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'''. | |||
== Complexity of a just interval == | |||
{{Main| Height }} | |||
The complexity of a just interval is often measured using height functions. | |||
Generally these can be thought of as measuring the size of the numerator and denominator when expressed in lowest terms. | |||
A simple example of a height function is the [[Weil height]] (or [[integer limit]]), which is simply the maximum of the numerator and denominator of the ratio. | |||
There are various measures of complexity for rational intervals. | |||
Commonly used 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 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 == | |||
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. | |||
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. | |||
== Links == | |||
* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19636.html Yahoo! Tuning Group | ''Complexity terminology wars''] | |||
[[Category:Complexity| ]] <!-- Main article --> | |||