Tempered monzos and vals: Difference between revisions

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A regular temperament T is an equivalence class of Z-module homomorphisms '''T''': J → K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same [[~~Abstract_regular_temperament|~~temperament]] if they differ only by unimodular transformation. An element of K is called a '''tmonzo''', and an element of the dual module K* is called a '''tval'''.

A regular temperament T is an equivalence class of Z-module homomorphisms '''T''': J → K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same [[temperament]] if they differ only by unimodular transformation. An element of K is called a '''tmonzo''', and an element of the dual module K* is called a '''tval'''.

Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they're linear functionals which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the [[~~Temperament_Mapping_Matrices_~~(M-maps)|mapping matrix]] for the temperament which is in [[~~Normal_lists~~|normal val list ~~form~~]].

Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they're linear functionals which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the [[Temperament Mapping Matrices (M-maps)|mapping matrix]] for the temperament which is in [[Normal lists #Normal val list|normal val list]] form.

=Example=

== Example ==

As an example, consider the mapping matrix

[~~<~~1 1 0|]

[{{val| 1 1 0 }}]

[~~<~~0 1 4|]

[{{val| 0 1 4 }}]

This matrix represents meantone temperament. If we right-multiply this matrix by the monzo |1 0 0~~>~~, representing 2/1, we get the tmonzo |1 0~~>~~. If we right-multiply it instead by |-1 1 0~~>~~, we get the tmonzo |0 1~~>~~. That 2/1 and 3/2 map to |1 0~~>~~ and |0 1~~>~~ respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now right-multiply the matrix by the monzo |-2 0 1~~>~~, representing 5/4, we get the tmonzo |-2 4~~>~~, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.

This matrix represents meantone temperament. If we right-multiply this matrix by the monzo {{monzo| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo| 1 0 }}. If we right-multiply it instead by {{monzo| -1 1 0 }}, we get the tmonzo {{monzo| 0 1 }}. That 2/1 and 3/2 map to {{monzo| 1 0 }} and {{monzo| 0 1 }} respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now right-multiply the matrix by the monzo {{monzo| -2 0 1 }}, representing 5/4, we get the tmonzo {{monzo| -2 4 }}, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.

[[Category:Theory]]

[[Category:Val]]

[[Category:Monzo]]

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-A regular temperament T is an equivalence class of Z-module homomorphisms '''T''': J → K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same [[Abstract_regular_temperament|temperament]] if they differ only by unimodular transformation. An element of K is called a '''tmonzo''', and an element of the dual module K* is called a '''tval'''.
+A regular temperament T is an equivalence class of Z-module homomorphisms '''T''': J → K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same [[temperament]] if they differ only by unimodular transformation. An element of K is called a '''tmonzo''', and an element of the dual module K* is called a '''tval'''.
-Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they're linear functionals which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the [[Temperament_Mapping_Matrices_(M-maps)|mapping matrix]] for the temperament which is in [[Normal_lists|normal val list form]].
+Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they're linear functionals which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the [[Temperament Mapping Matrices (M-maps)|mapping matrix]] for the temperament which is in [[Normal lists #Normal val list|normal val list]] form.
-=Example=
+== Example ==
 As an example, consider the mapping matrix
-[&lt;1 1 0|]
+[{{val| 1 1 0 }}]
-[&lt;0 1 4|]
+[{{val| 0 1 4 }}]
-This matrix represents meantone temperament. If we right-multiply this matrix by the monzo |1 0 0&gt;, representing 2/1, we get the tmonzo |1 0&gt;. If we right-multiply it instead by |-1 1 0&gt;, we get the tmonzo |0 1&gt;. That 2/1 and 3/2 map to |1 0&gt; and |0 1&gt; respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now right-multiply the matrix by the monzo |-2 0 1&gt;, representing 5/4, we get the tmonzo |-2 4&gt;, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.
+This matrix represents meantone temperament. If we right-multiply this matrix by the monzo {{monzo| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo| 1 0 }}. If we right-multiply it instead by {{monzo| -1 1 0 }}, we get the tmonzo {{monzo| 0 1 }}. That 2/1 and 3/2 map to {{monzo| 1 0 }} and {{monzo| 0 1 }} respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now right-multiply the matrix by the monzo {{monzo| -2 0 1 }}, representing 5/4, we get the tmonzo {{monzo| -2 4 }}, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.
+[[Category:Theory]]
+[[Category:Val]]
+[[Category:Monzo]]