Convex scale: Difference between revisions
m Moving from Category:Scale theory to Category:Scale using Cat-a-lot |
m I noticed the article had a todo:clarify tag in the article, but it wasn't in the todo:clarify category. So I added it to the todo:clarify category. |
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{{todo|clarify|comment=Is there maybe somebody who can explain this in plain English?|increase applicability|inline=1}} | {{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 [ | 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. 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== | ==Formal definition== | ||
The following definitions make sense in the context of any Z-[ | 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=== | ===Convex combination=== | ||
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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> | <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 [ | 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> | <math>$c = c_1 + c_2 + \dots + c_k$</math> | ||
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[[Category:Scale]] | [[Category:Scale]] | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Todo:clarify]] | |||