Periodic scale: Difference between revisions

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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.

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 ===

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>

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=== 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 epimorphism onto the integers (i.e. the map h is surjective).

=== Myhill's property ===

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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'''.

=== ~~Distributional evenness~~ ===

=== Interval 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 ===

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== See also ==

* [[Glossary of scale properties]]

== References ==

[[Category:Math]]

@@ Line 24: / Line 24: @@
 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 ===
-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>
@@ Line 54: / Line 54: @@
 === 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).
+{{Main|Detempering}}
+{{See also|Wikipedia: 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 epimorphism onto the integers (i.e. the map h is surjective).
 === Myhill's property ===
@@ Line 61: / Line 63: @@
 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 ===
+=== Interval variety ===
-{{Main| Distributional evenness }}
+{{Main|Interval 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 ===
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 == See also ==
 * [[Glossary of scale properties]]
+== References ==
+<references/>
 [[Category:Math]]