584edo

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 2.055 ¢ 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
Error	relative (%)	+38	-1	-50	-24	-31	-6	+38	-8	+21	-11	+24
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
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	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 it is distinct

Scales

Hemiluna14

Music

Francium

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