Cross-set - Revision history

Inthar: /* Examples */

2026-04-16T08:25:25Z

Examples

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	* cross-set(Pyth[5], 9:10:11)		* cross-set(Pyth[5], 9:10:11)
	** Note: detempered 2.3.5.11 Porcupine[15]; well-formed [[generator sequence]] GS(10/9, 11/10, 12/11)		** Note: detempered 2.3.5.11 Porcupine[15]; well-formed [[generator sequence]] GS(10/9, 11/10, 12/11)
	* cross-set(9/8-14/11-4/3-3/2-56/33-21/11-2/1, 9/7)		* {{nowrap\|cross-set(9/8-14/11-4/3-3/2-56/33-21/11-2/1, 9/7)}}

	=== Condition for interleaving ===		=== Condition for interleaving ===

Inthar: /* Examples */

2026-04-16T08:24:50Z

Examples

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	Interleavings can easily be built from a harmonic series mode as the strand: for example, if ''n''::2''n'' is the strand, then (2''n'' + 1)/''2n'' always works as the offset (e.g. strand 5:6:7:8:9:10, offset 10:11). Here are some other examples:		Interleavings can easily be built from a harmonic series mode as the strand: for example, if ''n''::2''n'' is the strand, then (2''n'' + 1)/''2n'' always works as the offset (e.g. strand 5:6:7:8:9:10, offset 10:11). Here are some other examples:
	* 12:14:16:18:21:24 by 5/4		* cross-set(12:14:16:18:21:24, 5/4)
	* 12:14:16:18:21:24 by 11:12		* cross-set(12:14:16:18:21:24, 11:12)
	* 12:14:16:18:21:24 by 12:13:22		* cross-set(12:14:16:18:21:24, 12:13:22)
	* 12:14:16:18:21:24 by 8:10:11		* cross-set(12:14:16:18:21:24, 8:10:11)
	* 12:14:16:18:21:24 by 9:10:11		* cross-set(12:14:16:18:21:24, 9:10:11)
	** Note: detempered 11-limit Porcupine[15]; well-formed [[generator sequence]] GS(10/9, 11/10, 12/11, 10/9, 11/10, 12/11, 10/9, 11/10, 189/176)		** Note: detempered 11-limit Porcupine[15]; well-formed [[generator sequence]] GS(10/9, 11/10, 12/11, 10/9, 11/10, 12/11, 10/9, 11/10, 189/176)
	* Pyth[5] by 8:10:11		* cross-set(Pyth[5], 8:10:11)
	* Pyth[5] by 9:10:11		* cross-set(Pyth[5], 9:10:11)
	** Note: detempered 2.3.5.11 Porcupine[15]; well-formed [[generator sequence]] GS(10/9, 11/10, 12/11)		** Note: detempered 2.3.5.11 Porcupine[15]; well-formed [[generator sequence]] GS(10/9, 11/10, 12/11)
	* 9/8-14/11-4/3-3/2-56/33-21/11-2/1 by 9/7		* cross-set(9/8-14/11-4/3-3/2-56/33-21/11-2/1, 9/7)

	=== Condition for interleaving ===		=== Condition for interleaving ===

Inthar: /* Notation */

2026-04-16T08:22:55Z

Notation

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	== Notation ==		== 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 ~~rhe~~ second chord is an interval ([https://en.xen.wiki/w/1-11-13-15_by_4/3_bihexany example]).		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]).

	== Examples ==		== Examples ==

Inthar: /* Conventions */

2026-04-16T08:22:44Z

Conventions

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	{{Adv\|This operation can be iterated over more than two chords (and is commutative and associative; mathematical parlance calls this operation a ''sumset'').}}		{{Adv\|This operation can be iterated over more than two chords (and is commutative and associative; mathematical parlance calls this operation a ''sumset'').}}

	== ~~Conventions~~ ==		== 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 rhe second chord is an interval ([https://en.xen.wiki/w/1-11-13-15_by_4/3_bihexany example]).		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 rhe second chord is an interval ([https://en.xen.wiki/w/1-11-13-15_by_4/3_bihexany example]).

Inthar: /* Conventions */

2026-04-16T08:22:31Z

Conventions

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	== Conventions ==		== Conventions ==
	~~''todo: phase out "by"''?~~		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 rhe second chord is an interval ([https://en.xen.wiki/w/1-11-13-15_by_4/3_bihexany example]).

	~~On this wiki, a cross-set of two chords may be denoted~~ "chord1 by chord2" as shorthand in lists or tables ~~(and the equave is usually understood)~~. This ~~convention~~ is borrowed from previous use on the Xen Wiki ([https://en.xen.wiki/w/1-11-13-15_by_4/3_bihexany example]).

	== Examples ==		== Examples ==

Inthar: /* Conventions */

2026-04-16T03:01:45Z

Conventions

← Older revision		Revision as of 03:01, 16 April 2026
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	''todo: phase out "by"''?		''todo: phase out "by"''?

	On this wiki, a cross-set of two chords may be denoted "chord1 by chord2" ~~(in typeset math, <math>C_1 \ \mathsf{by} \ C_2</math>) in more technical contexts or~~ as shorthand in lists or tables (and the equave is usually understood). This convention is borrowed from previous use on the Xen Wiki ([https://en.xen.wiki/w/1-11-13-15_by_4/3_bihexany example]).		On this wiki, a cross-set of two chords may be denoted "chord1 by chord2" as shorthand in lists or tables (and the equave is usually understood). This convention is borrowed from previous use on the Xen Wiki ([https://en.xen.wiki/w/1-11-13-15_by_4/3_bihexany example]).

	== Examples ==		== Examples ==

Inthar: /* Conventions */

2026-04-16T02:53:07Z

Conventions

← Older revision		Revision as of 02:53, 16 April 2026
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	== Conventions ==		== Conventions ==
	''todo: phase out "by"''		''todo: phase out "by"''?

	On this wiki, a cross-set of two chords may be denoted "chord1 by chord2" (in typeset math, <math>C_1 \ \mathsf{by} \ C_2</math>) in more technical contexts or as shorthand in lists or tables (and the equave is usually understood). This convention is borrowed from previous use on the Xen Wiki.		On this wiki, a cross-set of two chords may be denoted "chord1 by chord2" (in typeset math, <math>C_1 \ \mathsf{by} \ C_2</math>) in more technical contexts or as shorthand in lists or tables (and the equave is usually understood). This convention is borrowed from previous use on the Xen Wiki ([https://en.xen.wiki/w/1-11-13-15_by_4/3_bihexany example]).

	== Examples ==		== Examples ==

Inthar: /* Conventions */

2026-04-16T02:51:09Z

Conventions

← Older revision		Revision as of 02:51, 16 April 2026
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	''todo: phase out "by"''		''todo: phase out "by"''

	On this wiki, a cross-set of two chords may be denoted "chord1 by chord2" (in typeset math, <math>C_1 \ \mathsf{by} \ C_2</math>) in more technical contexts or as shorthand (and the equave is usually understood).		On this wiki, a cross-set of two chords may be denoted "chord1 by chord2" (in typeset math, <math>C_1 \ \mathsf{by} \ C_2</math>) in more technical contexts or as shorthand in lists or tables (and the equave is usually understood). This convention is borrowed from previous use on the Xen Wiki.

	== Examples ==		== Examples ==

Inthar: /* Conventions */

2026-04-16T02:50:34Z

Conventions

← Older revision		Revision as of 02:50, 16 April 2026
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	''todo: phase out "by"''		''todo: phase out "by"''

	On this wiki, a cross-set of two chords may be denoted "chord1 by chord2" (in typeset math, <math>C_1 \ \mathsf{by} \ C_2</math>) in more technical contexts or as shorthand (and the equave is usually understood). {{Adv\|An ''n''-fold iterated cross-set may be written chord<sup>by ''n''</sup>, and by convention chord<sup>by 0</sup> is a singleton consisting only of the unison (and equave multiples if equave reduction is applied).}}		On this wiki, a cross-set of two chords may be denoted "chord1 by chord2" (in typeset math, <math>C_1 \ \mathsf{by} \ C_2</math>) in more technical contexts or as shorthand (and the equave is usually understood).

	== Examples ==		== Examples ==

Inthar: /* Examples */

2026-04-16T02:50:09Z

Examples

← Older revision		Revision as of 02:50, 16 April 2026
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	=== Examples ===		=== Examples ===
	Pental blackdye is an example (pyth[5] by 9:10 = sLmLsLmLsL with L = 10/9). {{Adv\|More generally: If ''w''('''x''', '''y''') (a sequence of step sizes '''x''' and '''y''') is a binary scale, then the ternary scale ''w''('''z'''('''x'''-'''z'''), '''z'''('''y'''-'''z''')) (the same sequence but with substitutions '''x''' -> '''z'''('''x'''-'''z'''), '''y''' -> '''z'''('''y'''-'''z''')) is an interleaving, namely ''w''('''x''', '''y''') by '''z'''.}}		Pental blackdye is an example; it is cross-set(pyth[5], 9:10) = sLmLsLmLsL with L = 10/9). {{Adv\|More generally: If ''w''('''x''', '''y''') (a sequence of step sizes '''x''' and '''y''') is a binary scale, then the ternary scale ''w''('''z'''('''x'''-'''z'''), '''z'''('''y'''-'''z''')) (the same sequence but with substitutions '''x''' -> '''z'''('''x'''-'''z'''), '''y''' -> '''z'''('''y'''-'''z''')) is an interleaving, namely {{nowrap\|cross-set(''w''('''x''', '''y'''), '''z''').}}}}

	Pajara[10] is an interleaving of 2L3s (generated by a somewhat sharp 3/2) by 600c.		Pajara[10] is an interleaving of 2L3s (generated by a somewhat sharp 3/2) by 600c.