576edo: Difference between revisions

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~~576edo divides the octave into steps of 2.083 cents each.~~

~~==Theory==~~

576edo is an excellent 2.3.7 subgroup tuning.

== 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.

~~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.

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]].

~~Because of composition~~, ~~it may be preferrable to make references to smaller~~ EDOs ~~instead~~ of ~~using the best approximation~~. ~~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 best interval approximations often create circles of 1\576~~, ~~which may sound like an untempered comma.~~

=== Prime harmonics ===

=== 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 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: @@
-edo 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}}
-edo is an excellent 2.3.7 subgroup tuning.
+== 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.
-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.
+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]].
-Because of composition, it may be preferrable to make references to smaller EDOs instead of using the best approximation. 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 best interval approximations often create circles of 1\576, which may sound like an untempered comma.
+=== Prime harmonics ===
+{{Harmonics in equal|576|columns=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.
+[[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