810edo: Difference between revisions

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810 = 270 × 3, and 810edo has three copies of [[270edo]] in the 13-limit (and the 2.3.5.7.11.13.19 [[subgroup]]). It makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, only [[consistent]] to the [[9-odd-limit]]. [[11/9]], [[13/12]], [[13/9]], [[13/10]], and their [[octave complement]]s are all mapped inconsistently in this edo.

As an equal temperament, it [[tempering out|tempers out]] [[4914/4913]] in the 17-limit; and [[2024/2023]], [[2737/2736]], and [[3520/3519]] in the 23-limit. Although it does quite well in these limits, it is way less efficient as [[270edo]]'s or [[540edo]]'s 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 chords are worth the load of all the extra notes.

As an equal temperament, it [[tempering out|tempers out]] [[4914/4913]] in the 17-limit; and [[2024/2023]], [[2737/2736]], and [[3520/3519]] in the 23-limit. Although it does quite well in these limits, it is way less efficient than [[270edo]]'s or [[540edo]]'s 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 chords are worth the load of all the extra notes.

=== Prime harmonics ===

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=== Subsets and supersets ===

Since 810 factors into {{factorization|810}}, 810edo has subset edos {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405 }}.

== Regular temperament properties ==

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 = 270 × 3, and 810edo has three copies of [[270edo]] in the 13-limit (and the 2.3.5.7.11.13.19 [[subgroup]]). It makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, only [[consistent]] to the [[9-odd-limit]]. [[11/9]], [[13/12]], [[13/9]], [[13/10]], and their [[octave complement]]s are all mapped inconsistently in this edo.
-As an equal temperament, it [[tempering out|tempers out]] [[4914/4913]] in the 17-limit; and [[2024/2023]], [[2737/2736]], and [[3520/3519]] in the 23-limit. Although it does quite well in these limits, it is way less efficient as [[270edo]]'s or [[540edo]]'s 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 chords are worth the load of all the extra notes.
+As an equal temperament, it [[tempering out|tempers out]] [[4914/4913]] in the 17-limit; and [[2024/2023]], [[2737/2736]], and [[3520/3519]] in the 23-limit. Although it does quite well in these limits, it is way less efficient than [[270edo]]'s or [[540edo]]'s 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 chords are worth the load of all the extra notes.
 === Prime harmonics ===
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
 Since 810 factors into {{factorization|810}}, 810edo has subset edos {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405 }}.
 == Regular temperament properties ==