Consistent circle: Difference between revisions

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== Definition ==

We define a [[circle]]{{idiosyncratic}} of some [[interval]] ''a''/''b'' as an interval with such extremely low relative error (using the direct approximation) with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimal amount required to reach a whole number of octaves, ~~its [[direct approximation]]~~ is [[consistent]] with its actual size in [[JI]], 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 [[circle]]{{idiosyncratic}} of some [[interval]] ''a''/''b'' as an interval with such extremely low relative error (using the direct approximation) with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimal amount required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual size in [[JI]], 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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 == Definition ==
-We define a [[circle]]{{idiosyncratic}} of some [[interval]] ''a''/''b'' as an interval with such extremely low relative error (using the direct approximation) with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimal amount required to reach a whole number of octaves, its [[direct approximation]] is [[consistent]] with its actual size in [[JI]], 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 [[circle]]{{idiosyncratic}} of some [[interval]] ''a''/''b'' as an interval with such extremely low relative error (using the direct approximation) with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimal amount required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual size in [[JI]], 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.