User:Francium/5413edo: Difference between revisions

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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..."
 
Francium (talk | contribs)
4,816 edits
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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
|-
| 2.9.15.7
| 4747561509943/4746093750000, 27033764064453125/27021597764222976, 16689300537109375/16677181699666569
| {{mapping|5413 17159 21148 15196}}
| +0.0007
| 0.0108
| 4.87
|-
| 2.9.15.7.11
| 151263/151250, 246071287/246037500, 369140625/369098752, 409600000000000000/409370033518213779
| {{mapping|5413 17159 21148 15196 18726}}
| −0.0006
| 0.0100
| 4.51
|}
 
== Music ==
; [[Francium]]
* "Artificial Gorilla Blood" from ''Check Us Out'' (2026) – [https://open.spotify.com/track/7jvngwlo9OYUB34VFfG6ip Spotify] | [https://francium223.bandcamp.com/track/artificial-gorilla-blood Bandcamp] | [https://www.youtube.com/watch?v=hrS4KLSPimM YouTube] – in Godzilla, 5413edo tuning

Latest revision as of 11:29, 26 April 2026

← 5412edo 5413edo 5414edo →
Prime factorization 5413 (prime)
Step size 0.221689 ¢ 
Fifth 3166\5413 (701.866 ¢)
Semitones (A1:m2) 510:409 (113.1 ¢ : 90.67 ¢)
Dual sharp fifth 3167\5413 (702.088 ¢)
Dual flat fifth 3166\5413 (701.866 ¢)
Dual major 2nd 920\5413 (203.953 ¢)
Consistency limit 3
Distinct consistency limit 3

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

Approximation of odd harmonics in 5413edo
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
2.9.15.7 4747561509943/4746093750000, 27033764064453125/27021597764222976, 16689300537109375/16677181699666569 [⟨5413 17159 21148 15196]] +0.0007 0.0108 4.87
2.9.15.7.11 151263/151250, 246071287/246037500, 369140625/369098752, 409600000000000000/409370033518213779 [⟨5413 17159 21148 15196 18726]] −0.0006 0.0100 4.51

Music

Francium
  • "Artificial Gorilla Blood" from Check Us Out (2026) – Spotify | Bandcamp | YouTube – in Godzilla, 5413edo tuning
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