Consistent circle: Difference between revisions

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== Definitions ==

We define a [[circle]] of some (usually [[JI]]) [[interval]] ''a''/''b'' as an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.

We define a '''consistent''' [[circle]] (abbreviatable to just ''circle''{{idiosyncratic}}) of some (usually [[JI]]) [[interval]] ''a''/''b'' as: an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.

Note that when a/b ''does'' generate all notes of the edo (meaning ''N'' = ''m''), then that means that (''a''/''b'')''m'' = ''N'' reaches ''m'' = ''N'' octaves. This will always be true in a prime edo, such as 31edo, meaning we can easily deduce that stacking 35/32 31 times gets us at 4 octaves, because 35/32's direct mapping is 4\31. This same reasoning can be applied in general if you think instead in terms of the subset edo generated.

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In such a case, we say that ''N''-edo "'''has''' a circle of ~a/b's"; it is ''incorrect'' to say that ''N''-edo "'''is''' a circle of ~a/b's" because that would imply all notes are reached by repeatedly stacking ''a''/''b''.

=== Having ~~an ultraweak~~ circle ===

=== Having a sub-aweak circle ===

An "~~ultraweak~~ circle of ~a/b's" in ''N''-edo describes a case where the ~~subset~~ edo generated by a/b qualifies as a weak circle of a/b's.

A "'''sub'''-weak circle of ~a/b's" in ''N''-edo describes a case where the '''sub'''set edo generated by a/b qualifies as a weak circle of a/b's.

This can be a useful property to distinguish; for example 80edo has ~~an ultraweak~~ circle of ~10/9's, because (10/9)20 / 8 (~48.1{{cent}}) is smaller in JI than 1\20 = 60{{cent}}.

This can be a useful property to distinguish; for example 80edo has a sub-weak circle of ~10/9's, because (10/9)20 / 8 (~48.1{{cent}}) is smaller in JI than 1\20 = 60{{cent}}.

This means that if one is satisfied with the circle in the subset edo, one may find it to be sufficiently accurate for navigation with in the larger edo, because of familiarity with it the subset edo.

=== Having ~~an ultrastrong~~ circle ===

=== Having a super-strong circle ===

An "~~ultrastrong~~ circle of ~a/b's" in ''N''-edo describes a case where a/b generates a subset of ''N''-edo but is accurate enough that you can stack floor(''N''/2)-many a/b's and still have it be consistent.

A "'''super'''-strong circle of ~a/b's" in ''N''-edo describes a case where a/b generates a subset of ''N''-edo but is accurate enough that you can stack floor(''N''/2)-many a/b's and still have it be consistent w.r.t. the '''super'''set edo.

In other words, if ''N''-edo has ~~an ultrastrong~~ circle of ~a/b's, that means that were GCD(''k'', ''N'') = 1, it would still qualify as a ''weak circle''.

In other words, if ''N''-edo has a super-strong circle of ~a/b's, that means that were GCD(''k'', ''N'') = 1, it would still qualify as a ''weak circle''.

(We use this weaker/more generous bound rather than the default bound for closing error because such a circle is already going "above-and-beyond" in terms of what's necessary to produce a consistent circle.)

@@ Line 5: / Line 5: @@
 == Definitions ==
-We define a [[circle]] of some (usually [[JI]]) [[interval]] ''a''/''b'' as an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.
+We define a '''consistent''' [[circle]] (abbreviatable to just ''circle''{{idiosyncratic}}) of some (usually [[JI]]) [[interval]] ''a''/''b'' as: an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.
 Note that when a/b ''does'' generate all notes of the edo (meaning ''N'' = ''m''), then that means that (''a''/''b'')<sup>''m'' = ''N''</sup> reaches ''m'' = ''N'' octaves. This will always be true in a prime edo, such as 31edo, meaning we can easily deduce that stacking 35/32 31 times gets us at 4 octaves, because 35/32's direct mapping is 4\31. This same reasoning can be applied in general if you think instead in terms of the subset edo generated.
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 In such a case, we say that ''N''-edo "'''has''' a circle of ~a/b's"; it is ''incorrect'' to say that ''N''-edo "'''is''' a circle of ~a/b's" because that would imply all notes are reached by repeatedly stacking ''a''/''b''.
-=== Having an ultraweak circle ===
+=== Having a sub-aweak circle ===
-An "ultraweak circle of ~a/b's" in ''N''-edo describes a case where the subset edo generated by a/b qualifies as a weak circle of a/b's.
+A "'''sub'''-weak circle of ~a/b's" in ''N''-edo describes a case where the '''sub'''set edo generated by a/b qualifies as a weak circle of a/b's.
-This can be a useful property to distinguish; for example 80edo has an ultraweak circle of ~10/9's, because (10/9)<sup>20</sup> / 8 (~48.1{{cent}}) is smaller in JI than 1\20 = 60{{cent}}.
+This can be a useful property to distinguish; for example 80edo has a sub-weak circle of ~10/9's, because (10/9)<sup>20</sup> / 8 (~48.1{{cent}}) is smaller in JI than 1\20 = 60{{cent}}.
 This means that if one is satisfied with the circle in the subset edo, one may find it to be sufficiently accurate for navigation with in the larger edo, because of familiarity with it the subset edo.
-=== Having an ultrastrong circle ===
+=== Having a super-strong circle ===
-An "ultrastrong circle of ~a/b's" in ''N''-edo describes a case where a/b generates a subset of ''N''-edo but is accurate enough that you can stack floor(''N''/2)-many a/b's and still have it be consistent.
+A "'''super'''-strong circle of ~a/b's" in ''N''-edo describes a case where a/b generates a subset of ''N''-edo but is accurate enough that you can stack floor(''N''/2)-many a/b's and still have it be consistent w.r.t. the '''super'''set edo.
-In other words, if ''N''-edo has an ultrastrong circle of ~a/b's, that means that were GCD(''k'', ''N'') = 1, it would still qualify as a ''weak circle''.
+In other words, if ''N''-edo has a super-strong circle of ~a/b's, that means that were GCD(''k'', ''N'') = 1, it would still qualify as a ''weak circle''.
 (We use this weaker/more generous bound rather than the default bound for closing error because such a circle is already going "above-and-beyond" in terms of what's necessary to produce a consistent circle.)