Complexity: Difference between revisions

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Complexity and [[error]] are both usually treated as undesirable characteristics, but there is a trade-off between them in that very low complexity temperaments (e.g. small [[EDO]]s) cannot have low error, and very low error temperaments (e.g. [[microtemperament]]s or [[JI]] itself) cannot have low complexity. [[Badness]] is a way to combine complexity and error such that a search for low-badness temperaments yields results with a particularly good trade-off between complexity and error.

==Complexity of an interval in a temperament==

Besides saying that a temperament has a high or low complexity, we also speak of the '''complexity of an interval''' in a temperament. If an interval has a low complexity in a certain temperament, that means it can be reached in only a few [[generators]], so it is likely to appear frequently in scales of that temperament. For example, in [[meantone]] temperament, the generator represents 3/2, so clearly 3/2 has a very low complexity, since it can be reached in only one generator. In contrast, 45/32 can only be reached in 6 generators so it has a higher complexity and will tend to appear much less frequently in meantone scales.

The '''complexity of a chord''' likewise refers to the number of generator steps required to generate all the pitches of the chord.

Note that the concept of complexity applies not only to [[rank-2 temperaments]], but temperaments of any rank. For rank >2 temperaments, the lattice is a higher-dimensional space so there could be different ways of measuring the area/volume/etc. that a chord takes up.

==Complexity measures==

Some specific, mathematically rigorous definitions of "complexity" are...

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 Complexity and [[error]] are both usually treated as undesirable characteristics, but there is a trade-off between them in that very low complexity temperaments (e.g. small [[EDO]]s) cannot have low error, and very low error temperaments (e.g. [[microtemperament]]s or [[JI]] itself) cannot have low complexity. [[Badness]] is a way to combine complexity and error such that a search for low-badness temperaments yields results with a particularly good trade-off between complexity and error.
+==Complexity of an interval in a temperament==
+Besides saying that a temperament has a high or low complexity, we also speak of the '''complexity of an interval''' in a temperament. If an interval has a low complexity in a certain temperament, that means it can be reached in only a few [[generators]], so it is likely to appear frequently in scales of that temperament. For example, in [[meantone]] temperament, the generator represents 3/2, so clearly 3/2 has a very low complexity, since it can be reached in only one generator. In contrast, 45/32 can only be reached in 6 generators so it has a higher complexity and will tend to appear much less frequently in meantone scales.
+The '''complexity of a chord''' likewise refers to the number of generator steps required to generate all the pitches of the chord.
+Note that the concept of complexity applies not only to [[rank-2 temperaments]], but temperaments of any rank. For rank >2 temperaments, the lattice is a higher-dimensional space so there could be different ways of measuring the area/volume/etc. that a chord takes up.
+==Complexity measures==
 Some specific, mathematically rigorous definitions of "complexity" are...