Periodic scale: Difference between revisions

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~~== Definition ==~~

A '''periodic scale''' is a [[scale]] with a [[period]].

A '''periodic scale''' may be defined in mathematical language as a type of [~~http~~:~~//en.wikipedia.org/wiki/Quasiperiodic_function~~ quasiperiodic function] from the [~~http~~:~~//en.wikipedia.org/wiki/Integers~~ integers] to musical intervals; the integers in this case formalize the notion of "scale degrees." Musical intervals may be written either additively or multiplicatively, and we will assume an additive notation is used, and that intervals are given by positive or negative real numbers with values in cents. In this case, a periodic scale '''s''' has a nonzero quasiperiod '''P''' and repetition interval '''O''' satisfying the following conditions

== 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; the integers in this case formalize the notion of "scale degrees". Musical intervals may be written either additively or multiplicatively, and we will assume an additive notation is used, and that intervals are given by positive or negative real numbers with values in cents. In this case, a periodic scale '''s''' has a nonzero quasiperiod '''P''' and repetition interval '''O''' satisfying the following conditions

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Scales written in the widely used [http://www.huygens-fokker.org/scala/scl_format.html Scala format] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Informally, a periodic scale could be defined as the kind of scale a Scala .scl file is intended to denote. Of course, 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 [~~http~~:~~//en.wikipedia.org/wiki/Monotonic_function~~ monotonically strictly increasing], but this condition, giving a '''monotone periodic scale''', is often important to add:

Scales written in the widely used [http://www.huygens-fokker.org/scala/scl_format.html Scala format] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Informally, a periodic scale could be defined as the kind of scale a Scala .scl file is intended to denote. Of course, 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|monotonically strictly increasing]], but this condition, giving a '''monotone periodic scale''', is often important to add:

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== Scale properties ==

=== ~~[[constant~~ structure~~|Constant Structure]]~~ ===

=== Constant structure ===

If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by [[Erv Wilson]]) means that i≠j implies class(i) ∩ class(j) = ∅. In academic music theory, this is called the ''partitioning property''.

=== ~~[[Rothenberg propriety|~~Propriety]] ===

=== Propriety ===

If s is monotone, and if i ≤ j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i < j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. In academic music theory circles, strict propriety is most often called ''coherence''. Note that strict propriety implies constant structure.

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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.

=== ~~[http://en.wikipedia.org/wiki/Myhill%27s_property~~ Myhill's property] ===

=== Myhill's property ===

A monotone scale in which every class but classes nP have exactly two elements has Myhill's property. If every such class has exactly three elements, it has the '''trivalence property'''. Myhill's property is synonymous with '''strict [[MOS]]''', though some authors prefer to identify MOS itself with Myhill's property.

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=== Convexity ===

The scale is ~~[[Convex scale|~~convex]] if every convex combination of notes, meaning every ℕ-linear combination of scale notes, is a scale note. If the quasiperiod '''P''' is normalized so as to be positive and minimal, this is equivalent to the condition that the equivalence classes of the notes modulo the repetition interval '''O''' is a [http://en.wikipedia.org/wiki/Convex_lattice_polytope ℤ-polytope] in the lattice defined by a basis for G mod '''O'''.

The scale is convex if every convex combination of notes, meaning every ℕ-linear combination of scale notes, is a scale note. If the quasiperiod '''P''' is normalized so as to be positive and minimal, this is equivalent to the condition that the equivalence classes of the notes modulo the repetition interval '''O''' is a [http://en.wikipedia.org/wiki/Convex_lattice_polytope ℤ-polytope] in the lattice defined by a basis for G mod '''O'''.

=== Maximal evenness ===

~~=== [[Maximal evenness]] ===~~

Maximally even scales of n notes in m edo are any mode of the sequence ME(n, m) = [floor(i*m/n) | i=1..n], where the "floor" function rounds down to the nearest integer.

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-== Definition ==
+A '''periodic scale''' is a [[scale]] with a [[period]].
-A '''periodic scale''' may be defined in mathematical language as a type of [http://en.wikipedia.org/wiki/Quasiperiodic_function quasiperiodic function] from the [http://en.wikipedia.org/wiki/Integers integers] to musical intervals; the integers in this case formalize the notion of "scale degrees." Musical intervals may be written either additively or multiplicatively, and we will assume an additive notation is used, and that intervals are given by positive or negative real numbers with values in cents. In this case, a periodic scale '''s''' has a nonzero quasiperiod '''P''' and repetition interval '''O''' satisfying the following conditions
+== 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; the integers in this case formalize the notion of "scale degrees". Musical intervals may be written either additively or multiplicatively, and we will assume an additive notation is used, and that intervals are given by positive or negative real numbers with values in cents. In this case, a periodic scale '''s''' has a nonzero quasiperiod '''P''' and repetition interval '''O''' satisfying the following conditions
 <math>(1)\ s[0] = 0</math>
@@ Line 6: / Line 8: @@
 <math>(2)\ s[i + P] = s[i] + O</math>
-Scales written in the widely used [http://www.huygens-fokker.org/scala/scl_format.html Scala format] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Informally, a periodic scale could be defined as the kind of scale a Scala .scl file is intended to denote. Of course, 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 [http://en.wikipedia.org/wiki/Monotonic_function monotonically strictly increasing], but this condition, giving a '''monotone periodic scale''', is often important to add:
+Scales written in the widely used [http://www.huygens-fokker.org/scala/scl_format.html Scala format] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Informally, a periodic scale could be defined as the kind of scale a Scala .scl file is intended to denote. Of course, 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|monotonically strictly increasing]], but this condition, giving a '''monotone periodic scale''', is often important to add:
 <math>(3)\ i < j\text{ implies }s[i] < s[j]</math>
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 == Scale properties ==
-=== [[constant structure|Constant Structure]] ===
+=== Constant structure ===
+{{Main| Constant structure }}
 If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by [[Erv Wilson]]) means that i≠j implies class(i) ∩ class(j) = ∅. In academic music theory, this is called the ''partitioning property''.
-=== [[Rothenberg propriety|Propriety]] ===
+=== Propriety ===
+{{Main| Rothenberg propriety }}
 If s is monotone, and if i ≤ j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i &lt; j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. In academic music theory circles, strict propriety is most often called ''coherence''. Note that strict propriety implies constant structure.
@@ Line 29: / Line 35: @@
 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.
-=== [http://en.wikipedia.org/wiki/Myhill%27s_property Myhill's property] ===
+=== Myhill's property ===
+{{See also| Wikipedia: Myhill's property }}
 A monotone scale in which every class but classes nP have exactly two elements has Myhill's property. If every such class has exactly three elements, it has the '''trivalence property'''. Myhill's property is synonymous with '''strict [[MOS]]''', though some authors prefer to identify MOS itself with Myhill's property.
@@ Line 36: / Line 44: @@
 === Convexity ===
-The scale is [[Convex scale|convex]] if every convex combination of notes, meaning every ℕ-linear combination of scale notes, is a scale note. If the quasiperiod '''P''' is normalized so as to be positive and minimal, this is equivalent to the condition that the equivalence classes of the notes modulo the repetition interval '''O''' is a [http://en.wikipedia.org/wiki/Convex_lattice_polytope ℤ-polytope] in the lattice defined by a basis for G mod '''O'''.
+{{Main| Convex scale }}
+The scale is convex if every convex combination of notes, meaning every ℕ-linear combination of scale notes, is a scale note. If the quasiperiod '''P''' is normalized so as to be positive and minimal, this is equivalent to the condition that the equivalence classes of the notes modulo the repetition interval '''O''' is a [http://en.wikipedia.org/wiki/Convex_lattice_polytope ℤ-polytope] in the lattice defined by a basis for G mod '''O'''.
+=== Maximal evenness ===
+{{Main| Maximal evenness }}
-=== [[Maximal evenness]] ===
 Maximally even scales of n notes in m edo are any mode of the sequence ME(n, m) = [floor(i*m/n) | i=1..n], where the "floor" function rounds down to the nearest integer.