View source for Cross-set scale

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>

== Example ==
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.
{| class="wikitable center-all"
|-
|1/1 × 1/1<br>1/1
|5/4 × 1/1<br>5/4
|3/2 × 1/1<br>3/2
|7/4 × 1/1<br>7/4
|-
|1/1 × 5/4<br>5/4
|5/4 × 5/4<br>25/16
|3/2 × 5/4<br>15/8
|7/4 × 5/4<br>35/32
|-
|1/1 × 3/2<br>3/2
|5/4 × 3/2<br>15/8
|3/2 × 3/2<br>9/8
|7/4 × 3/2<br>21/16
|-
|1/1 × 7/4<br>7/4
|5/4 × 7/4<br>35/32
|3/2 × 7/4<br>21/16
|7/4 × 7/4<br>49/32
|}

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 scale ===
; [[Nick Vuci]]
* [[:File:NV-20210528-4567CrossSet-PreludeAndFugue.mp3|Prelude and Fugue]]
* [[:File:NV-20210523-4567crossSet-jam-synthOne padVKAnalyzer.mp3|Short ambient piece]]
* [[:File:NV-20210508-4567CrossSet-SynthOne Garageband.mp3|Improv]]

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

== References ==
<references />

[[Category:Scale]]