Tempered monzos and vals
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- This revision was by author clumma and made on 2014-12-14 22:41:01 UTC.
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Original Wikitext content:
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**. 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]]. =Example= As an example, consider the mapping matrix [<1 1 0|] [<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.
Original HTML content:
<html><head><title>Tmonzos and Tvals</title></head><body>A regular temperament T is an equivalence class of Z-module homomorphisms <strong>T</strong>: 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 <a class="wiki_link" href="/abstract%20regular%20temperament">temperament</a> if they differ only by unimodular transformation. An element of K is called a <strong>tmonzo</strong>, and an element of the dual module K* is called a <strong>tval</strong>.<br /> <br /> 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 <a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29">mapping matrix</a> for the temperament which is in <a class="wiki_link" href="/Normal%20lists">normal val list form</a>.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:0 -->Example</h1> As an example, consider the mapping matrix<br /> [<1 1 0|]<br /> [<0 1 4|]<br /> <br /> 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.</body></html>