Mathematical theory of regular temperaments: Difference between revisions
→Characterizing a regular temperament: Update descriptions of normal forms to reflect changes to these pages |
m →Characterizing a regular temperament: if you just hate the word *val*.. |
||
| Line 29: | Line 29: | ||
== Characterizing a regular temperament == | == Characterizing a regular temperament == | ||
=== Normal | === Normal map lists === | ||
{{Main| Normal | {{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 <math> | 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> | ||
| Line 40: | Line 40: | ||
=== Normal comma lists === | === Normal comma lists === | ||
{{Main| Normal forms#Normal forms for commas }} | {{Main| 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. | 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. | ||
Using commas has the advantage of showing family relationships more clearly. | |||
=== Plücker coordinates === | === Plücker coordinates === | ||