584edo

← 583edo

584edo

585edo →

Prime factorization	23 × 73
Step size	2.05479¢
Fifth	342\584 (702.74¢) (→171\292)
Semitones (A1:m2)	58:42 (119.2¢ : 86.3¢)
Dual sharp fifth	342\584 (702.74¢) (→171\292)
Dual flat fifth	341\584 (700.685¢)
Dual major 2nd	99\584 (203.425¢)
Consistency limit	5
Distinct consistency limit	5

584 equal divisions of the octave (abbreviated 584edo or 584ed2), also called 584-tone equal temperament (584tet) or 584 equal temperament (584et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 584 equal parts of about 2.05 ¢ each. Each step represents a frequency ratio of 2^1/584, or the 584th root of 2.

Theory

584edo is only consistent to the 5-odd-limit and the error of harmonic 3 is quite large. With reasonable approximations to harmonics 5, 7, 9, 11, 13, and 17, it commends itself as a 2.9.5.7.11.13.17 subgroup tuning.

If we use the harmonic 3 instead, we notice the better-tuned 584d val is enfactored, with the same tuning as 292edo. Therefore, we are left with the patent val, which tempers out 48828125/48771072 and 67108864/66976875, supporting hemiluna.

Odd harmonics

Approximation of odd harmonics in 584edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.785	-0.012	-1.018	-0.485	-0.633	-0.117	+0.772	-0.161	+0.432	-0.233	+0.493
	Relative (%)	+38.2	-0.6	-49.5	-23.6	-30.8	-5.7	+37.6	-7.8	+21.0	-11.3	+24.0
Steps (reduced)		926 (342)	1356 (188)	1639 (471)	1851 (99)	2020 (268)	2161 (409)	2282 (530)	2387 (51)	2481 (145)	2565 (229)	2642 (306)

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[463 -292⟩	[⟨584 926]]	−0.2476	0.2475	12.05
2.3.5	[3 -18 11⟩, [21 -20 4⟩	[⟨584 926 1356]]	−0.1633	0.2346	11.42
2.3.5.7	1500625/1492992, 1605632/1594323, 235298/234375	[⟨584​ 926 ​1356 ​1639]]	−0.0319	0.3052	14.85
2.3.5.7.11	5632/5625, 160083/160000, 26411/26244, 968000/964467	[⟨584​ 926 ​1356 ​1639​ 2020​]]	+0.0111	0.2862	13.93
2.3.5.7.11.13	2080/2079, 1001/1000, 4096/4095, 85750/85293, 983125/979776	[⟨584​ 926 ​1356​ 1639 ​2020 ​2161]]	+0.0145	0.2613	12.72

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	47\584	96.58	200/189	Hemiluna (584, 7-limit)

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

Scales

• Hemiluna14

Music

Francium

• Are You From The Moon? (2023) – hemiluna in 584edo tuning