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.

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 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.

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 [[generator]]s, 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.

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.

Note that the concept of complexity applies not only to [[rank-2 temperament]]s, but temperaments of any rank. For higher-rank 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...~~

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

* [[Graham complexity]]

* [[Graham complexity]] (chord complexity)

* [[Generator complexity]]

* [[Generator complexity]] (temperament complexity)

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

* [[Tenney-Euclidean temperament_measures #TE complexity|TE complexity]] (temperament complexity)

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

* Various [[height]] functions (interval complexity in JI)

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

* [[Tenney-Euclidean_metrics #Temperamental complexity|Interval temperamental complexity]] (interval complexity in temperaments)

[[Category:Complexity| ]]

[[Category:Theory]]

[[Category:Overview]]

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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.
-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 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.
+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 [[generator]]s, 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.
+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.
+Note that the concept of complexity applies not only to [[rank-2 temperament]]s, but temperaments of any rank. For higher-rank 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...
+Some specific, mathematically rigorous definitions of "complexity" are…
-* [[Graham complexity]]
+* [[Graham complexity]] (chord complexity)
-* [[Generator complexity]]
+* [[Generator complexity]] (temperament complexity)
-* [[Tenney-Euclidean_temperament_measures#TE complexity|TE complexity]]
+* [[Tenney-Euclidean temperament_measures #TE complexity|TE complexity]] (temperament complexity)
-* [[Tenney-Euclidean_metrics#Temperamental complexity|Temperamental complexity]]
+* Various [[height]] functions (interval complexity in JI)
-* [[Benedetti height]]</li></ul>
+* [[Tenney-Euclidean_metrics #Temperamental complexity|Interval temperamental complexity]] (interval complexity in temperaments)
 [[Category:Complexity| ]] <!-- main article -->
+[[Category:Theory]]
+[[Category:Overview]]