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| 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.
| | {{Todo|inline=1|expand|comment=explain musical application --[[User:Hkm|hkm]] ([[User talk:Hkm|talk]]) 20:39, 25 June 2025 (UTC)}}[[File:Lattice Marvel Convex12.png|400px|thumb|A convex set of 12 tones from the marvel lattice.]] |
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| {{todo|clarify|comment=Is there maybe somebody who can explain this in plain English?|increase applicability|inline=1}}
| | 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. |
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| A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [http://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition. | | 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. |
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| 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. | | 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. |
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| ==Formal definition==
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| The following definitions make sense in the context of any Z-[http://en.wikipedia.org/wiki/Module_%28mathematics%29 module], which is the same concept as an [http://en.wikipedia.org/wiki/Abelian_group abelian group].
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| ===Convex combination===
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| 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
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| <math>$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$</math>
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| 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 [http://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
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| <math>$c = c_1 + c_2 + \dots + c_k$</math>
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| we obtain
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| <math>$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$</math>
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| 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.
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| ===Convex set===
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| 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.
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| ==Examples== | | ==Examples== |
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| * Every [[Fokker block]] is convex. | | * Every [[Fokker block]] is convex. |
| * Every untempered [[tonality diamond]] is convex. | | * Every untempered [[tonality diamond]] is convex. |
| * [[Gallery of Z-polygon transversals]]
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| | [[Category:Scale]] |
| [[Category:Math]] | | [[Category:Math]] |
| [[Category:Scale theory]] | | [[Category:Todo:clarify]] |