Complexity

In tuning, complexity can be said with respect to individual intervals, chords, scales 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 temperaments and of just or tempered intervals.

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

Complexity of a just interval

Main article: 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 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 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.

An example of temperament interval complexity is the 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 temperaments, 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 temperaments, 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 generators 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 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 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 ⟨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

• Yahoo! Tuning Group | Complexity terminology wars