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 just intervals
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 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.
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 edos) typically do not have low error, and very low error temperaments (e.g. 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 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 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.