Convex scale: Difference between revisions

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[[File:Lattice Marvel Convex12.png|400px|thumb|A convex set of 12 tones from the marvel lattice.]]

In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.

In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' (also called a Z-polytope) in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.

~~{{todo|clarify|comment=Is there maybe somebody who can explain this~~ in ~~plain English?|increase applicability|inline=1}}~~

A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [https://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. Alternatively, a convex set in a lattice is a set where any weighted average of elements (where no element has negative weight) is within the set if it is on the lattice.

A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [https://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition.

The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. (Every scale has a unique convex hull.) See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.

The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.

~~==Formal definition==~~

The following definitions make sense in the context of any Z-[https://en.wikipedia.org/wiki/Module_%28mathematics%29 module], which is the same concept as an [https://en.wikipedia.org/wiki/Abelian_group abelian group].

~~===Convex combination===~~

A '''convex combination''' of a set of vectors is, intuitively, a weighted average of the vectors where all the weights are non-negative. Formally, b is a convex combination of a1, a2... whenever there exist non-negative integers c1, c2... such that

~~<math>$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$</math>~~

Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent definition can be given in terms of the [https://en.wikipedia.org/wiki/Injective_hull injective hull] of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by

~~<math>$c = c_1 + c_2 + \dots + c_k$</math>~~

~~we obtain~~

~~<math>$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$</math>~~

~~where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers.~~

~~===Convex set===~~

A convex set (sometimes called a Z-polytope) in a lattice is a set that includes all convex combinations of its elements. The convex closure of a set of lattice elements is the smallest convex set containing the set.

==Examples==

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* Every [[Fokker block]] is convex.

* Every untempered [[tonality diamond]] is convex.

* [[Gallery of Z-polygon transversals]]

[[Category:Scale]]

[[Category:Math]]

[[Category:Todo:clarify]]

@@ Line 1: / Line 1: @@
 [[File:Lattice Marvel Convex12.png|400px|thumb|A convex set of 12 tones from the marvel lattice.]]
-In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.
+In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' (also called a Z-polytope) in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.
-{{todo|clarify|comment=Is there maybe somebody who can explain this in plain English?|increase applicability|inline=1}}
+A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [https://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. Alternatively, a convex set in a lattice is a set where any weighted average of elements (where no element has negative weight) is within the set if it is on the lattice.
-A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [https://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition.
+The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. (Every scale has a unique convex hull.) See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.
-The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches.
-==Formal definition==
-The following definitions make sense in the context of any Z-[https://en.wikipedia.org/wiki/Module_%28mathematics%29 module], which is the same concept as an [https://en.wikipedia.org/wiki/Abelian_group abelian group].
-===Convex combination===
-A '''convex combination''' of a set of vectors is, intuitively, a weighted average of the vectors where all the weights are non-negative. Formally, b is a convex combination of a1, a2... whenever there exist non-negative integers c1, c2... such that
-<math>$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$</math>
-Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent definition can be given in terms of the [https://en.wikipedia.org/wiki/Injective_hull injective hull] of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by
-<math>$c = c_1 + c_2 + \dots + c_k$</math>
-we obtain
-<math>$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$</math>
-where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers.
-===Convex set===
-A convex set (sometimes called a Z-polytope) in a lattice is a set that includes all convex combinations of its elements. The convex closure of a set of lattice elements is the smallest convex set containing the set.
 ==Examples==
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 * Every [[Fokker block]] is convex.
 * Every untempered [[tonality diamond]] is convex.
-* [[Gallery of Z-polygon transversals]]
 [[Category:Scale]]
 [[Category:Math]]
 [[Category:Todo:clarify]]