Mathematical theory of regular temperaments: Difference between revisions

Line 29:

== Characterizing a regular temperament ==

=== Normal ~~val lists~~ ===

=== Normal form ===

~~Given a list of vals~~, we ~~may~~ [[Mathematical theory of saturation|saturate]] it and reduce it ~~using~~ the [[~~Normal lists|~~Hermite normal form]] ~~to a normal val list~~, which ~~canonically represents the abstract temperament~~. Applying the ~~vals successively (an operation we may regard as a matrix multiplication if we like) to~~ a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal ~~val list~~ for 7-limit miracle is [{~~{val|~~ 1 1 3 3 ~~}},~~&~~nbsp;{{val| 0~~ 6 -7 -2 }}] and applying this to the ~~monzo~~ for either 16/15 or 15/14 leads to [0 1].

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 <math>

\begin{bmatrix}

1 & 1 & 3 & 3 \\

0 & 6 & -7 & -2 \\

\end{bmatrix}

</math> and applying this to the vector for either 16/15 or 15/14 leads to [0 1].

=== Normal comma lists ===

~~The~~ normal ~~comma list uniquely defines~~ the ~~abstract temperament, and~~ has the advantage of showing family relationships ~~even~~ more clearly ~~than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.~~

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.

~~When specifying a temperament by the list of commas it tempers out, the list should be [[defactoring|defactored]] so it presents the intervals in their simplest, most direct form~~.

=== Plücker coordinates ===

@@ Line 29: / Line 29: @@
 == Characterizing a regular temperament ==
-=== Normal val lists ===
+=== Normal form ===
-{{Main| Normal lists #Normal val lists }}
+{{Main| Normal form }}
-Given a list of vals, we may [[Mathematical theory of saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [{{val| 1 1 3 3 }},&nbsp;{{val| 0 6 -7 -2 }}] and applying this to the monzo for either 16/15 or 15/14 leads to [0&nbsp;1].
+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 <math>
+	\begin{bmatrix}
+& 1 & 3 & 3 \\
+& 6 & -7 & -2 \\
+	\end{bmatrix}
+</math> and applying this to the vector for either 16/15 or 15/14 leads to [0&nbsp;1].
 === Normal comma lists ===
-{{Main| Normal lists #Normal interval lists }}
+{{Main| Normal forms#Normal forms for commas }}
-The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.
+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.
-When specifying a temperament by the list of commas it tempers out, the list should be [[defactoring|defactored]] so it presents the intervals in their simplest, most direct form.
 === Plücker coordinates ===