Mathematical theory of regular temperaments: Difference between revisions
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{{Main| Normal forms #Normal forms for mappings }} | {{Main| 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 | ||
$$ | |||
\begin{bmatrix} | |||
1 & 1 & 3 & 3 \\ | |||
0 & 6 & -7 & -2 \\ | |||
\end{bmatrix} | |||
$$ | |||
and applying this to the vector for either 16/15 or 15/14 leads to [0 1]. | |||
=== Normal comma lists === | === Normal comma lists === | ||