Cross-set scale: Difference between revisions

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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>

== 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 ==

@@ Line 4: / Line 4: @@
 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 ==