540edo: Difference between revisions

Line 1:

== Theory ==

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 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.

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|540}}
+{{ED intro}}
 == Theory ==
-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 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.