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]\displaystyle{ \mathrm{Div} (S) = - \log_N \left( \sum \limits_{x \in X} x^2 \right) }[/math]

[math]\displaystyle{ 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.