57edo: Difference between revisions

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== Theory ==

57edo ~~can be used to tune the [[mothra]] temperament, and~~ is an excellent tuning for the 2.5/3.7.11.13.17.19 [[just intonation subgroup]]. One way to describe 57edo is that it has a [[5-limit]] part consisting of three [[ring number|ring]]s of 19edo, plus a no-threes no-fives part which is much more accurate~~. A good generator to exploit the 2.5/3.7.11.13.17.19 aspect of 57 is the approximate [[11/8]], which is 26\57. This gives the [[19-limit]] 46 & 57 temperament [[heinz]]~~.

57edo is an excellent tuning for the 2.5/3.7.11.13.17.19 [[just intonation subgroup]]. One way to describe 57edo is that it has a [[5-limit]] part consisting of three [[ring number|ring]]s of 19edo, plus a no-threes no-fives part which is much more accurate.

[[~~5-limit|5-limit~~]] [[~~comma~~]]s: [[81/80]], [[3125/3072]]

Using the full prime-limit [[patent val]], the equal temperament tempers out [[81/80]], [[1029/1024]], and [[3125/3072]] in the 7-limit; and [[99/98]], [[385/384]], [[441/440]], and [[625/616]] in the [[11-limit]]. A good generator to exploit the 2.5/3.7.11.13.17.19 aspect of 57 is the approximate [[11/8]], which is 26\57. This gives the [[19-limit]] 46 & 57 temperament [[heinz]]. It can also be used to tune [[mothra]] as well as [[trismegistus]].

~~[[7-limit|~~7-limit]] ~~commas: 81~~/80, ~~3125~~/~~3072~~, [[~~1029~~/~~1024~~]]

[[~~11-limit|~~11-limit]] ~~commas:~~ [[99/98]], [[~~385/384~~]], [[~~441/440~~]], [[~~625/616~~]]

=== Odd harmonics ===

=== Subsets and supersets ===

57edo contains [[3edo]] and [[19edo]] as subsets.

== Intervals ==

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 == Theory ==
-edo can be used to tune the [[mothra]] temperament, and is an excellent tuning for the 2.5/3.7.11.13.17.19 [[just intonation subgroup]]. One way to describe 57edo is that it has a [[5-limit]] part consisting of three [[ring number|ring]]s of 19edo, plus a no-threes no-fives part which is much more accurate. A good generator to exploit the 2.5/3.7.11.13.17.19 aspect of 57 is the approximate [[11/8]], which is 26\57. This gives the [[19-limit]] 46 &amp; 57 temperament [[heinz]].
+edo is an excellent tuning for the 2.5/3.7.11.13.17.19 [[just intonation subgroup]]. One way to describe 57edo is that it has a [[5-limit]] part consisting of three [[ring number|ring]]s of 19edo, plus a no-threes no-fives part which is much more accurate.
-[[5-limit|5-limit]] [[comma]]s: [[81/80]], [[3125/3072]]
+Using the full prime-limit [[patent val]], the equal temperament tempers out [[81/80]], [[1029/1024]], and [[3125/3072]] in the 7-limit; and [[99/98]], [[385/384]], [[441/440]], and [[625/616]] in the [[11-limit]]. A good generator to exploit the 2.5/3.7.11.13.17.19 aspect of 57 is the approximate [[11/8]], which is 26\57. This gives the [[19-limit]] 46 &amp; 57 temperament [[heinz]]. It can also be used to tune [[mothra]] as well as [[trismegistus]].
-[[7-limit|7-limit]] commas: 81/80, 3125/3072, [[1029/1024]]
-[[11-limit|11-limit]] commas: [[99/98]], [[385/384]], [[441/440]], [[625/616]]
 === Odd harmonics ===
 {{Harmonics in equal|57}}
+=== Subsets and supersets ===
+edo contains [[3edo]] and [[19edo]] as subsets.
 == Intervals ==