Tempered monzos and vals: Difference between revisions

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A regular temperament is a homomorphism (a kind of function) from the space of just intervals~~, J,~~ to the space of tempered intervals generated by that temperament~~, which can be called K~~, where both ~~J and K~~ are abelian groups. Technically, a regular temperament refers to an equivalence class of functions separated by unimodular transformations. An element of K is called a '''tempered monzo''', or for short "tmonzo", and represents the number of steps of each generator required to reach a tempered interval, and an element of the dual module K* is called a '''tval'''.

A regular temperament is a homomorphism (a kind of function) from the space of just intervals to the space of tempered intervals generated by that temperament, where both these spaces are [[Stacking|abelian groups]]. Technically, a regular temperament refers to an equivalence class of functions separated by unimodular transformations (which are functionally the same temperament). An element of the space of tempered intervals is called a '''tempered monzo''', or for short "tmonzo", and represents the number of steps of each generator required to reach a tempered interval, 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|normal val list]] form.

== Example ==

As an example, consider the mapping ~~matrix~~

As an example, consider the mapping

[{{val| 1 1 0 }}]

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[{{val| 0 1 4 }}]

This matrix represents meantone temperament. If we ~~right-multiply~~ this ~~matrix by~~ the monzo {{monzo| 1 0 0 }} ~~(which corresponds to applying the mapping to the monzo)~~, 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.

This mapping, which is referred to in mathematical terms as a "matrix", represents meantone temperament. If we [[Mathematical guide/Matrix multiplication|apply]] this mapping to the monzo {{monzo| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo| 1 0 }} (one tempered 2/1). If we instead apply it to {{monzo| -1 1 0 }}, we get the tmonzo {{monzo| 0 1 }} (one tempered 3/2). 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 apply this mapping to 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.

== See also ==

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-A regular temperament is a homomorphism (a kind of function) from the space of just intervals, J, to the space of tempered intervals generated by that temperament, which can be called K, where both J and K are abelian groups. Technically, a regular temperament refers to an equivalence class of functions separated by unimodular transformations. An element of K is called a '''tempered monzo''', or for short "tmonzo", and represents the number of steps of each generator required to reach a tempered interval, and an element of the dual module K* is called a '''tval'''.
+A regular temperament is a homomorphism (a kind of function) from the space of just intervals to the space of tempered intervals generated by that temperament, where both these spaces are [[Stacking|abelian groups]]. Technically, a regular temperament refers to an equivalence class of functions separated by unimodular transformations (which are functionally the same temperament). An element of the space of tempered intervals is called a '''tempered monzo''', or for short "tmonzo", and represents the number of steps of each generator required to reach a tempered interval, 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|normal val list]] form.
 == Example ==
-As an example, consider the mapping matrix
+As an example, consider the mapping
 [{{val| 1 1 0 }}]
@@ Line 10: / Line 10: @@
 [{{val| 0 1 4 }}]
-This matrix represents meantone temperament. If we right-multiply this matrix by the monzo {{monzo| 1 0 0 }} (which corresponds to applying the mapping to the monzo), 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.
+This mapping, which is referred to in mathematical terms as a "matrix", represents meantone temperament. If we [[Mathematical guide/Matrix multiplication|apply]] this mapping to the monzo {{monzo| 1 0 0 }}, representing 2/1, we get the tmonzo {{monzo| 1 0 }} (one tempered 2/1). If we instead apply it to {{monzo| -1 1 0 }}, we get the tmonzo {{monzo| 0 1 }} (one tempered 3/2). 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 apply this mapping to 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.
 == See also ==