53edo: Difference between revisions

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=== Prime harmonics ===

~~There's also a cluster of usable higher primes starting at 71; even 89 (4.84{{cent}} flat), 97 (4.63{{cent}} sharp) and 101 (2.6{{cent}} sharp) are usable if placed in just the right context.~~

~~{{Harmonics in equal|53|columns=4|start=20}}~~

~~This make [[53edo]] excellent (for its size) in the 2.3.5.7.11.13.19.23.37.41.71.73.79.83 subgroup, although some higher error primes like 11 and 23 require the right context to be convincing.~~

~~Note that the high primes, in rooted (/2<sup>n</sup>) position, essentially act as alternate interpretations of [[LCJI]] intervals, if you want to force a rooted interpretation; namely:~~

[[71/64]] as ~[[10/9]], [[73/64]] as ~[[8/7]], [[79/64]] as ~[[16/13]], and perhaps most questionably in the context of [[53edo]], [[83/64]] as ~[[13/10]]. (Note that [[8edo]] offers a very good approximation of [[83/64]], so if you are working with a system that maps 13/10 to 3\8 = 450.000{{cent}} it makes more sense to think of 83/64 as the rooted approximation of 13/10 in that context.)

=== Subsets and supersets ===

53edo is the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]].

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[[File:53-edo spiral.png|588x588px]]

== JI ~~approximation~~ ==

== Approximation to JI ==

53edo provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.

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|}

=== Higher-limit JI ===

There is also a cluster of usable higher primes starting at 71; even 89 (4.84{{cent}} flat), 97 (4.63{{cent}} sharp) and 101 (2.6{{cent}} sharp) are usable if placed in just the right context.

This make [[53edo]] excellent (for its size) in the 2.3.5.7.11.13.19.23.37.41.71.73.79.83 subgroup, although some higher error primes like 11 and 23 require the right context to be convincing.

Note that the high primes, in rooted (/2<sup>n</sup>) position, essentially act as alternate interpretations of [[LCJI]] intervals, if you want to force a rooted interpretation; namely:

[[71/64]] as ~[[10/9]], [[73/64]] as ~[[8/7]], [[79/64]] as ~[[16/13]], and perhaps most questionably in the context of 53edo, [[83/64]] as ~[[13/10]]. (Note that [[8edo]] offers a very good approximation of [[83/64]], so if you are working with a system that maps 13/10 to 3\8 = 450.000{{cent}} it makes more sense to think of 83/64 as the rooted approximation of 13/10 in that context.)

== Regular temperament properties ==

@@ Line 18: / Line 18: @@
 === Prime harmonics ===
 {{Harmonics in equal|53|columns=13}}
-There's also a cluster of usable higher primes starting at 71; even 89 (4.84{{cent}} flat), 97 (4.63{{cent}} sharp) and 101 (2.6{{cent}} sharp) are usable if placed in just the right context.
-{{Harmonics in equal|53|columns=4|start=20}}
-This make [[53edo]] excellent (for its size) in the 2.3.5.7.11.13.19.23.37.41.71.73.79.83 subgroup, although some higher error primes like 11 and 23 require the right context to be convincing.
-Note that the high primes, in rooted (/2<sup>n</sup>) position, essentially act as alternate interpretations of [[LCJI]] intervals, if you want to force a rooted interpretation; namely:
-[[71/64]] as ~[[10/9]], [[73/64]] as ~[[8/7]], [[79/64]] as ~[[16/13]], and perhaps most questionably in the context of [[53edo]], [[83/64]] as ~[[13/10]]. (Note that [[8edo]] offers a very good approximation of [[83/64]], so if you are working with a system that maps 13/10 to 3\8 = 450.000{{cent}} it makes more sense to think of 83/64 as the rooted approximation of 13/10 in that context.)
 === Subsets and supersets ===
 edo is the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]].
@@ Line 660: / Line 654: @@
 [[File:53-edo spiral.png|588x588px]]
-== JI approximation ==
+== Approximation to JI ==
 edo provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.
@@ Line 809: / Line 803: @@
 |}
 {{15-odd-limit|53}}
+=== Higher-limit JI ===
+There is also a cluster of usable higher primes starting at 71; even 89 (4.84{{cent}} flat), 97 (4.63{{cent}} sharp) and 101 (2.6{{cent}} sharp) are usable if placed in just the right context.
+{{Harmonics in equal|53|columns=4|start=20}}
+This make [[53edo]] excellent (for its size) in the 2.3.5.7.11.13.19.23.37.41.71.73.79.83 subgroup, although some higher error primes like 11 and 23 require the right context to be convincing.
+Note that the high primes, in rooted (/2<sup>n</sup>) position, essentially act as alternate interpretations of [[LCJI]] intervals, if you want to force a rooted interpretation; namely:
+[[71/64]] as ~[[10/9]], [[73/64]] as ~[[8/7]], [[79/64]] as ~[[16/13]], and perhaps most questionably in the context of 53edo, [[83/64]] as ~[[13/10]]. (Note that [[8edo]] offers a very good approximation of [[83/64]], so if you are working with a system that maps 13/10 to 3\8 = 450.000{{cent}} it makes more sense to think of 83/64 as the rooted approximation of 13/10 in that context.)
 == Regular temperament properties ==