Complexity: Difference between revisions

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Being a characteristic of [[temperament]]s, complexity 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.

For an [[equal temperament]], a ~~simple~~ definition of the complexity is the number of notes per octave~~. Which~~ means that [[12edo]] has a complexity of 12, etc. ~~This notion can be generalized~~ to ~~temperaments~~ of ~~higher rank~~.

A commonly used temperament complexity measure is [[Tenney–Euclidean temperament measures #TE complexity|Tenney–Euclidean complexity]], which works nicely for multirank temperaments and equal temperaments alike.

For an [[equal temperament]], a simpler definition of the complexity is the number of notes per octave, which means that [[12edo|12et]] has a complexity of 12, etc. For unusual mappings where 2 is mapped to a strange number of steps, that does not work. Norm-based complexities such as TE complexity are foolproof and equave-agnostic, however. For example, the TE complexity of 31et is 30.98, which is close to the edo number as expected for a patent val. But if one were to take the TE complexity of {{val| 1 1900 2785 3370 }}, which is technically a tuning of 1et, they would get 1038.83, which matches the complexity of the tuning much better than the naive approach of simply taking 1 for the complexity, and means that that val is roughly equivalent to 1039et in complexity.

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|edos]]) typically do not have low error, and very low error temperaments (e.g. [[microtemperament|microtemperaments]]) typically do not 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.

~~A commonly used definition of temperament complexity is [[Tenney-Euclidean temperament_measures #TE complexity|Tenney-Euclidean complexity]].~~

== Complexity of an interval in a temperament ==

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 Being a characteristic of [[temperament]]s, complexity 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.
-For an [[equal temperament]], a simple definition of the complexity is the number of notes per octave. Which means that [[12edo]] has a complexity of 12, etc. This notion can be generalized to temperaments of higher rank.
+A commonly used temperament complexity measure is [[Tenney–Euclidean temperament measures #TE complexity|Tenney–Euclidean complexity]], which works nicely for multirank temperaments and equal temperaments alike.
+For an [[equal temperament]], a simpler definition of the complexity is the number of notes per octave, which means that [[12edo|12et]] has a complexity of 12, etc. For unusual mappings where 2 is mapped to a strange number of steps, that does not work. Norm-based complexities such as TE complexity are foolproof and equave-agnostic, however. For example, the TE complexity of 31et is 30.98, which is close to the edo number as expected for a patent val. But if one were to take the TE complexity of {{val| 1 1900 2785 3370 }}, which is technically a tuning of 1et, they would get 1038.83, which matches the complexity of the tuning much better than the naive approach of simply taking 1 for the complexity, and means that that val is roughly equivalent to 1039et in complexity.
 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|edos]]) typically do not have low error, and very low error temperaments (e.g. [[microtemperament|microtemperaments]]) typically do not 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.
-A commonly used definition of temperament complexity is [[Tenney-Euclidean temperament_measures #TE complexity|Tenney-Euclidean complexity]].
 == Complexity of an interval in a temperament ==