Mathematical theory of regular temperaments: Difference between revisions
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== Characterizing a regular temperament == | == Characterizing a regular temperament == | ||
=== Normal | === Normal mapping matrices === | ||
{{Main| Normal forms #Normal forms for mappings }} | {{Main| Temperament mapping matrix }} | ||
{{See also| Normal forms #Normal forms for mappings }} | |||
Since an abstract temperament corresponds to some linear map, we can represent it as a matrix. We can [[Mathematical theory of saturation|saturate]] it and reduce it to the [[Hermite normal form]], which gives a unique representation. Applying this map to the vector representation of a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal form for 7-limit miracle is | Since an abstract temperament corresponds to some linear map, we can represent it as a matrix. We can [[Mathematical theory of saturation|saturate]] it and reduce it to the [[Hermite normal form]], which gives a unique representation. Applying this map to the vector representation of a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal form for 7-limit miracle is | ||
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and applying this to the vector for either 16/15 or 15/14 leads to [0 1]. | and applying this to the vector for either 16/15 or 15/14 leads to [0 1]. | ||
=== Normal comma | === Normal comma bases === | ||
{{Main| Normal forms #Normal forms for commas }} | {{Main| Comma basis }} | ||
{{See also| Normal forms #Normal forms for commas }} | |||
A temperament may also be defined by a list of commas. By putting these into a normal form, the representation is also unique. Using commas has the advantage of showing family relationships more clearly. | A temperament may also be defined by a list of commas. By putting these into a normal form, the representation is also unique. Using commas has the advantage of showing family relationships more clearly. | ||