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 ~~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-, 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, 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.

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.

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

=== Prime harmonics ===

=== ~~Divisors~~ ===

=== Subsets and supersets ===

540 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 }}.

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 == 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-, 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, 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.
+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.
+Its step is known as a '''dexl''', proposed by [[Joe Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia | Dexl / 540edo]</ref>.
 === Prime harmonics ===
 {{Harmonics in equal|540|columns=11}}
-=== Divisors ===
+=== 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 }}.