Glossary of scale properties - Revision history

BudjarnLambeth at 01:54, 20 July 2025

2025-07-20T01:54:42Z

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	A simplified explanation of the various properties of [[periodic scale]]s.		A simplified explanation of the various properties of [[periodic scale]]s. Also check the main [[Glossary]].

	{{TOC Horizontal		{{TOC Horizontal

FloraC: - interval variety (not a property of scales)

2025-07-07T14:00:49Z

- interval variety (not a property of scales)

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	=== H ===		=== H ===
	=== I ===		=== I ===
	~~; [[interval variety]]~~
	~~: The number of different interval qualities available for an interval class in a [[#P\|periodic scale]].~~

	=== J ===		=== J ===
	=== K ===		=== K ===
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	; [[maximum variety]] (MV)		; [[maximum variety]] (MV)
	: The maximum [[~~#I\|~~interval variety]] from all interval classes of a [[#P\|periodic scale]].		: The maximum [[interval variety]] from all interval classes of a [[#P\|periodic scale]].

	; MOS		; MOS
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	=== S ===		=== S ===
	; [[strict variety]] (SV)		; [[strict variety]] (SV)
	: The [[~~#I\|~~interval variety]] of all interval classes of a [[#P\|periodic scale]], when all interval classes have the same interval variety.		: The [[interval variety]] of all interval classes of a [[#P\|periodic scale]], when all interval classes have the same interval variety.

	; symmetry		; symmetry

FloraC: Note trivalence property = SV3

2025-07-07T13:55:04Z

Note trivalence property = SV3

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	; trivalence property		; trivalence property
	: Same as Myhill's property, but replace "two interval sizes" with "three interval sizes." The scale formed from the notes of a dominant 7th chord (e.g. C-E-G-Bb-C) is an example of a trivalent scale.		: Same as [[#M\|Myhill's property]], but replace "two interval sizes" with "three interval sizes". A.k.a. strict variety 3. The scale formed from the notes of a dominant 7th chord (e.g. C-E-G-Bb-C) is an example of a trivalent scale.

	=== U ===		=== U ===

FloraC: Split mos and Myhill's property (even tho mos isn't a property, it's fully defined by one property (MV2))

2025-07-07T13:51:03Z

Split mos and Myhill's property (even tho mos isn't a property, it's fully defined by one property (MV2))

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	=== D ===		=== D ===
	; [[distributional evenness]] (DE)		; [[distributional evenness]] (DE)
	: A scale with two step sizes is ''distributionally even'' if it has its two step sizes distributed as evenly as possible		: A scale with two step sizes is ''distributionally even'' if it has its two step sizes distributed as evenly as possible.

	=== E ===		=== E ===
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	: The maximum [[#I\|interval variety]] from all interval classes of a [[#P\|periodic scale]].		: The maximum [[#I\|interval variety]] from all interval classes of a [[#P\|periodic scale]].

	; ~~Myhill's property and~~ MOS		; MOS
	* A scale ~~has ''Myhill's property''~~ if there are ''~~exactly~~'' two interval sizes for each ~~(reduced)~~ interval class not including the equave. A ~~scale is~~ a ~~[[MOS~~ scale]] if there are ''~~no more than~~'' two interval sizes for each ~~generic~~ interval class not including the equave. ~~This is equivalent to~~ a scale ~~being Myhill~~ with ~~a smaller equave.~~ Myhill's property is ~~sometimes~~ called "strict ~~MOS"~~.		* A scale is a [[mos scale]] if there are ''no more than'' two interval sizes for each generic interval class not including the equave. A.k.a. maximum variety 2.

			; Myhill's property
			* A scale has ''Myhill's property'' if there are ''exactly'' two interval sizes for each interval class not including the equave. A.k.a. strict variety 2. A scale with Myhill's property is called a ''strict mos''.

	=== N ===		=== N ===

FloraC: Copypaste some important properties from the Glossary page. + chirality

2025-07-07T13:00:33Z

Copypaste some important properties from the Glossary page. + chirality

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	=== C ===		=== C ===
			; [[chirality]]
			: A scale is chiral if reversing the order of the steps results in a different scale up to rotation.

	; [[constant structure]] (CS)		; [[constant structure]] (CS)
	: A scale is a constant structure if all intervals of the same size are also within the same generic interval class. A single interval cannot be a part of two classes. The 12-tone diatonic scale does not have this property, since tritones can either be augmented fourths or diminished fifths. This is referred to as the ''partitioning property'' in most academic literature.		: A scale is a constant structure if all intervals of the same size are also within the same generic interval class. A single interval cannot be a part of two classes. The 12-tone diatonic scale does not have this property, since tritones can either be augmented fourths or diminished fifths. This is referred to as the ''partitioning property'' in most academic literature.
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	=== H ===		=== H ===
	=== I ===		=== I ===
			; [[interval variety]]
			: The number of different interval qualities available for an interval class in a [[#P\|periodic scale]].

	=== J ===		=== J ===
	=== K ===		=== K ===
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	; [[maximal evenness]]		; [[maximal evenness]]
	: A [[#P\|periodic]] [[#B\|binary]] scale is maximally even with respect to an [[equal-step tuning]] if it is the result of rounding a smaller equal tuning to the nearest notes of the parent equal tuning with the same equave.		: A [[#P\|periodic]] [[#B\|binary]] scale is maximally even with respect to an [[equal-step tuning]] if it is the result of rounding a smaller equal tuning to the nearest notes of the parent equal tuning with the same equave.

			; [[maximum variety]] (MV)
			: The maximum [[#I\|interval variety]] from all interval classes of a [[#P\|periodic scale]].

	; Myhill's property and MOS		; Myhill's property and MOS
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	=== S ===		=== S ===
			; [[strict variety]] (SV)
			: The [[#I\|interval variety]] of all interval classes of a [[#P\|periodic scale]], when all interval classes have the same interval variety.

	; symmetry		; symmetry
	: A scale is symmetrical if at least one mode of the scale is symmetrical. Therefore, every interval of that mode must have an inverse. These scales will always have an odd number of notes ''per period''. They may not always have an odd number of notes ''per octave'', however. The diatonic scale is symmetrical, but so is 12edo.		: A scale is symmetrical if at least one mode of the scale is symmetrical. Therefore, every interval of that mode must have an inverse. These scales will always have an odd number of notes ''per period''. They may not always have an odd number of notes ''per octave'', however. The diatonic scale is symmetrical, but so is 12edo.

FloraC: Adopt property names if possible (periodic scale -> periodicity). Style

2025-07-07T12:52:37Z

Adopt property names if possible (periodic scale -> periodicity). Style

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	: A scale is a constant structure if all intervals of the same size are also within the same generic interval class. A single interval cannot be a part of two classes. The 12-tone diatonic scale does not have this property, since tritones can either be augmented fourths or diminished fifths. This is referred to as the ''partitioning property'' in most academic literature.		: A scale is a constant structure if all intervals of the same size are also within the same generic interval class. A single interval cannot be a part of two classes. The 12-tone diatonic scale does not have this property, since tritones can either be augmented fourths or diminished fifths. This is referred to as the ''partitioning property'' in most academic literature.

	; [[~~Convex~~ scale\|convexity]]		; [[convex scale\|convexity]]
	: A scale in a [[regular temperament]] is convex if its representation on a [[harmonic lattice diagram]] forms a convex polygon.		: A scale in a [[regular temperament]] is convex if its representation on a [[harmonic lattice diagram]] forms a convex polygon.

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	=== L ===		=== L ===
	=== M ===		=== M ===
	; [[~~Maximal~~ evenness]]		; [[maximal evenness]]
	: A [[#P\|periodic]] [[#B\|binary]] scale is maximally even with respect to an [[equal-step tuning]] if it is the result of rounding a smaller equal tuning to the nearest notes of the parent equal tuning with the same equave.		: A [[#P\|periodic]] [[#B\|binary]] scale is maximally even with respect to an [[equal-step tuning]] if it is the result of rounding a smaller equal tuning to the nearest notes of the parent equal tuning with the same equave.

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	=== O ===		=== O ===
	=== P ===		=== P ===
	; [[~~Periodic~~ scale]]		; [[periodic scale\|periodicity]]
	: A scale is periodic if its [[step pattern]] repeats after a certain [[#I\|interval]].		: A scale is periodic if its [[step pattern]] repeats after a certain [[#I\|interval]].

	; [[~~Pepper~~ ambiguity]]		; [[pepper ambiguity]]
	: The Pepper ambiguity of an [[interval]] in an [[equal-step tuning]] is the ratio of the best approximation to the second best approximation.		: The Pepper ambiguity of an [[interval]] in an [[equal-step tuning]] is the ratio of the best approximation to the second best approximation.

FloraC: Cleanup

2025-07-07T12:29:40Z

Cleanup

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	== Examples ==		== Examples ==
	The ~~12-tone~~ diatonic scale has Myhill's property, and is also distributionally even.		The [[5L 2s\|diatonic scale]] in 12edo has Myhill's property, and is also distributionally even.

	The diminished scale is ~~an MOS~~ with a 1/4-octave period. Because there is only one interval size at the period, it does not have exactly two interval sizes per interval class. Therefore, it is ~~MOS~~, but ~~doesn't~~ have Myhill's property.		The [[diminished scale]] is a mos with a 1/4-octave period. Because there is only one interval size at the period, it does not have exactly two interval sizes per interval class. Therefore, it is a mos, but does not have Myhill's property.

	An ~~EDO~~ is a kind of degenerate ~~MOS~~, in that it is distributionally even. It does not have Myhill's property. In other words, it has no more than two interval sizes for each generic interval class, but does not have exactly two interval sizes.		An edo is a kind of degenerate mos, in that it is distributionally even. It does not have Myhill's property. In other words, it has no more than two interval sizes for each generic interval class, but does not have exactly two interval sizes.

	== See also ==		== See also ==
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	[[Category:Terms\| ]]		[[Category:Terms\| ]]
	[[Category:Scale]]		[[Category:Scale]]
	~~[[Category:~~Todo~~:add examples]]~~		{{Todo\| add illustration }}
	~~[[Category:Todo:~~add illustration]]

FloraC at 11:38, 7 July 2025

2025-07-07T11:38:28Z

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	\| a21=[[#U\|U]]		\| a21=[[#U\|U]]
	\| a22=[[#V\|V]]		\| a22=[[#V\|V]]
	\| a23=[[#W\|W–Z]]		\| a23=[[#W–Z\|W–Z]]
	}}		}}
	=== A ===		=== A ===

FloraC: Fix

2025-07-07T11:38:12Z

Fix

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	\| a21=[[#U\|U]]		\| a21=[[#U\|U]]
	\| a22=[[#V\|V]]		\| a22=[[#V\|V]]
	\| a23=[[#W\|~~W–Z~~]]		\| a23=[[#W\|W–Z]]
	}}		}}
	=== A ===		=== A ===
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	=== Q ===		=== Q ===
	; [[quaternary]] scale		; [[quaternary]] scale
	: A scale of [[#A\|arity]] 3, i.e. with ~~three~~ distinct step sizes.		: A scale of [[#A\|arity]] 4, i.e. with four distinct step sizes.

	=== R ===		=== R ===
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	=== U ===		=== U ===
	=== V ===		=== V ===
	=== ~~W-Z~~ ===		=== W–Z ===

	== Examples ==		== Examples ==

Fredg999: Sort key

2025-04-18T03:20:52Z

Sort key

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	* [[Periodic scale]] – contains mathematical definitions of several scale properties		* [[Periodic scale]] – contains mathematical definitions of several scale properties

			[[Category:Terms\| ]]
	[[Category:Scale]]		[[Category:Scale]]
	~~[[Category:Terms]]~~
	[[Category:Todo:add examples]]		[[Category:Todo:add examples]]
	[[Category:Todo:add illustration]]		[[Category:Todo:add illustration]]