576edo: Difference between revisions

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== 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]], ~~assining~~ [[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 [[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.

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]].

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=== 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.

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"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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|-

| 12

| 239\576 (1\576)

| 239\576 (1\576)

| 497.916 (2.083)

| 497.916 (2.083)

| 4/3 (32805/32768)

| 4/3 (32805/32768)

| [[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

<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}}
-{{EDO intro|576}}
+{{ED intro}}
 == 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]], assining [[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.
+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.
 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]].
@@ Line 11: / Line 11: @@
 === 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.
+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"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 33: / Line 33: @@
 |-
 | 12
-| 239\576<br>(1\576)
+| 239\576<br />(1\576)
-| 497.916<br>(2.083)
+| 497.916<br />(2.083)
-| 4/3<br>(32805/32768)
+| 4/3<br />(32805/32768)
 | [[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
+<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