Cross-set scale: Difference between revisions
No edit summary |
Improve lead section, move math theory in a separate section, misc. edits |
||
| Line 1: | Line 1: | ||
A '''cross-set scale''' is a [[scale]] produced by taking every ordered pair in the [[Wikipedia:Cartesian product|Cartesian product]] of two | 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. | ||
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. | ||
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 | |||
{| class="wikitable center-all" | {| 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. | 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>\text{Cross-set}(A, B, ..., Z) = \{ a + b + \cdots + z : (a, b, ..., z) \in A \times B \times \cdots \times Z\}.</math> | |||
In combinatorics, this operation is called a [[wikipedia:Sumset|sumset]]. | |||
== Music == | == Music == | ||
| Line 41: | Line 47: | ||
; [[Frédéric Gagné]] | ; [[Frédéric Gagné]] | ||
* [https://musescore.com/user/5995996/scores/11287339 | * [https://musescore.com/user/5995996/scores/11287339 ''Floating in Outer Space''] | ||
== References == | == References == | ||
<references /> | |||
[[Category:Scale]] | [[Category:Scale]] | ||
Revision as of 05:25, 25 July 2023
A cross-set scale (or simply cross-set) is a scale produced by taking every ordered pair in the 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.
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.[1]
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-reducing as necessary. It contains 10 distinct intervals out of 16 combinations.
| 1/1 × 1/1 1/1 |
5/4 × 1/1 5/4 |
3/2 × 1/1 3/2 |
7/4 × 1/1 7/4 |
| 1/1 × 5/4 5/4 |
5/4 × 5/4 25/16 |
3/2 × 5/4 15/8 |
7/4 × 5/4 35/32 |
| 1/1 × 3/2 3/2 |
5/4 × 3/2 15/8 |
3/2 × 3/2 9/8 |
7/4 × 3/2 21/16 |
| 1/1 × 7/4 7/4 |
5/4 × 7/4 35/32 |
3/2 × 7/4 21/16 |
7/4 × 7/4 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]\displaystyle{ \text{Cross-set}(A, B, ..., Z) = \{ a + b + \cdots + z : (a, b, ..., z) \in A \times B \times \cdots \times Z\}. }[/math]
In combinatorics, this operation is called a sumset.
Music
4:5:6:7 cross-set tuning
References
- ↑ Narushima, T. (2017). Microtonality and the tuning systems of Erv Wilson. Routledge.