576edo: Difference between revisions

Line 3:

== Theory ==

576 is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It is ~~what's~~ known as a [[~~oeis:A033833~~|highly factorable edo]] ~~and is best~~ played through JI-agnostic approaches that make use of its divisors (see Subsets and supersets section below). 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.

576 is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It is known as a [[Highly composite equal division #Highly factorable numbers|highly factorable edo]], which enables it to be played through JI-agnostic approaches that make use of its divisors (see [[#Subsets and supersets]] section below). This approach may be preferrable since the [[patent val]] will create sequences that fall aside by 1\576 of each other{{clarify}}, which may not "live up to the spirit" of a composite number like 576.

Nonetheless, 576edo does offer simple interpretations. Despite having bad 5/4, 576edo is [[consistent]] in the 7-odd-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 – {{monzo| 99 -66 2 }}, {{monzo| 110 -57 -7 }}, and {{monzo| 88 -75 11 }}. The associated rank-2 temperaments are 94 & 576, 41 & 535, and 229 & 347.

In the 5-limit, 576edo ~~supports~~ 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|~~garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.

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 supports a messed-up variant of the [[rectified hebrew]] scale[which?], but with step hardness of 5:3 instead of 3:2, and in which 5/4 is reached via 359 third-tone generators down instead of 6 generators up. The relationship that 7/4 is 15 generators and 13/8 is 13 steps is still preserved.

576edo supports a messed-up version of the [[Rectified Hebrew]] scale, but with step hardness of 5:3 instead of 3:2, and from regular temperament theory perspective, 5/4 is reached via 359 third-tone generators down instead of 6 generators up. The relationship that 7/4 is 15 generators and 13/8 is 13 steps is still preserved.

~~=== Subsets and supersets ===~~

Its 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.

=== Prime harmonics ===

[[~~Category:Equal divisions~~ of the ~~octave|###]]~~

=== Subsets and supersets ===

576edo's nontrivial divisors are {{EDOs| 2, 3, 4, 6, 8, 9, 12, 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 {{w|Byzantine music|Byzantine chanting}}, has been theoreticized by {{w|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 the compositeness, it may be preferrable to make references to smaller edos instead of using the best approximation.

== Regular temperament properties ==

Line 23:

|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperaments

|-

Line 32:

| 339.583

| 243/200

| [[Amity]]

| [[Amity]] (576)

|-

| 12

Line 38:

| 497.916 (2.083)

| 4/3 (32805/32768)

| [[Atomic]]

| [[Atomic]] (576)

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

<~~!-- 3-digit number --~~>

[[~~Category:Amity~~]]

[[~~Category:Atomic~~]]

@@ Line 3: / Line 3: @@
 == Theory ==
-is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It is what's known as a [[oeis:A033833|highly factorable edo]] and is best played through JI-agnostic approaches that make use of its divisors (see Subsets and supersets section below). 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.
+is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It is known as a [[Highly composite equal division #Highly factorable numbers|highly factorable edo]], which enables it to be played through JI-agnostic approaches that make use of its divisors (see [[#Subsets and supersets]] section below). This approach may be preferrable since the [[patent val]] will create sequences that fall aside by 1\576 of each other{{clarify}}, which may not "live up to the spirit" of a composite number like 576.
 Nonetheless, 576edo does offer simple interpretations. Despite having bad 5/4, 576edo is [[consistent]] in the 7-odd-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 – {{monzo| 99 -66 2 }}, {{monzo| 110 -57 -7 }}, and {{monzo| 88 -75 11 }}. The associated rank-2 temperaments are 94 & 576, 41 & 535, and 229 & 347.
-In the 5-limit, 576edo supports 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|garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.
+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.
+edo supports a messed-up variant of the [[rectified hebrew]] scale<sup>[which?]</sup>, but with step hardness of 5:3 instead of 3:2, and in which 5/4 is reached via 359 third-tone generators down instead of 6 generators up. The relationship that 7/4 is 15 generators and 13/8 is 13 steps is still preserved.
-edo supports a messed-up version of the [[Rectified Hebrew]] scale, but with step hardness of 5:3 instead of 3:2, and from regular temperament theory perspective, 5/4 is reached via 359 third-tone generators down instead of 6 generators up. The relationship that 7/4 is 15 generators and 13/8 is 13 steps is still preserved.
-=== Subsets and supersets ===
-Its 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.
 === Prime harmonics ===
 {{Harmonics in equal|576|columns=11}}
-[[Category:Equal divisions of the octave|###]]
+=== Subsets and supersets ===
+edo's nontrivial divisors are {{EDOs| 2, 3, 4, 6, 8, 9, 12, 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 {{w|Byzantine music|Byzantine chanting}}, has been theoreticized by {{w|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 the compositeness, it may be preferrable to make references to smaller edos instead of using the best approximation.
 == Regular temperament properties ==
@@ Line 23: / Line 23: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 32: / Line 32: @@
 | 339.583
 | 243/200
-| [[Amity]]
+| [[Amity]] (576)
 |-
 | 12
@@ Line 38: / Line 38: @@
 | 497.916<br>(2.083)
 | 4/3<br>(32805/32768)
-| [[Atomic]]
+| [[Atomic]] (576)
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
-<!-- 3-digit number -->
-[[Category:Amity]]
-[[Category:Atomic]]