Consistent circle: Difference between revisions

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Intuitively, a ~~'''consistent'''~~ [[circle]] of ~a/b's in an [[edo]] describes a case where ''a''/''b'' is so accurately approximated that you can rely on navigating with it being [[consistent]] with respect to the "circle of notes" defined. It is closely related to the concept of [[telicity]], except that a circle can be of ''any'' [[JI]] interval, as long as it is [[consistency#mathematical definition|consistently mapped]] to the unison when octave-reduced. (It also admits generalization to any [[equave]].) Note that the precise term ''consistent circle of ~a/b's'' may be shortened to ''circle of a/b's''{{idiosyncratic}} in writing for the sake of brevity wherever it's unambiguous (or in informal writing).

Intuitively, a [[consistent circle]] of ~a/b's in an [[edo]] describes a case where ''a''/''b'' is so accurately approximated that you can rely on navigating with it being [[consistent]] with respect to the "circle of notes" defined. It is closely related to the concept of [[telicity]], except that a circle can be of ''any'' [[JI]] interval, as long as it is [[consistency#mathematical definition|consistently mapped]] to the unison when octave-reduced. (It also admits generalization to any [[equave]].) Note that the precise term ''consistent circle of ~a/b's'' may be shortened to ''circle of a/b's''{{idiosyncratic}} in writing for the sake of brevity wherever it's unambiguous (or in informal writing).

== Motivation ==

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== Definitions ==

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.

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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-Intuitively, a '''consistent''' [[circle]] of ~a/b's in an [[edo]] describes a case where ''a''/''b'' is so accurately approximated that you can rely on navigating with it being [[consistent]] with respect to the "circle of notes" defined. It is closely related to the concept of [[telicity]], except that a circle can be of ''any'' [[JI]] interval, as long as it is [[consistency#mathematical definition|consistently mapped]] to the unison when octave-reduced. (It also admits generalization to any [[equave]].) Note that the precise term ''consistent circle of ~a/b's'' may be shortened to ''circle of a/b's''{{idiosyncratic}} in writing for the sake of brevity wherever it's unambiguous (or in informal writing).
+Intuitively, a [[consistent circle]] of ~a/b's in an [[edo]] describes a case where ''a''/''b'' is so accurately approximated that you can rely on navigating with it being [[consistent]] with respect to the "circle of notes" defined. It is closely related to the concept of [[telicity]], except that a circle can be of ''any'' [[JI]] interval, as long as it is [[consistency#mathematical definition|consistently mapped]] to the unison when octave-reduced. (It also admits generalization to any [[equave]].) Note that the precise term ''consistent circle of ~a/b's'' may be shortened to ''circle of a/b's''{{idiosyncratic}} in writing for the sake of brevity wherever it's unambiguous (or in informal writing).
 == Motivation ==
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 == Definitions ==
-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.
+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.

Consistent circle: Difference between revisions

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