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]].

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>

== Notation ==

The notation "chord1 by chord2" has been proposed as shorthand in lists or tables. This is borrowed from previous use on the Xen Wiki where the second chord is an interval ([https://en.xen.wiki/w/1-11-13-15_by_4/3_bihexany example]).

== Example ==

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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]].

Subtlety: The cross-set of two ''chords'' is properly an unreduced chord, the sumset of two finite subsets of <math>\mathbb{R},</math> whereas the cross-set of two ''scales'' with the same equave is best thought of as the sumset of two finite subsets of <math>\mathbb{R}/(\text{equave})\mathbb{Z}.</math> (While you can theoretically take a cross-set of scales with incommensurable equaves, that requires thinking of the scales as infinite albeit periodically repeating subsets of <math>\mathbb{R}.</math> In fact, the resulting cross-set is dense in <math>\mathbb{R},</math> thus not properly a scale.)

== Music ==

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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]].
 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>
+== Notation ==
+The notation "chord1 by chord2" has been proposed as shorthand in lists or tables. This is borrowed from previous use on the Xen Wiki where the second chord is an interval ([https://en.xen.wiki/w/1-11-13-15_by_4/3_bihexany example]).
 == Example ==
@@ Line 35: / Line 38: @@
 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]].
+Subtlety: The cross-set of two ''chords'' is properly an unreduced chord, the sumset of two finite subsets of <math>\mathbb{R},</math> whereas the cross-set of two ''scales'' with the same equave is best thought of as the sumset of two finite subsets of <math>\mathbb{R}/(\text{equave})\mathbb{Z}.</math> (While you can theoretically take a cross-set of scales with incommensurable equaves, that requires thinking of the scales as infinite albeit periodically repeating subsets of <math>\mathbb{R}.</math> In fact, the resulting cross-set is dense in <math>\mathbb{R},</math> thus not properly a scale.)
 == Music ==
@@ Line 47: / Line 52: @@
 ; [[Frédéric Gagné]]
-* [https://musescore.com/user/5995996/scores/11287339 ''Floating in Outer Space'']
+* [https://youtu.be/MkfA_mtfrRQ ''Floating in Outer Space'']
 == References ==