Scale diversity
Diversity is a scale measurement which categorizes scales according to the "diversity" of available intervals. As a general rule of thumb, scales with many unique interval sizes will have a high diversity. Similarly, scales with many redundant intervals will be assigned a low diversity rating.
Properties:
- Degenerate scales (scales with no intervals smaller than an octave and larger than a unison) have a diversity of 0.
- EDOs have a diversity of 1.
- Div(S) ≥ 0 since there are no intervals larger than an octave.
Definition:
[math]\mathrm{Div} (S) = - \log_N \left( \sum \limits_{x \in X} x^2 \right)[/math]
[math]X = \mathrm{steps} ( \mathrm{sort} ( \mathrm{dia} (S)))[/math]
Where:
S is a multiset.
N is the cardinality of S.
dia(S) is the diamond function.
sort(S) returns a tuple with all of the elements of S in non-decreasing order.
steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T
The scale diversity measure was proposed by Ryan Avella.