Complexity: Difference between revisions

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In tuning, '''complexity''' can be said with respect to individual [[interval]]s, [[chord]]s, [[scale]]s as well as the entire [[tuning system]]. While mathematically rigorous measurements of complexity are not available for all contexts and purposes, some of them have been extensively studied, including those of [[regular temperament]]s and of just or tempered [[interval]]s.

When a complexity measures is defined in terms of a vector space, it is usually called a '''norm'''.

== Comexity of just intervals ==

:''Main article: [[height]]''

The complexity of a just interval is often called the '''height'''.

There are various measures of complexity for rational intervals. Generally these can be tought of as measuring the size of the numerator and denominator when expressed in lowest terms.

Specific examples of these are [[Benedetti height]], [[Tenney height]], [[Wilson height]] and the [[Tenney-Euclidean metrics#TE_norm|Tenney-Euclidean norm]].

=== Relationship to consonance ===

It is usually true that simpler (i.e. less complex) JI intervals are more consonant, however the converse does not hold.

Examples of this are easy to find. Consider for example an interval such as 3001/2001, which is very complex but still sounds consonant due to its proximity to [[3/2]].

== Complexity of a temperament ==

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) typically do not have low error, and very low error temperaments (e.g. [[microtemperament]]~~s or [[JI]] itself~~) 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.

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.

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 ==

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.

An example of temperament interval complexity is the [[Tenney-Euclidean_metrics #TE temperamental norm|Tenney-Euclidean temperamental norm]].

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

For an example of this, see [[Graham complexity]].

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…~~

* [[Graham complexity]] (chord complexity)

* [[Generator complexity]] (temperament complexity)

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

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

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

[[Category:Complexity| ]]

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 In tuning, '''complexity''' can be said with respect to individual [[interval]]s, [[chord]]s, [[scale]]s as well as the entire [[tuning system]]. While mathematically rigorous measurements of complexity are not available for all contexts and purposes, some of them have been extensively studied, including those of [[regular temperament]]s and of just or tempered [[interval]]s.
+When a complexity measures is defined in terms of a vector space, it is usually called a '''norm'''.
+== Comexity of just intervals ==
+:''Main article: [[height]]''
+The complexity of a just interval is often called the '''height'''.
+There are various measures of complexity for rational intervals. Generally these can be tought of as measuring the size of the numerator and denominator when expressed in lowest terms.
+Specific examples of these are [[Benedetti height]], [[Tenney height]], [[Wilson height]] and the [[Tenney-Euclidean metrics#TE_norm|Tenney-Euclidean norm]].
+=== Relationship to consonance ===
+It is usually true that simpler (i.e. less complex) JI intervals are more consonant, however the converse does not hold.
+Examples of this are easy to find. Consider for example an interval such as 3001/2001, which is very complex but still sounds consonant due to its proximity to [[3/2]].
 == Complexity of a temperament ==
 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) typically do not have low error, and very low error temperaments (e.g. [[microtemperament]]s or [[JI]] itself) 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.
+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.
+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 ==
 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.
+An example of temperament interval complexity is the [[Tenney-Euclidean_metrics #TE temperamental norm|Tenney-Euclidean temperamental norm]].
 The ''complexity of a chord'' likewise refers to the number of generator steps required to generate all the pitches of the chord.
+For an example of this, see [[Graham complexity]].
 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…
-* [[Graham complexity]] (chord complexity)
-* [[Generator complexity]] (temperament complexity)
-* [[Tenney-Euclidean temperament_measures #TE complexity|TE complexity]] (temperament complexity)
-* Various [[height]] functions (interval complexity in JI)
-* [[Tenney-Euclidean_metrics #TE temperamental norm|Interval temperamental complexity]] (interval complexity in temperaments)
 [[Category:Complexity| ]] <!-- Main article -->