Periodic scale: Difference between revisions

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== 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'' (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.

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=== Modes ===

A mode 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''] < ''O'', 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''.

=== Classes ===

A class is a category of all intervals spanning a specified number of scale degrees, such as ~~"2-mosstep"~~, ~~"3-mosstep"~~ etc in ~~a MOS~~, 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 ===

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 ~~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.<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 ==

* [[~~Scale~~ properties ~~simplified~~]]

* [[Glossary of scale properties]]

== References ==

[[Category:Math]]

@@ Line 4: / Line 4: @@
 == 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'' (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.
@@ Line 19: / Line 19: @@
 === Modes ===
-A mode 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''] &lt; ''O'', which entails 0 ≤ ''i'' &lt; ''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''] &lt; ''s[P]'', which entails 0 ≤ ''i'' &lt; ''P''.
 === Classes ===
-A class is a category of all intervals spanning a specified number of scale degrees, such as "2-mosstep", "3-mosstep" etc in a MOS, 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 ===
-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 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.
+{{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 ===
@@ Line 82: / Line 86: @@
 == See also ==
-* [[Scale properties simplified]]
+* [[Glossary of scale properties]]
+== References ==
+<references/>
 [[Category:Math]]