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Complexity: Difference between revisions

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→Complexity of a temperament: move some of Vector's stuff here
Complexity of a just interval and complexity of a tempered interval should always go together
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When a complexity measures is defined in terms of a vector space, it is usually called a '''norm'''.   
When a complexity measures is defined in terms of a vector space, it is usually called a '''norm'''.   


== Complexity of just intervals ==
== Complexity of a just interval ==
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The complexity of a just interval is often measured using height functions.
The complexity of a just interval is often measured using height functions.
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It is usually true that simpler (i.e. less complex) JI intervals are more consonant, however the converse does not hold.
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]].
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 an interval in a temperament ==
Besides saying that an interval 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 of a temperament ==
== Complexity of a temperament ==
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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|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.
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.


== Complexity of an interval in a temperament ==
== Links ==
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.
* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19636.html Yahoo! Tuning Group | ''Complexity terminology wars'']
 
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.


[[Category:Complexity| ]] <!-- Main article -->
[[Category:Complexity| ]] <!-- Main article -->
== Links ==
* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19636.html Yahoo! Tuning Group | ''Complexity terminology wars'']
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