540edo: Difference between revisions

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The '''540 equal divisions of the octave''' ('''540edo'''), or the '''540(-tone) equal temperament''' ('''540tet''', '''540et'''), divides the [[octave]] in 540 [[equal]] steps of about 2.22 [[cent]]s each.

== Theory ==

Since 540 = 2 × 270 and 540 = 45 × 12, it contains [[270edo]] and [[12edo]] as subsets, both ~~belonging to~~ [[~~The Riemann Zeta Function and Tuning #Zeta EDO lists|the ''zeta peak edos'', ''zeta integral edos'' and ''~~zeta ~~gap edos'' sequences~~]]. It is [[enfactoring|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17- and 19-limit system, and ~~perhaps~~ beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.

Since {{nowrap|540 {{=}} 2 × 270}} and {{nowrap|540 {{=}} 45 × 12}}, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.

The equal temperament [[tempering out|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]] and [[1729/1728]] in the 19-limit; [[1105/1104]] and [[1496/1495]] in the 23-limit. Although it does quite well in these limits, it is not as ''efficient'' as 270edo's original mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied [[essentially tempered chord]]s are worth the load of all the extra notes.

The approximated [[29/1|29]] and [[31/1|31]] are relatively weak, but [[37/1|37]], [[41/1|41]] and [[43/1|43]] are quite spot on, with the 43 coming from 270edo. For this reason, we may consider it as a full [[43-limit]] system. For all the primes starting with 29, it removes the distinction of otonal and utonal [[superparticular ratio|superparticular]] pairs (e.g. 29/28 vs 30/29 for prime 29) by tempering out the corresponding [[square superparticular]]s, which is responsible for its slightly flat-tending tuning profile. Prime [[47/1|47]] does not have that privilege and falls practically halfway between, though the sharp mapping might be preferred to keep [[47/46]] wider than [[48/47]]. As a compensation, you do get a spot-on prime [[53/1|53]] for free.

=== Prime harmonics ===

{{Harmonics in equal|540|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 540edo (continued)}}

=== Subsets and supersets ===

540 is a very composite number. The [[prime factorization]] of 540 is {{factorization|540}}. Its nontrivial divisors are {{EDOs| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.

A step of 540edo is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia | Dexl / 540edo]</ref>.

== Approximation to JI ==

=== ~~Divisors~~ ===

== Regular temperament properties ==

~~540 is a very composite number. The prime factorization of 540 is~~ 2<~~sup~~>2~~</sup> ×~~ 3~~<sup>3</sup> ×~~ 5. ~~Its divisors are 2~~, 3, 4, 5, ~~6, 9, 10, 12~~, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, ~~108~~, ~~135~~, ~~180~~, ~~and 270~~.

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list|Comma List]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve Stretch (¢)

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3.5.7.11.13.17

| 676/675, 1001/1000, 1156/1155, 1716/1715, 3025/3024, 4096/4095

| {{mapping| 540 856 1254 1516 1868 1998 2207 }}

| -0.0022

| 0.1144

| 5.15

|-

| 2.3.5.7.11.13.17.19

| 676/675, 1001/1000, 1156/1155, 1216/1215, 1331/1330, 1445/1444, 1729/1728

| {{mapping| 540 856 1254 1516 1868 1998 2207 2294 }}

| -0.0098

| 0.1088

| 4.90

|-

| 2.3.5.7.11.13.17.19.23

| 676/675, 1001/1000, 1105/1104, 1156/1155, 1216/1215, 1331/1330, 1445/1444, 1496/1495

| {{mapping| 540 856 1254 1516 1868 1998 2207 2294 2443 }}

| -0.024

| 0.1100

| 4.95

|}

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

== Notes ==

@@ Line 1: / Line 1: @@
-The '''540 equal divisions of the octave''' ('''540edo'''), or the '''540(-tone) equal temperament''' ('''540tet''', '''540et'''), divides the [[octave]] in 540 [[equal]] steps of about 2.22 [[cent]]s each.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-Since 540 = 2 × 270 and 540 = 45 × 12, it contains [[270edo]] and [[12edo]] as subsets, both belonging to [[The Riemann Zeta Function and Tuning #Zeta EDO lists|the ''zeta peak edos'', ''zeta integral edos'' and ''zeta gap edos'' sequences]]. It is [[enfactoring|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17- and 19-limit system, and perhaps beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.
+Since {{nowrap|540 {{=}} 2 × 270}} and {{nowrap|540 {{=}} 45 × 12}}, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.
+The equal temperament [[tempering out|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]] and [[1729/1728]] in the 19-limit; [[1105/1104]] and [[1496/1495]] in the 23-limit. Although it does quite well in these limits, it is not as ''efficient'' as 270edo's original mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied [[essentially tempered chord]]s are worth the load of all the extra notes.
+The approximated [[29/1|29]] and [[31/1|31]] are relatively weak, but [[37/1|37]], [[41/1|41]] and [[43/1|43]] are quite spot on, with the 43 coming from 270edo. For this reason, we may consider it as a full [[43-limit]] system. For all the primes starting with 29, it removes the distinction of otonal and utonal [[superparticular ratio|superparticular]] pairs (e.g. 29/28 vs 30/29 for prime 29) by tempering out the corresponding [[square superparticular]]s, which is responsible for its slightly flat-tending tuning profile. Prime [[47/1|47]] does not have that privilege and falls practically halfway between, though the sharp mapping might be preferred to keep [[47/46]] wider than [[48/47]]. As a compensation, you do get a spot-on prime [[53/1|53]] for free.
 === Prime harmonics ===
-{{Primes in edo|540|columns=11}}
+{{Harmonics in equal|540|columns=12}}
+{{Harmonics in equal|540|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 540edo (continued)}}
+=== Subsets and supersets ===
+is a very composite number. The [[prime factorization]] of 540 is {{factorization|540}}. Its nontrivial divisors are {{EDOs| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.
+A step of 540edo is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia | Dexl / 540edo]</ref>.
+== Approximation to JI ==
+{{Q-odd-limit intervals|540|23}}
-=== Divisors ===
+== Regular temperament properties ==
-is a very composite number. The prime factorization of 540 is 2<sup>2</sup> × 3<sup>3</sup> × 5. Its divisors are 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270.
+{| class="wikitable center-4 center-5 center-6"
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! colspan="2" | Tuning Error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3.5.7.11.13.17
+| 676/675, 1001/1000, 1156/1155, 1716/1715, 3025/3024, 4096/4095
+| {{mapping| 540 856 1254 1516 1868 1998 2207 }}
+| -0.0022
+| 0.1144
+| 5.15
+|-
+| 2.3.5.7.11.13.17.19
+| 676/675, 1001/1000, 1156/1155, 1216/1215, 1331/1330, 1445/1444, 1729/1728
+| {{mapping| 540 856 1254 1516 1868 1998 2207 2294 }}
+| -0.0098
+| 0.1088
+| 4.90
+|-
+| 2.3.5.7.11.13.17.19.23
+| 676/675, 1001/1000, 1105/1104, 1156/1155, 1216/1215, 1331/1330, 1445/1444, 1496/1495
+| {{mapping| 540 856 1254 1516 1868 1998 2207 2294 2443 }}
+| -0.024
+| 0.1100
+| 4.95
+|}
-[[Category:Equal divisions of the octave]]
+== Notes ==