# 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 21/584, 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 it is distinct

Francium