Cross-set scale: Difference between revisions
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Cited Narushima's book which apparently attributes the term "cross-set" to Wilson. |
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In combinatorics, this operation is called a [[wikipedia:Sumset|sumset]]. | 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> | |||
== Examples == | == Examples == | ||
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. | 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. | ||
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== See also == | == See also == | ||
* [[Tonality diamond]] | * [[Tonality diamond]] | ||
== References == | |||
[[Category:Scale]] | [[Category:Scale]] | ||
Revision as of 18:20, 24 July 2023
A cross-set scale is a scale generated by taking every ordered pair in the 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):
[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.
The term cross-set goes back to Erv Wilson.[1]
Examples
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-reducing as necessary. It contains 10 distinct pitches 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.
Music
4:5:6:7 cross-set tuning
See also
References
- ↑ Narushima, T. (2017). Microtonality and the tuning systems of Erv Wilson. Routledge.