577edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|577}}
{{ED intro}}


== Theory ==
== Theory ==
577et is consistent to the [[7-odd-limit]] and its [[harmonic]] [[3/1|3]] is about halfway its steps. Using the patent val, it tempers out 26873856/26796875, 184528125/184473632 and [[1640558367/1638400000]] in the 7-limit; [[5632/5625]], 151263/151250, 472392/471625 and 102487/102400 in the 11-limit.
577edo is [[consistent]] to the [[7-odd-limit]], but its [[harmonic]] [[3/1|3]] is about halfway its steps. In the 2.9.5.7.11 [[subgroup]] interpretation, it tempers out 26873856/26796875, 184528125/184473632 and [[1640558367/1638400000]] in the 7-limit; [[5632/5625]], 102487/102400, 151263/151250, and 472392/471625 in the 11-limit.


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|577}}
{{Harmonics in equal|577}}


===Subsets and supersets===
=== Subsets and supersets ===
577edo is the 106th [[prime EDO]]. [[1154edo]], which doubles it, gives a good correction to the harmonic 3, but it does poorly in the harmonics 5 and 7. [[2308edo]], which quadruples it, also gives a good correction to the harmonic 3 and its consistent to the [[11-odd-limit]].
577edo is the 106th [[prime edo]]. [[1154edo]], which doubles it, gives a good correction to the harmonic 3, but it does poorly in the harmonics 5 and 7. [[2308edo]], which quadruples it, also gives a good correction to the harmonic 3 and is consistent to the [[11-odd-limit]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| 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
! rowspan="2" | [[Subgroup]]
|{{monzo|-1829 577}}
! rowspan="2" | [[Comma list]]
|{{mapping|577 1829}}
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.9
| {{monzo| -1829 577 }}
| {{mapping| 577 1829 }}
| +0.0153
| +0.0153
| 0.0153
| 0.0153
| 0.74
| 0.74
|-
|-
|2.9.5
| 2.9.5
|{{monzo|-7 11 -12}}, {{monzo|125 -5 -47}}
| {{monzo| -7 11 -12 }}, {{monzo| 125 -5 -47 }}
|{{mapping|577 1829 1340}}
| {{mapping| 577 1829 1340 }}
| -0.0637
| −0.0637
| 0.1124
| 0.1124
| 5.40
| 5.40
|-
|-
|2.9.5.7
| 2.9.5.7
|26873856/26796875, 184528125/184473632, 1640558367/1638400000
| 26873856/26796875, 184528125/184473632, 1640558367/1638400000
|{{mapping|577 1829 1340 1620}}
| {{mapping| 577 1829 1340 1620 }}
| -0.0767
| −0.0767
| 0.0999
| 0.0999
| 4.80
| 4.80
|-
|-
|2.9.5.7.11
| 2.9.5.7.11
|5632/5625, 151263/151250, 472392/471625, 102487/102400
| 5632/5625, 102487/102400, 151263/151250, 472392/471625
|{{mapping|577 1829 1340 1620 1996}}
| {{mapping| 577 1829 1340 1620 1996 }}
| -0.0503
| −0.0503
| 0.1038
| 0.1038
| 4.99
| 4.99
|-
|-
|2.9.5.7.11.13
| 2.9.5.7.11.13
|1001/1000, 10648/10647, 10985/10976, 75712/75625, 472392/471625
| 1001/1000, 10648/10647, 10985/10976, 75712/75625, 472392/471625
|{{mapping|577 1829 1340 1620 1996 2135}}
| {{mapping| 577 1829 1340 1620 1996 2135 }}
| -0.0275
| −0.0275
| 0.1076
| 0.1076
| 5.17
| 5.17
|}
|}
== Music ==
; [[Francium]]
* "iloveinstallations" from ''albumwithoutspaces'' (2024) – [https://open.spotify.com/track/0vmk6yMSwTdBI53cx8XsVM Spotify] | [https://francium223.bandcamp.com/track/iloveinstallations Bandcamp] | [https://www.youtube.com/watch?v=Buv51YyfYdo YouTube]
* "Gore Pile" from ''You Are A...'' (2024) – [https://open.spotify.com/track/0P0WNf7bAm0WHgDlGlGgY5 Spotify] | [https://francium223.bandcamp.com/track/gore-pile Bandcamp] | [https://www.youtube.com/watch?v=DLJdsP5AvdA YouTube]
* "QUACK" from ''CAPSLOCK'' (2024) – [https://open.spotify.com/track/2JVr8aUTYoM4pqSDZ2v8Jc Spotify] | [https://francium223.bandcamp.com/track/quack Bandcamp] | [https://www.youtube.com/watch?v=jnDMYcyEFmk YouTube]
* "Rolling My Eye Ball" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/4M9rZmFwo7Wh7E2FUWcxR3 Spotify] | [https://francium223.bandcamp.com/track/rolling-my-eye-ball Bandcamp] | [https://www.youtube.com/watch?v=j6LEETAtTos YouTube]

Latest revision as of 12:43, 21 February 2025

← 576edo 577edo 578edo →
Prime factorization 577 (prime)
Step size 2.07972 ¢ 
Fifth 338\577 (702.946 ¢)
Semitones (A1:m2) 58:41 (120.6 ¢ : 85.27 ¢)
Dual sharp fifth 338\577 (702.946 ¢)
Dual flat fifth 337\577 (700.867 ¢)
Dual major 2nd 98\577 (203.813 ¢)
Consistency limit 7
Distinct consistency limit 7

577 equal divisions of the octave (abbreviated 577edo or 577ed2), also called 577-tone equal temperament (577tet) or 577 equal temperament (577et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 577 equal parts of about 2.08 ¢ each. Each step represents a frequency ratio of 21/577, or the 577th root of 2.

Theory

577edo is consistent to the 7-odd-limit, but its harmonic 3 is about halfway its steps. In the 2.9.5.7.11 subgroup interpretation, it tempers out 26873856/26796875, 184528125/184473632 and 1640558367/1638400000 in the 7-limit; 5632/5625, 102487/102400, 151263/151250, and 472392/471625 in the 11-limit.

Odd harmonics

Approximation of odd harmonics in 577edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.991 +0.515 +0.325 -0.097 -0.191 -0.320 -0.574 -0.969 -0.113 -0.764 -0.198
Relative (%) +47.7 +24.7 +15.6 -4.7 -9.2 -15.4 -27.6 -46.6 -5.4 -36.7 -9.5
Steps
(reduced) 		915
(338) 		1340
(186) 		1620
(466) 		1829
(98) 		1996
(265) 		2135
(404) 		2254
(523) 		2358
(50) 		2451
(143) 		2534
(226) 		2610
(302)

Subsets and supersets

577edo is the 106th prime edo. 1154edo, which doubles it, gives a good correction to the harmonic 3, but it does poorly in the harmonics 5 and 7. 2308edo, which quadruples it, also gives a good correction to the harmonic 3 and is consistent to the 11-odd-limit.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.9 [-1829 577⟩ [⟨577 1829]] +0.0153 0.0153 0.74
2.9.5 [-7 11 -12⟩, [125 -5 -47⟩ [⟨577 1829 1340]] −0.0637 0.1124 5.40
2.9.5.7 26873856/26796875, 184528125/184473632, 1640558367/1638400000 [⟨577 1829 1340 1620]] −0.0767 0.0999 4.80
2.9.5.7.11 5632/5625, 102487/102400, 151263/151250, 472392/471625 [⟨577 1829 1340 1620 1996]] −0.0503 0.1038 4.99
2.9.5.7.11.13 1001/1000, 10648/10647, 10985/10976, 75712/75625, 472392/471625 [⟨577 1829 1340 1620 1996 2135]] −0.0275 0.1076 5.17

Music

Francium
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