540edo: Difference between revisions
m Adopt template: Factorization; misc. cleanup |
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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. | 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. | ||
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. | 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. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === 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 }}. | 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>. | |||
== Regular temperament properties == | == Regular temperament properties == | ||