Cross-set scale: Difference between revisions

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A '''cross-set scale''' (or simply '''cross-set''') is a [[scale]] produced by taking every ordered pair in the [[Wikipedia:Cartesian product|Cartesian product]] of two scales, or of a scale with itself, and stacking ~~all~~ elements in each ordered pair. Cross-set scales may also be generalized to more than two initial scales.

A '''cross-set scale''' (or simply '''cross-set''') is a [[scale]] produced by taking every ordered pair in the [[Wikipedia:Cartesian product|Cartesian product]] of two scales, or of a scale with itself, and stacking both elements in each ordered pair. Cross-set scales may also be generalized to more than two initial scales.

If the second scale is the inverse of the first scale (e.g. ''a'' becomes 1/''a''), the result is a reciprocal cross-set (scale). If additionally the first scale is a sequence of odd harmonics starting from 1, the result is a [[tonality diamond]].

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In mathematical notation, the cross-set of scales ''A'', ''B'', ..., ''Z'' is (note that interval stacking has been written as addition):

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

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

In combinatorics, this operation is called a [[wikipedia:Sumset|sumset]].

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; [[Frédéric Gagné]]

* [https://~~musescore~~.~~com~~/~~user/5995996/scores/11287339~~ ''Floating in Outer Space'']

* [https://youtu.be/MkfA_mtfrRQ ''Floating in Outer Space'']

== References ==

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-A '''cross-set scale''' (or simply '''cross-set''') is a [[scale]] produced by taking every ordered pair in the [[Wikipedia:Cartesian product|Cartesian product]] of two scales, or of a scale with itself, and stacking all elements in each ordered pair. Cross-set scales may also be generalized to more than two initial scales.
+A '''cross-set scale''' (or simply '''cross-set''') is a [[scale]] produced by taking every ordered pair in the [[Wikipedia:Cartesian product|Cartesian product]] of two scales, or of a scale with itself, and stacking both elements in each ordered pair. Cross-set scales may also be generalized to more than two initial scales.
 If the second scale is the inverse of the first scale (e.g. ''a'' becomes 1/''a''), the result is a reciprocal cross-set (scale). If additionally the first scale is a sequence of odd harmonics starting from 1, the result is a [[tonality diamond]].
@@ Line 35: / Line 35: @@
 In mathematical notation, the cross-set of scales ''A'', ''B'', ..., ''Z'' is (note that interval stacking has been written as addition):
-<math>\text{Cross-set}(A, B, ..., Z) = A + B + \cdots + Z = \{ a + b + \cdots + z : (a, b, ..., z) \in A \times B \times \cdots \times Z\}.</math>
+<math>\begin{align*}\text{Cross-set}(A, B, ..., Z) &= A + B + \cdots + Z \\ &= \{ a + b + \cdots + z : (a, b, ..., z) \in A \times B \times \cdots \times Z\}.\end{align*}</math>
 In combinatorics, this operation is called a [[wikipedia:Sumset|sumset]].
@@ Line 47: / Line 47: @@
 ; [[Frédéric Gagné]]
-* [https://musescore.com/user/5995996/scores/11287339 ''Floating in Outer Space'']
+* [https://youtu.be/MkfA_mtfrRQ ''Floating in Outer Space'']
 == References ==