569edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo| 902 -569 }} | | {{monzo| 902 -569 }} | ||
| {{mapping| 569 902 }} | | {{mapping| 569 902 }} | ||
| | | −0.1040 | ||
| 0.1040 | | 0.1040 | ||
| 4.93 | | 4.93 | ||
| Line 24: | Line 33: | ||
| 1600000/1594323, {{monzo| -94 -11 48 }} | | 1600000/1594323, {{monzo| -94 -11 48 }} | ||
| {{mapping| 569 902 1321 }} | | {{mapping| 569 902 1321 }} | ||
| | | −0.0157 | ||
| 0.1510 | | 0.1510 | ||
| 7.16 | | 7.16 | ||
| Line 41: | Line 50: | ||
| 0.1834 | | 0.1834 | ||
| 8.70 | | 8.70 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 51: | Line 67: | ||
| 243/200 | | 243/200 | ||
| [[Paramity]] | | [[Paramity]] | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Music == | == Music == | ||
; [[Francium]] | ; [[Francium]] | ||
* "Cuckoo Pickpocket" from ''Cursed Cuckoo Creations'' (2024) | * "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) | * "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 06:02, 21 February 2025
| ← 568edo | 569edo | 570edo → |
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
| 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
| 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