it, gives a good correction to its harmonics 3 and 5.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [17159 -5413⟩ | [⟨5413 17159]] | −0.0069 | 0.0069 | 3.11 |
| 2.9.15 | [200 -52 -9⟩, [-49 -93 88⟩ | [⟨5413 17159 21148]] | −0.0046 | 0.0064 | 2.89 |