576edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,382 edits
MisterShafXen (talk | contribs)
1,074 edits
m Fixed typo
Tags: Visual edit Mobile edit Mobile web edit Advanced mobile edit
 
(28 intermediate revisions by 7 users not shown)
Line 1: Line 1:
576edo divides the octave into steps of 2.08<SPAN STYLE="text-decoration:overline">3</SPAN> cents each.
{{Infobox ET}}
==Theory==
{{ED intro}}
{{primes in edo|576|columns=14}}


576 is a highly composite number which is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It's xenharmonic divisors are {{EDOs|8, 9, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288}}. Some of these like 72 and 96 have been put into historical use. Its approximation to the perfect fifth is just one step above the 12edo fifth.
== Theory ==
576edo is [[consistent]] in the 7-odd-limit, though the error on harmonic [[5/4|5]] is quite large. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. It tempers out the [[septimal ennealimma]], assigning [[7/6]] to 2\9, as well as {{monzo| 99 -66 0 2 }}, {{monzo| 110 -57 0 -7 }} , and {{monzo| 88 -75 0 11 }}. In the 5-limit, the patent val of 576edo [[support]]s the [[atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{val| 576 913 1336 1618 }}, is a tuning for the [[garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.


576edo is an excellent 2.3.7 subgroup tuning. Using the patent val, it tempers out the [[septimal ennealimma]], 40353607/40310784, and assigns 7/6 to 2\9 of the octave, property that ultimately derives from [[9edo]]. However, other commas being tempered out are far more complex - [99, -66, 2⟩, [110, -57, -7⟩, and [88, -75, 11⟩.
In higher limits, the 2.3.7 subgroup can be used with optional additions of [[19/16|19]] or [[29/16|29]], or fractional subgroups using [[13/10]].


In the 11-limit, using the 576c and 576e vals, it tempers out [[9801/9800]]. Specifying on 576c removes [[3136/3125]], and on 576e removes [[1375/1372]]. In the 19-limit, 576e val eliminates [[715/714]] and [[1225/1224]]. In addition, using the 576e val preserves the [[24edo]] mapping for [[11/8]].
=== Prime harmonics ===
{{Harmonics in equal|576|columns=11}}


Because of composition, it may be preferrable to make references to smaller EDOs instead of using the best approximation, that is the patent val. For example, using the ⟨576 912 1616 1992] [[val]] for representing the 2.3.7.11 subgroup makes references to 24edo and 36edo: ⟨1\1 7\12 11\24 29\36] when octave reduced. In fact, this approach may be preferrable since the patent val will create sequences that fall aside by 1\576 of each other, which may sound like an untempered comma.
=== Subsets and supersets ===
Since 576 factors as {{factorization|576}}, 576edo has subset edos {{EDOs| 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}, of which {{EDOs|12, 24, 72, and 96}} are particularly notable. Overall, 576edo contains a number of notable divisions that are multiples of 12, and it is a [[Highly composite equal division#Highly factorable numbers|highly factorable]] edo.
 
[[1152edo]], which is also a highly factorable edo, divides the edostep in two and corrects the mapping for 5.
 
== Regular temperament properties ==
=== Rank-2 temperaments ===
{| 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
| 163\576
| 339.583
| 243/200
| [[Amity]] (576)
|-
| 12
| 239\576<br />(1\576)
| 497.916<br />(2.083)
| 4/3<br />(32805/32768)
| [[Atomic]] (576)
|}
<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

Latest revision as of 15:45, 10 April 2026

← 575edo 576edo 577edo →
Prime factorization 26 × 32
Step size 2.08333 ¢ 
Fifth 337\576 (702.083 ¢)
Semitones (A1:m2) 55:43 (114.6 ¢ : 89.58 ¢)
Consistency limit 7
Distinct consistency limit 7

576 equal divisions of the octave (abbreviated 576edo or 576ed2), also called 576-tone equal temperament (576tet) or 576 equal temperament (576et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 576 equal parts of about 2.08 ¢ each. Each step represents a frequency ratio of 21/576, or the 576th root of 2.

Theory

576edo is consistent in the 7-odd-limit, though the error on harmonic 5 is quite large. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. It tempers out the septimal ennealimma, assigning 7/6 to 2\9, as well as [99 -66 0 2⟩, [110 -57 0 -7⟩ , and [88 -75 0 11⟩. In the 5-limit, the patent val of 576edo supports the atomic temperament and the amity temperament. The 576c val supports maquila. The 576ccd val, ⟨576 913 1336 1618], is a tuning for the garibaldi temperament in the 7-limit. In addition, in this case 5/4 comes from 72edo, and 7/4 comes form 288edo.

In higher limits, the 2.3.7 subgroup can be used with optional additions of 19 or 29, or fractional subgroups using 13/10.

Prime harmonics

Approximation of prime harmonics in 576edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.128 -0.897 -0.076 +0.765 -0.944 -0.789 +0.404 +0.892 -0.411 +0.798
Relative (%) +0.0 +6.2 -43.1 -3.6 +36.7 -45.3 -37.9 +19.4 +42.8 -19.7 +38.3
Steps
(reduced) 		576
(0) 		913
(337) 		1337
(185) 		1617
(465) 		1993
(265) 		2131
(403) 		2354
(50) 		2447
(143) 		2606
(302) 		2798
(494) 		2854
(550)

Subsets and supersets

Since 576 factors as 26 × 32, 576edo has subset edos 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288, of which 12, 24, 72, and 96 are particularly notable. Overall, 576edo contains a number of notable divisions that are multiples of 12, and it is a highly factorable edo.

1152edo, which is also a highly factorable edo, divides the edostep in two and corrects the mapping for 5.

Regular temperament properties

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 163\576 339.583 243/200 Amity (576)
12 239\576
(1\576) 		497.916
(2.083) 		4/3
(32805/32768) 		Atomic (576)

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

Retrieved from "https://en.xen.wiki/index.php?title=576edo&oldid=227572"