569edo: Difference between revisions

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.

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
Error	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)

569edo is the 104th prime edo.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	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

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 it is distinct

"Cuckoo Pickpocket" from Cursed Cuckoo Creations (2024) – Spotify | Bandcamp | YouTube
"ME" from CAPSLOCK (2024) – Spotify | Bandcamp | YouTube

@@ Line 12: / Line 12: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| 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
@@ Line 41: / Line 50: @@
 | 0.1834
 | 8.70
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| 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
@@ Line 51: / Line 67: @@
 | 243/200
 | [[Paramity]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == Music ==