Periodic scale - Revision history

Inthar: /* Interval variety */

2025-12-18T04:33:03Z

Interval variety

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	A monotone scale in which every class but classes ''nP'' have exactly two elements is a MOS with period P (as opposed to a fraction of P; that is, a strict MOS), and thus has Myhill's property. If every such class has exactly three elements, it has the '''trivalence property'''.		A monotone scale in which every class but classes ''nP'' have exactly two elements is a MOS with period P (as opposed to a fraction of P; that is, a strict MOS), and thus has Myhill's property. If every such class has exactly three elements, it has the '''trivalence property'''.

	=== ~~Variety~~ ===		=== Interval variety ===
	{{Main\|~~Strict~~ variety}}		{{Main\|Interval variety}}

	A monotone scale in which every class comes in at most ''n'' elements is maximum variety ''n'', or MV''n''. If ''n'' = 2, then it is a MOS.		A monotone scale in which every class comes in at most ''n'' elements is maximum variety ''n'', or MV''n''. If ''n'' = 2, then it is a MOS.

Inthar: /* Variety */

2025-12-18T04:32:22Z

Variety

← Older revision		Revision as of 04:32, 18 December 2025
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	A monotone scale in which every class but classes ''nP'' have exactly two elements is a MOS with period P (as opposed to a fraction of P; that is, a strict MOS), and thus has Myhill's property. If every such class has exactly three elements, it has the '''trivalence property'''.		A monotone scale in which every class but classes ''nP'' have exactly two elements is a MOS with period P (as opposed to a fraction of P; that is, a strict MOS), and thus has Myhill's property. If every such class has exactly three elements, it has the '''trivalence property'''.

	=== ~~Distributional evenness~~ ===		=== Variety ===
	{{Main\| ~~Distributional evenness~~ }}		{{Main\|Strict variety}}

	A monotone scale in which every class comes in ~~exactly~~ ''n'' elements is ''n''~~-distributionally even~~, or ''n''~~-DE~~. If ''n'' = 2, then ~~we can simply say that~~ it is ~~distributionally even and is thus~~ a MOS ~~(of a more general form)~~. ~~Some authors prefer~~ a ~~stricter definition of~~ MOS ~~identifying it~~ with Myhill's property.		A monotone scale in which every class comes in at most ''n'' elements is maximum variety ''n'', or MV''n''. If ''n'' = 2, then it is a MOS.

			A monotone scale in which every class comes in exactly ''n'' elements is ''strict variety n'', or SV''n''. If ''n'' = 2, then it is a 1-period MOS or equivalently a scale with Myhill's property.

	=== Convexity ===		=== Convexity ===

VectorGraphics at 23:56, 24 April 2025

2025-04-24T23:56:50Z

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	A class is a category of all intervals spanning a specified number of scale degrees, such as seconds, thirds, fourths etc in diatonic, or the generalization to any kind of scale.		A class is a category of all intervals spanning a specified number of scale degrees, such as seconds, thirds, fourths etc in diatonic, or the generalization to any kind of scale.

	In mathematical terms, we can define a function class(''k'') on the integers which gives sets representing the ''generic intervals'' of a periodic scale. For some integer ''k'', the set class(''k'') consists of all intervals <math>s[k+i] - s[i]</math>. Since ''s'' is quasiperiodic, class(''P'') only contains the period ''O'', but the rest may contain multiple intervals.		In mathematical terms, we can define a function class(''k'') on the integers which gives sets representing the ''generic intervals'' of a periodic scale. For some integer ''k'', the set class(''k'') consists of all intervals <math>s[k+i] - s[i]</math>. Equivalently, it is all the intervals found on the same degree of the different modes of the scale, or all the intervals between notes a given number of scale steps apart. Since ''s'' is quasiperiodic, class(''P'') only contains the period ''O'', but the rest may contain multiple intervals.

	=== Step form and cumulative form ===		=== Step form and cumulative form ===
	Given a periodic scale as defined above, we may define its ''step form'' as		Given a periodic scale, we may call the function defined above the "cumulative form", and we may define its ''step form'' as

	<math>\Delta s[i] = s[i+1] - s[i],</math>		<math>\Delta s[i] = s[i+1] - s[i],</math>

Fredg999: /* Epimorphism */ Use Main and See also templates

2025-04-24T21:51:56Z

Epimorphism: Use Main and See also templates

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	=== Epimorphism ===		=== Epimorphism ===
	If there exists a linear map h: G → ℤ so that h(s[''i'']) = ''i'', then s is weakly epimorphic with the map h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were first considered by Yves Hellegouarch.<ref>Yves Hellegouarch, ''A Mathematical Interpretation of Expressive Intonation'', in ''Mathematics and Art'', p. 141-148, Springer-Verlag, 2002</ref> The name comes from the fact that h is an ~~{{w\|~~epimorphism}} onto the integers (i.e. the map h is surjective).		{{Main\|Detempering}}
			{{See also\|Wikipedia: Epimorphism}}
	~~Epimorphism is explained in more detail on [[detempering]]~~.		If there exists a linear map h: G → ℤ so that h(s[''i'']) = ''i'', then s is weakly epimorphic with the map h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were first considered by Yves Hellegouarch.<ref>Yves Hellegouarch, ''A Mathematical Interpretation of Expressive Intonation'', in ''Mathematics and Art'', p. 141-148, Springer-Verlag, 2002</ref> The name comes from the fact that h is an epimorphism onto the integers (i.e. the map h is surjective).

	=== Myhill's property ===		=== Myhill's property ===

Sintel: /* Epimorphism */ Leave a link to detempering here

2025-04-24T21:36:06Z

Epimorphism: Leave a link to detempering here

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	=== Epimorphism ===		=== Epimorphism ===
	If there exists a linear map h: G → ℤ so that h(s[''i'']) = ''i'', then s is weakly epimorphic with the map h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were first considered by Yves Hellegouarch.<ref>Yves Hellegouarch, ''A Mathematical Interpretation of Expressive Intonation'', in ''Mathematics and Art'', p. 141-148, Springer-Verlag, 2002</ref> The name comes from the fact that h is an {{w\|epimorphism}} onto the integers (i.e. the map h is surjective).		If there exists a linear map h: G → ℤ so that h(s[''i'']) = ''i'', then s is weakly epimorphic with the map h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were first considered by Yves Hellegouarch.<ref>Yves Hellegouarch, ''A Mathematical Interpretation of Expressive Intonation'', in ''Mathematics and Art'', p. 141-148, Springer-Verlag, 2002</ref> The name comes from the fact that h is an {{w\|epimorphism}} onto the integers (i.e. the map h is surjective).

			Epimorphism is explained in more detail on [[detempering]].

	=== Myhill's property ===		=== Myhill's property ===

Sintel: /* Epimorphism */ add ref

2025-04-23T01:15:16Z

Epimorphism: add ref

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	=== Epimorphism ===		=== Epimorphism ===
	If there exists a linear map h: G → ℤ so that h(s[''i'']) = ''i'', then s is weakly epimorphic with the map h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were first considered by [[Yves Hellegouarch]].~~{{cn}}~~ The name comes from the fact that h is an {{w\|epimorphism}} onto the integers (i.e. the map h is surjective).		If there exists a linear map h: G → ℤ so that h(s[''i'']) = ''i'', then s is weakly epimorphic with the map h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were first considered by Yves Hellegouarch.<ref>Yves Hellegouarch, ''A Mathematical Interpretation of Expressive Intonation'', in ''Mathematics and Art'', p. 141-148, Springer-Verlag, 2002</ref> The name comes from the fact that h is an {{w\|epimorphism}} onto the integers (i.e. the map h is surjective).

	=== Myhill's property ===		=== Myhill's property ===
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	== See also ==		== See also ==
	* [[Glossary of scale properties]]		* [[Glossary of scale properties]]

			== References ==
			<references/>

	[[Category:Math]]		[[Category:Math]]

VectorGraphics: added link for stacking

2025-04-23T00:08:19Z

added link for stacking

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	== Mathematical definition ==		== Mathematical definition ==
	A periodic scale may be defined in mathematical language as a type of [[Wikipedia: Quasiperiodic function\|quasiperiodic function]] from the [[Wikipedia: Integer\|integers]] to musical intervals, or in layman's terms, a "table" that maps integers (which formalize the notion of "scale degrees") to intervals given in cents (hence, an additive notation will be used, with the stacking of intervals notated by addition). In this case, a periodic scale ''s'' has a nonzero quasiperiod ''P'' (the period in scale steps) and repetition interval ''O'', also notated s[P] (the period in cents) where by adding P to the scale degree, O is always added to the resulting interval.		A periodic scale may be defined in mathematical language as a type of [[Wikipedia: Quasiperiodic function\|quasiperiodic function]] from the [[Wikipedia: Integer\|integers]] to musical intervals, or in layman's terms, a "table" that maps integers (which formalize the notion of "scale degrees") to intervals given in cents (hence, an additive notation will be used, with the [[stacking]] of intervals notated by addition). In this case, a periodic scale ''s'' has a nonzero quasiperiod ''P'' (the period in scale steps) and repetition interval ''O'', also notated s[P] (the period in cents) where by adding P to the scale degree, O is always added to the resulting interval.

	Since arbitrarily high and low pitches go beyond the [[human hearing range\|range of human hearing]], this definition is a mathematical idealization, but it is much simpler to adopt the idealization than to worry about that. Neither Scala nor the above definition assumes that the scales are [[Wikipedia: Monotonic function\|strictly increasing]], but this condition, giving a '''monotone periodic scale''', is often important to add.		Since arbitrarily high and low pitches go beyond the [[human hearing range\|range of human hearing]], this definition is a mathematical idealization, but it is much simpler to adopt the idealization than to worry about that. Neither Scala nor the above definition assumes that the scales are [[Wikipedia: Monotonic function\|strictly increasing]], but this condition, giving a '''monotone periodic scale''', is often important to add.

Sintel: /* Epimorphism */ do not use the term 'homomorphism' please

2025-04-22T23:24:49Z

Epimorphism: do not use the term 'homomorphism' please

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	=== Epimorphism ===		=== Epimorphism ===
	If there exists a ~~homomorphism~~ h: G → ℤ so that h(s[''i'']) = ''i'', then s is weakly epimorphic with the ~~homomorphism~~ h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were ~~apparently~~ first considered by [[Yves Hellegouarch]]. The name comes from the fact that h is an epimorphism onto ℤ.		If there exists a linear map h: G → ℤ so that h(s[''i'']) = ''i'', then s is weakly epimorphic with the map h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were first considered by [[Yves Hellegouarch]].{{cn}} The name comes from the fact that h is an {{w\|epimorphism}} onto the integers (i.e. the map h is surjective).

	=== Myhill's property ===		=== Myhill's property ===

Fredg999: Text replacement - "Scale properties simplified" to "Glossary of scale properties"

2025-04-18T02:14:03Z

Text replacement - "Scale properties simplified" to "Glossary of scale properties"

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	== See also ==		== See also ==
	* [[~~Scale~~ properties ~~simplified~~]]		* [[Glossary of scale properties]]

	[[Category:Math]]		[[Category:Math]]

Hkm: In termso f mo

2025-04-08T00:45:10Z

In termso f mo

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	=== Modes ===		=== Modes ===
	A mode, or "rotation", of a periodic scale is a scale r such that r[''i''] = s[''i'' + ''N''] - s[''N''], where ''N'' is a fixed integer; in other words, it is the same scale pattern, but starting on a different scale degree. Since s[''i'' + ''P''] - s[''P''] = s[''i''], there are only a finite number of rotations, equal to the number of notes of the scale reduced to the range of the interval of equivalence, 0 ≤ s[''i''] < ''s[P]'', which entails 0 ≤ ''i'' < ''P''.		A mode, or "rotation", of a periodic scale is a scale r such that r[''i''] = s[''i'' + ''N''] - s[''N''], where ''N'' is a fixed integer; in other words, it is the same scale pattern, but starting on a different scale degree. Since s[''i'' + ''P''] - s[''P''] = s[''i''], there are only a finite number of rotations, equal to the number of notes of the scale reduced to the range of the interval of equivalence, 0 ≤ s[''i''] < ''s[P]'', which entails 0 ≤ ''i'' < ''P''.

	~~In termso f mo~~

	=== Classes ===		=== Classes ===