540edo: Difference between revisions

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== Theory ==

Since 540 = 2 × 270 and 540 = 45 × 12, it contains [[270edo]] and [[12edo]] as subsets, both being important zeta ~~edos~~. 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 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 540 = 2 × 270 and 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 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.

It 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.

It [[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.

~~Its~~ step 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>.

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>.

=== Prime harmonics ===

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=== Subsets and supersets ===

540 is a very composite number. The prime factorization of 540 is 22 × 33 × 5. Its 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 }}.

540 is a very composite number. The prime factorization of 540 is 22 × 33 × 5. 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 }}.

== Regular temperament properties ==

Line 28:

| 2.3.5.7.11.13.17

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

| [{{~~val~~| 540 856 1254 1516 1868 1998 2207 }}]

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

| -0.0022

| 0.1144

Line 35:

| 2.3.5.7.11.13.17.19

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

| [{{~~val~~| 540 856 1254 1516 1868 1998 2207 2294 }}]

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

| -0.0098

| 0.1088

Line 42:

| 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

| [{{~~val~~| 540 856 1254 1516 1868 1998 2207 2294 2443 }}]

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

| -0.024

| 0.1100

| 4.95

|}

== Notes ==

@@ Line 3: / Line 3: @@
 == Theory ==
-Since 540 = 2 × 270 and 540 = 45 × 12, it contains [[270edo]] and [[12edo]] as subsets, both being important zeta edos. 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 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 540 = 2 × 270 and 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 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.
-It 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.
+It [[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.
-Its step 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>.
+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>.
 === Prime harmonics ===
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
-is a very composite number. The prime factorization of 540 is 2<sup>2</sup> × 3<sup>3</sup> × 5. Its 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 }}.
+is a very composite number. The prime factorization of 540 is 2<sup>2</sup> × 3<sup>3</sup> × 5. 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 }}.
 == Regular temperament properties ==
@@ Line 28: / Line 28: @@
 | 2.3.5.7.11.13.17
 | 676/675, 1001/1000, 1156/1155, 1716/1715, 3025/3024, 4096/4095
-| [{{val| 540 856 1254 1516 1868 1998 2207 }}]
+| {{mapping| 540 856 1254 1516 1868 1998 2207 }}
 | -0.0022
 | 0.1144
@@ Line 35: / Line 35: @@
 | 2.3.5.7.11.13.17.19
 | 676/675, 1001/1000, 1156/1155, 1216/1215, 1331/1330, 1445/1444, 1729/1728
-| [{{val| 540 856 1254 1516 1868 1998 2207 2294 }}]
+| {{mapping| 540 856 1254 1516 1868 1998 2207 2294 }}
 | -0.0098
 | 0.1088
@@ Line 42: / Line 42: @@
 | 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
-| [{{val| 540 856 1254 1516 1868 1998 2207 2294 2443 }}]
+| {{mapping| 540 856 1254 1516 1868 1998 2207 2294 2443 }}
 | -0.024
 | 0.1100
 | 4.95
 |}
+== Notes ==