Consistent circle: Difference between revisions
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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 [[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. | ||
Note that when | 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. | ||
=== Closing error === | === Closing error === | ||
The ''closing error'' of ''a''/''b'' is defined as follows: If ''k''\''N'' is the [[direct approximation|best approximation]] of ''a''/''b'', meaning that: | The ''closing error'' of ''a''/''b'' is defined as follows: If ''k''\''N'' is the [[direct approximation|best approximation]] of ''a''/''b'', meaning that: | ||
: ''k'' = round(''N'' log<sub>2</sub>( | : ''k'' = round(''N'' log<sub>2</sub>(a/b)) | ||
...then the ''closing error'' of | ...then the ''closing error'' of a/b in ''N''-edo is: | ||
: ''N''<sup>2</sup> log<sub>2</sub>( | : ''N''<sup>2</sup> log<sub>2</sub>(a/b) / GCD(''k'',''N'') | ||
Therefore, ''N''-[[edo]] is a ''circle of a | Therefore, ''N''-[[edo]] is a ''circle of ~a/b'''s iff: | ||
: |''N''<sup>2</sup> log<sub>2</sub>( | : |''N''<sup>2</sup> log<sub>2</sub>(a/b) / GCD(''k'',''N'')| < 1/2 | ||
And is a ''weak circle'' if it is < 1 instead. | And is a ''weak circle'' if it is < 1 instead. | ||