584edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
15,213 edits
mNo edit summary
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
(4 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|584}}
{{ED intro}}


== Theory ==
== Theory ==
Line 11: Line 11:


== Regular temperament properties ==
== 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
| 2.3
| {{monzo| 463 -292 }}
| {{monzo| 463 -292 }}
| {{mapping| 584 926 }}
| {{mapping| 584 926 }}
| &minus;0.2476
| −0.2476
| 0.2475
| 0.2475
| 12.05
| 12.05
Line 23: Line 32:
| {{monzo| 3 -18 11 }}, {{monzo| 21 -20 4 }}
| {{monzo| 3 -18 11 }}, {{monzo| 21 -20 4 }}
| {{mapping| 584 926 1356 }}
| {{mapping| 584 926 1356 }}
| &minus;0.1633
| −0.1633
| 0.2346
| 0.2346
| 11.42
| 11.42
Line 30: Line 39:
| 1500625/1492992, 1605632/1594323, 235298/234375
| 1500625/1492992, 1605632/1594323, 235298/234375
| {{mapping| 584​ 926 ​1356 ​1639 }}
| {{mapping| 584​ 926 ​1356 ​1639 }}
| &minus;0.0319
| −0.0319
| 0.3052
| 0.3052
| 14.85
| 14.85
Line 47: Line 56:
| 0.2613
| 0.2613
| 12.72
| 12.72
|{{comma basis end}}
|}


=== Rank-2 temperaments ===
=== 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
| 1
Line 57: Line 73:
| 200/189
| 200/189
| [[Hemiluna]] (584, 7-limit)
| [[Hemiluna]] (584, 7-limit)
{{rank-2 end}}
|}
{{orf}}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==

Latest revision as of 13:33, 13 March 2026

← 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 distinct

Scales

Music

Francium
Retrieved from "https://en.xen.wiki/index.php?title=584edo&oldid=225917"