Scale diversity: Difference between revisions
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Properties: | 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:= | =Definition:= | ||
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N is the cardinality of S. | N is the cardinality of S. | ||
dia(S) is the [[ | dia(S) is the [[diamond function]]. | ||
sort(S) returns a tuple with all of the elements of S in non-decreasing order. | sort(S) returns a tuple with all of the elements of S in non-decreasing order. | ||
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steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T | steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T | ||
The scale diversity measure was proposed by [[ | The scale diversity measure was proposed by [[Ryan Avella]]. | ||
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[[Category: | [[Category:Math]] | ||
[[Category:Scale]] | |||
{{todo|clarify}} | |||
Latest revision as of 16:32, 26 February 2022
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.