576edo: Difference between revisions

(18 intermediate revisions by 5 users not shown)

Line 1:

{{Infobox ET

~~| Prime factorization = 26 × 32~~

~~| Step size = 2.08333¢~~

~~| Fifth = 337\576 (702.08333¢)~~

~~| Major 2nd = 98\576 (204.16666¢)~~

~~| Consistency = 7~~

}}

== 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.

~~==Theory==~~

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]].

~~{{Harmonics in equal~~|~~576}}~~

~~576 is a near-highly composite number which is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It's xenharmonic divisors (that is, besides 12edo and its subsets) are~~ {{~~EDOs~~|~~8, 9, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288~~}}. Some of these have been put into practical use. 72edo has been used in [[Wikipedia:Byzantine music|Byzantine chanting]], has been theoreticized by [[wikipedia:Alois Hába|Alois Haba]] and [[Ivan Wyschnegradsky]], and has been used by jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]]. Because of composition, it may be preferrable to make references to smaller EDOs instead of using the best approximation, that is the patent val. 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 not "live up to the spirit" of a composite number like 576.

=== Prime harmonics ===

=== ~~Regular temperament-based approach~~ ===

=== Subsets and supersets ===

~~Nonetheless~~, 576edo ~~does offer simple interpretations~~.

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.

~~Despite having bad 5/4, 576edo is~~ [[~~consistent~~]] ~~in the 7-limit. As a corollary~~, ~~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⟩~~.

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

~~In the~~ 5-~~limit, the patent val supports~~ [[~~amity~~]] ~~temperament, and the 576c val supports~~ [[~~maquila~~]].

== Regular temperament properties ==

=== Rank-2 temperaments ===

[[~~Category:Equal divisions of the octave|~~#~~##]] <!~~-- 3-~~digit number~~ -->

{| class="wikitable center-all left-5"

[[~~Category:Amity~~]]

|+ style="font-size: 105%;" | 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)

|}

<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

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 2<sup>6</sup> × 3<sup>2</sup>
+{{ED intro}}
-| Step size = 2.08333¢
-| Fifth = 337\576 (702.08333¢)
-| Major 2nd = 98\576 (204.16666¢)
-| Consistency = 7
-}}
-{{EDO intro|576}}
+== Theory ==
+edo 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.
-==Theory==
+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]].
-{{Harmonics in equal|576}}
-is a near-highly composite number which is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It's xenharmonic divisors (that is, besides 12edo and its subsets) are {{EDOs|8, 9, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288}}. Some of these have been put into practical use. 72edo has been used in [[Wikipedia:Byzantine music|Byzantine chanting]], has been theoreticized by [[wikipedia:Alois Hába|Alois Haba]] and [[Ivan Wyschnegradsky]], and has been used by jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]]. Because of composition, it may be preferrable to make references to smaller EDOs instead of using the best approximation, that is the patent val. 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 not "live up to the spirit" of a composite number like 576.
+=== Prime harmonics ===
+{{Harmonics in equal|576|columns=11}}
-=== Regular temperament-based approach ===
+=== Subsets and supersets ===
-Nonetheless, 576edo does offer simple interpretations.
+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.
-Despite having bad 5/4, 576edo is [[consistent]] in the 7-limit. As a corollary, 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⟩.
+[[1152edo]], which is also a highly factorable edo, divides the edostep in two and corrects the mapping for 5.
-In the 5-limit, the patent val supports [[amity]] temperament, and the 576c val supports [[maquila]].
+== Regular temperament properties ==
+=== Rank-2 temperaments ===
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+{| class="wikitable center-all left-5"
-[[Category:Amity]]
+|+ 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