Cross-set scale - Revision history

Inthar at 13:12, 16 April 2026

2026-04-16T13:12:23Z

← Older revision		Revision as of 13:12, 16 April 2026
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	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>

			== Notation ==
			The notation "chord1 by chord2" has been proposed as shorthand in lists or tables. This is borrowed from previous use on the Xen Wiki where the second chord is an interval ([https://en.xen.wiki/w/1-11-13-15_by_4/3_bihexany example]).

	== Example ==		== Example ==

Inthar: /* Theory */

2025-12-26T03:52:14Z

Theory

← Older revision		Revision as of 03:52, 26 December 2025
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	In combinatorics, this operation is called a [[wikipedia:Sumset\|sumset]].		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 <math>\mathbb{R},</math> whereas the cross-set of two ''scales'' with the same equave is best thought of as the sumset of two finite subsets of <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 properly a scale.		Subtlety: The cross-set of two ''chords'' is properly an unreduced chord, the sumset of two finite subsets of <math>\mathbb{R},</math> whereas the cross-set of two ''scales'' with the same equave is best thought of as the sumset of two finite subsets of <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 properly a scale.)

	== Music ==		== Music ==

Inthar: /* Theory */

2025-12-26T03:48:32Z

Theory

← Older revision		Revision as of 03:48, 26 December 2025
Line 39:		Line 39:
	In combinatorics, this operation is called a [[wikipedia:Sumset\|sumset]].		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 <math>\mathbb{R},</math> whereas the cross-set of two ''scales'' with the same equave is best thought of as the sumset of two finite subsets of <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 ~~very practical~~.		Subtlety: The cross-set of two ''chords'' is properly an unreduced chord, the sumset of two finite subsets of <math>\mathbb{R},</math> whereas the cross-set of two ''scales'' with the same equave is best thought of as the sumset of two finite subsets of <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 properly a scale.

	== Music ==		== Music ==

Inthar at 03:43, 26 December 2025

2025-12-26T03:43:00Z

← Older revision		Revision as of 03:43, 26 December 2025
Line 39:		Line 39:
	In combinatorics, this operation is called a [[wikipedia:Sumset\|sumset]].		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 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 very practical.		Subtlety: The cross-set of two ''chords'' is properly an unreduced chord, the sumset of two finite subsets of <math>\mathbb{R},</math> whereas the cross-set of two ''scales'' with the same equave is best thought of as the sumset of two finite subsets of <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 very practical.

	== Music ==		== Music ==

Inthar: /* Theory */

2025-12-26T03:40:45Z

Theory

← Older revision		Revision as of 03:40, 26 December 2025
Line 39:		Line 39:
	In combinatorics, this operation is called a [[wikipedia:Sumset\|sumset]].		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 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~~.		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 very practical.

	== Music ==		== Music ==

Inthar: /* Theory */

2025-12-26T03:37:57Z

Theory

← Older revision		Revision as of 03:37, 26 December 2025
Line 39:		Line 39:
	In combinatorics, this operation is called a [[wikipedia:Sumset\|sumset]].		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 ==		== Music ==

Inthar at 03:37, 26 December 2025

2025-12-26T03:37:09Z

← Older revision		Revision as of 03:37, 26 December 2025
Line 39:		Line 39:
	In combinatorics, this operation is called a [[wikipedia:Sumset\|sumset]].		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>		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.

	== Music ==		== Music ==

Inthar: /* Theory */

2025-12-26T03:34:49Z

Theory

← Older revision		Revision as of 03:34, 26 December 2025
Line 38:		Line 38:

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

	== Music ==		== Music ==

Fredg999: /* 4:5:6:7 cross-set scale */ Replace MuseScore link with YouTube link

2024-01-02T22:28:50Z

4:5:6:7 cross-set scale: Replace MuseScore link with YouTube link

← Older revision		Revision as of 22:28, 2 January 2024
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	; [[Frédéric Gagné]]		; [[Frédéric Gagné]]
	* [https://~~musescore~~.~~com~~/~~user/5995996/scores/11287339~~ ''Floating in Outer Space'']		* [https://youtu.be/MkfA_mtfrRQ ''Floating in Outer Space'']

	== References ==		== References ==

Inthar: /* Theory */

2023-11-04T18:32:34Z

Theory

← Older revision		Revision as of 18:32, 4 November 2023
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	In mathematical notation, the cross-set of scales ''A'', ''B'', ..., ''Z'' is (note that interval stacking has been written as addition):		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 + \cdots + z : (a, b, ..., z) \in A \times B \times \cdots \times Z\}.</math>		<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]].		In combinatorics, this operation is called a [[wikipedia:Sumset\|sumset]].