569edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|569}} == Theory == 569et is only consistent to the 5-odd-limit. Using the patent val, it tempers out 1220703125/1219784832, 33554432/33480783..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|569}}
{{ED intro}}


== Theory ==
== Theory ==
569et is only consistent to the [[5-odd-limit]]. Using the patent val, it tempers out 1220703125/1219784832, [[33554432/33480783]], [[65625/65536]] and 200120949/200000000 in the 7-limit; 95703125/95664294, 26214400/26198073, 1953125/1951488, 151263/151250, [[200704/200475]], 1879453125/1879048192, 422576/421875, 160083/160000, 16808715/16777216, 42592/42525 and 43923/43904 in the 11-limit.
569edo is only [[consistent]] to the [[5-odd-limit]]. The equal temperament [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) in the 5-limit. Using the [[patent val]], it [[tempering out|tempers out]] [[65625/65536]], 823543/820125, and [[33554432/33480783]] in the 7-limit; 42592/42525, 42875/42768, 43923/43904, and 151263/151250 in the 11-limit.


=== Prime harmonics ===
=== Prime harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
569edo is the 104th [[prime EDO]].
569edo is the 104th [[prime edo]].


== 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]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|902 -569}}
! rowspan="2" | [[Comma list]]
|{{mapping|569 902}}
! rowspan="2" | [[Mapping]]
| -0.1040
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 902 -569 }}
| {{mapping| 569 902 }}
| −0.1040
| 0.1040
| 0.1040
| 4.93
| 4.93
|-
|-
|2.3.5
| 2.3.5
|1600000/1594323, {{monzo|-94 -11 48}}
| 1600000/1594323, {{monzo| -94 -11 48 }}
|{{mapping|569 902 1321}}
| {{mapping| 569 902 1321 }}
| -0.0157
| −0.0157
| 0.1510
| 0.1510
| 7.16
| 7.16
|-
|-
|2.3.5.7
| 2.3.5.7
|823543/820125, 65625/65536, 1600000/1594323
| 65625/65536, 823543/820125, 1600000/1594323
|{{mapping|569 902 1321 1597}}
| {{mapping| 569 902 1321 1597 }}
| +0.0605
| +0.0605
| 0.1858
| 0.1858
| 8.81
| 8.81
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|42875/42768, 42592/42525, 160083/160000, 151263/151250
| 42592/42525, 42875/42768, 43923/43904, 151263/151250
|{{mapping|569 902 1321 1597 1968}}
| {{mapping| 569 902 1321 1597 1968 }}
| +0.0992
| +0.0992
| 0.1834
| 0.1834
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|161\569
| 161\569
|339.54
| 339.54
|243/200
| 243/200
|[[Amity]]
| [[Paramity]]
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
== Music ==
; [[Francium]]
* "Cuckoo Pickpocket" from ''Cursed Cuckoo Creations'' (2024) – [https://open.spotify.com/track/6YZU6zGbFwGcEt5WDEzRFA Spotify] | [https://francium223.bandcamp.com/track/cuckoo-pickpocket Bandcamp] | [https://www.youtube.com/watch?v=DWNS4bGwFKg YouTube]
* "ME" from ''CAPSLOCK'' (2024) – [https://open.spotify.com/track/6taed4kRYHcc2woLVzjNj7 Spotify] | [https://francium223.bandcamp.com/track/me Bandcamp] | [https://www.youtube.com/watch?v=OlvXsg9GJF8 YouTube]

Latest revision as of 13:33, 13 March 2026

← 568edo 569edo 570edo →
Prime factorization 569 (prime)
Step size 2.10896 ¢ 
Fifth 333\569 (702.285 ¢)
Semitones (A1:m2) 55:42 (116 ¢ : 88.58 ¢)
Consistency limit 5
Distinct consistency limit 5

569 equal divisions of the octave (abbreviated 569edo or 569ed2), also called 569-tone equal temperament (569tet) or 569 equal temperament (569et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 569 equal parts of about 2.11 ¢ each. Each step represents a frequency ratio of 21/569, or the 569th root of 2.

Theory

569edo is only consistent to the 5-odd-limit. The equal temperament tempers out 1600000/1594323 (amity comma) in the 5-limit. Using the patent val, it tempers out 65625/65536, 823543/820125, and 33554432/33480783 in the 7-limit; 42592/42525, 42875/42768, 43923/43904, and 151263/151250 in the 11-limit.

Prime harmonics

Approximation of prime harmonics in 569edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.330 -0.373 -0.812 -0.879 +0.949 +0.493 -0.149 +0.197 -0.403 +0.131
Relative (%) +0.0 +15.6 -17.7 -38.5 -41.7 +45.0 +23.4 -7.1 +9.3 -19.1 +6.2
Steps
(reduced) 		569
(0) 		902
(333) 		1321
(183) 		1597
(459) 		1968
(261) 		2106
(399) 		2326
(50) 		2417
(141) 		2574
(298) 		2764
(488) 		2819
(543)

Subsets and supersets

569edo is the 104th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [902 -569⟩ [⟨569 902]] −0.1040 0.1040 4.93
2.3.5 1600000/1594323, [-94 -11 48⟩ [⟨569 902 1321]] −0.0157 0.1510 7.16
2.3.5.7 65625/65536, 823543/820125, 1600000/1594323 [⟨569 902 1321 1597]] +0.0605 0.1858 8.81
2.3.5.7.11 42592/42525, 42875/42768, 43923/43904, 151263/151250 [⟨569 902 1321 1597 1968]] +0.0992 0.1834 8.70

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 161\569 339.54 243/200 Paramity

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Francium
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