Complexity: Difference between revisions

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In tuning, '''complexity''' is...

In tuning, '''complexity''' is a characteristic of [[temperament]]s, which can be used to evaluate and compare them. Generally speaking, if a temperament has high complexity, that means that interesting pitches (e.g. ones [[consonant]] with each other) are many [[generator]]s apart, so useful scales tend to have many notes. If a temperament has low complexity, fewer generators are required, and scales with fewer notes are more likely to be useful.

~~<ul><li>~~[[~~Graham_complexity|~~Graham complexity]]~~</li><li>~~[[~~Generator_complexity|~~Generator complexity]]~~</li><li>~~[[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]]~~</li><li>~~[[Tenney-Euclidean_metrics#Temperamental complexity|Temperamental complexity]]~~</li><li>~~[[~~Benedetti_height|~~Benedetti height]]</li></ul>

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.

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

* [[Graham complexity]]

* [[Generator complexity]]

* [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]]

* [[Tenney-Euclidean_metrics#Temperamental complexity|Temperamental complexity]]

* [[Benedetti height]]</li></ul>

==See also==

~~<ul><li>~~[http://en.wikipedia.org/wiki/Complexity http://en.wikipedia.org/wiki/Complexity~~]</li></ul> [[Category:complexity]~~]

* [http://en.wikipedia.org/wiki/Complexity http://en.wikipedia.org/wiki/Complexity]

~~[[Category:todo:add_definition]]~~

[[Category:~~todo:expand~~]]

[[Category:complexity]]

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-In tuning, '''complexity''' is...
+In tuning, '''complexity''' is a characteristic of [[temperament]]s, which can be used to evaluate and compare them. Generally speaking, if a temperament has high complexity, that means that interesting pitches (e.g. ones [[consonant]] with each other) are many [[generator]]s apart, so useful scales tend to have many notes. If a temperament has low complexity, fewer generators are required, and scales with fewer notes are more likely to be useful.
-<ul><li>[[Graham_complexity|Graham complexity]]</li><li>[[Generator_complexity|Generator complexity]]</li><li>[[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]]</li><li>[[Tenney-Euclidean_metrics#Temperamental complexity|Temperamental complexity]]</li><li>[[Benedetti_height|Benedetti height]]</li></ul>
+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.
+Some specific, mathematically rigorous definitions of "complexity" are...
+* [[Graham complexity]]
+* [[Generator complexity]]
+* [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]]
+* [[Tenney-Euclidean_metrics#Temperamental complexity|Temperamental complexity]]
+* [[Benedetti height]]</li></ul>
 ==See also==
-<ul><li>[http://en.wikipedia.org/wiki/Complexity http://en.wikipedia.org/wiki/Complexity]</li></ul>      [[Category:complexity]]
+* [http://en.wikipedia.org/wiki/Complexity http://en.wikipedia.org/wiki/Complexity]
-[[Category:todo:add_definition]]
-[[Category:todo:expand]]
+[[Category:complexity]]