Cross-set scale: Difference between revisions

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A '''cross-set scale''' is a [[scale]] ~~generated~~ by taking every ordered pair in the [[Wikipedia:Cartesian product|Cartesian product]] of two or more scales, or of a scale with itself, and stacking all elements in each ordered pair. In mathematical notation, the cross-set of scales ''A'', ''B'', ..., ''Z'' is (note that stacking has been written as addition):

A '''cross-set scale''' is a [[scale]] produced by taking every ordered pair in the [[Wikipedia:Cartesian product|Cartesian product]] of two or more scales, or of a scale with itself, and stacking all elements in each ordered pair. In mathematical notation, the cross-set of scales ''A'', ''B'', ..., ''Z'' is (note that stacking has been written as addition):

<math>\text{Cross-set}(A, B, ..., Z) = \{ a + b + \cdots + z : (a, b, ..., z) \in A \times B \times \cdots \times Z\}.</math>

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The term ''cross-set'' goes back to [[Erv Wilson]].<ref name="Narushima 2017">Narushima, T. (2017). Microtonality and the tuning systems of Erv Wilson. Routledge.</ref>

== Examples ==

The 4:5:6:7 cross-set scale is ~~generated~~ by multiplying every pair of intervals from the 4:5:6:7 tetrad ([[1/1]] - [[5/4]] - [[3/2]] - [[7/4]]), including an interval with itself, and [[Octave reduction|octave-reducing]] as necessary. It contains 10 distinct pitches out of 16 combinations.

The 4:5:6:7 cross-set scale is produced by multiplying every pair of intervals from the 4:5:6:7 tetrad ([[1/1]] - [[5/4]] - [[3/2]] - [[7/4]]), including an interval with itself, and [[Octave reduction|octave-reducing]] as necessary. It contains 10 distinct pitches out of 16 combinations.

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-A '''cross-set scale''' is a [[scale]] generated by taking every ordered pair in the [[Wikipedia:Cartesian product|Cartesian product]] of two or more scales, or of a scale with itself, and stacking all elements in each ordered pair. In mathematical notation, the cross-set of scales ''A'', ''B'', ..., ''Z'' is (note that stacking has been written as addition):
+A '''cross-set scale''' is a [[scale]] produced by taking every ordered pair in the [[Wikipedia:Cartesian product|Cartesian product]] of two or more scales, or of a scale with itself, and stacking all elements in each ordered pair. In mathematical notation, the cross-set of scales ''A'', ''B'', ..., ''Z'' is (note that stacking has been written as addition):
 <math>\text{Cross-set}(A, B, ..., Z) = \{ a + b + \cdots + z : (a, b, ..., z) \in A \times B \times \cdots \times Z\}.</math>
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 The term ''cross-set'' goes back to [[Erv Wilson]].<ref name="Narushima 2017">Narushima, T. (2017). Microtonality and the tuning systems of Erv Wilson. Routledge.</ref>
 == Examples ==
-The 4:5:6:7 cross-set scale is generated by multiplying every pair of intervals from the 4:5:6:7 tetrad ([[1/1]] - [[5/4]] - [[3/2]] - [[7/4]]), including an interval with itself, and [[Octave reduction|octave-reducing]] as necessary. It contains 10 distinct pitches out of 16 combinations.
+The 4:5:6:7 cross-set scale is produced by multiplying every pair of intervals from the 4:5:6:7 tetrad ([[1/1]] - [[5/4]] - [[3/2]] - [[7/4]]), including an interval with itself, and [[Octave reduction|octave-reducing]] as necessary. It contains 10 distinct pitches out of 16 combinations.
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