Cross-set scale: Difference between revisions

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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 the number line <math>\mathbb{R},</math> whereas the cross-set of two ''scales'' with the same equave is properly the sumset of two finite subsets of the circle whose circumference is the equave, <math>\mathbb{R}/(\text{equave})\mathbb{Z}.</math> While you can ~~of course~~ 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>, ~~and~~ the resulting cross-set is dense in <math>\mathbb{R},</math> thus not being of so much practicality.

Subtlety: The cross-set of two ''chords'' is properly an unreduced chord, the sumset of two finite subsets of the number line <math>\mathbb{R},</math> whereas the cross-set of two ''scales'' with the same equave is properly the sumset of two finite subsets of the circle whose circumference is the equave, <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 being of so much practicality.

== Music ==

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 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 the number line <math>\mathbb{R},</math> whereas the cross-set of two ''scales'' with the same equave is properly the sumset of two finite subsets of the circle whose circumference is the equave, <math>\mathbb{R}/(\text{equave})\mathbb{Z}.</math> While you can of course 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>, and the resulting cross-set is dense in <math>\mathbb{R},</math> thus not being of so much practicality.
+Subtlety: The cross-set of two ''chords'' is properly an unreduced chord, the sumset of two finite subsets of the number line <math>\mathbb{R},</math> whereas the cross-set of two ''scales'' with the same equave is properly the sumset of two finite subsets of the circle whose circumference is the equave, <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 being of so much practicality.
 == Music ==