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''' (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.

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

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]].

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

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>

~~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>~~

== Example ==

=~~= Examples~~ ==

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 intervals out of 16 combinations.

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.

{| class="wikitable center-all"

|-

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The starting scales do not need to be in [[just intonation]]; a cross-set scale could be constructed from any kind of scale.

== Theory ==

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

<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]].

== Music ==

=== 4:5:6:7 cross-set ~~tuning~~ ===

=== 4:5:6:7 cross-set scale ===

; [[Nick Vuci]]

* [[:File:NV-20210528-4567CrossSet-PreludeAndFugue.mp3|Prelude and Fugue]]

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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'']

~~== See also ==~~

* [[Tonality diamond]]

== References ==

[[Category:Scale]]

@@ Line 1: / Line 1: @@
-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''' (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.
-<math>\text{Cross-set}(A, B, ..., Z) = \{ a + b + \cdots + z : (a, b, ..., z) \in A \times B \times \cdots \times Z\}.</math>
+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]].
-In combinatorics, this operation is called a [[wikipedia:Sumset|sumset]].
+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>
-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>
+== Example ==
-== Examples ==
+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 intervals out of 16 combinations.
-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.
 {| class="wikitable center-all"
 |-
@@ Line 32: / Line 31: @@
 The starting scales do not need to be in [[just intonation]]; a cross-set scale could be constructed from any kind of scale.
+== Theory ==
+In mathematical notation, the cross-set of scales ''A'', ''B'', ..., ''Z'' is (note that interval stacking has been written as addition):
+<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]].
 == Music ==
-=== 4:5:6:7 cross-set tuning ===
+=== 4:5:6:7 cross-set scale ===
 ; [[Nick Vuci]]
 * [[:File:NV-20210528-4567CrossSet-PreludeAndFugue.mp3|Prelude and Fugue]]
@@ Line 41: / Line 47: @@
 ; [[Frédéric Gagné]]
-* [https://musescore.com/user/5995996/scores/11287339|''Floating in Outer Space'']
+* [https://youtu.be/MkfA_mtfrRQ ''Floating in Outer Space'']
-== See also ==
-* [[Tonality diamond]]
 == References ==
+<references />
 [[Category:Scale]]