|
|
| (26 intermediate revisions by 8 users not shown) |
| Line 1: |
Line 1: |
| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | '''Tempered monzos''' and '''tempered vals''' are like regular [[monzo]]s and [[val]]s, except they work in a space of tempered intervals (for example, the intervals found in meantone) rather than in [[just intonation]]. A tempered val (for short, "'''tval'''") specifies a tuning or further temperament of the [[generator]]s of a temperament. For example, the 31edo tval for meantone is {{val| 31 49 }}, assume the generators are ~2 and ~3. A tempered monzo (for short, "'''tmonzo'''") specifies a particular tempered interval in terms of stacking the temperament's generators. For example, the tmonzo form of the major third in meantone is {{monzo| -6 4 }}. Taking the {{w|dot product}} of these (multiplying corresponding elements and adding up the results) yields the tuning of the major third in 31edo, 10\31. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-04 12:16:38 UTC</tt>.<br>
| |
| : The original revision id was <tt>356331752</tt>.<br>
| |
| : The revision comment was: <tt></tt><br>
| |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| |
| <h4>Original Wikitext content:</h4>
| |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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 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]]. | | Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator [[tuning map]], which maps from tmonzos to tunings in cents (or another logarithmic measure). |
|
| |
|
| =Example= | | Note that there is no restriction on which bases tmonzos can be written in (i.e. what intervals should be considered the generators of the temperament), but one option is to use the basis corresponding to the [[temperament mapping matrix|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 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.</pre></div> | | $$ |
| <h4>Original HTML content:</h4>
| | \begin{bmatrix} |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tmonzos and Tvals</title></head><body>A regular temperament T is an equivalence class of Z-module homomorphisms <strong>T</strong>: J → K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same <a class="wiki_link" href="/abstract%20temperament">temperament</a> if they differ only by unimodular transformation. An element of K is called a <strong>tmonzo</strong>, and an element of the dual module K* is called a <strong>tval</strong>.<br />
| | 1 & 1 & 0 \\ |
| <br />
| | 0 & 1 & 4 |
| Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they're linear functionals which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the <a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29">mapping matrix</a> for the temperament which is in <a class="wiki_link" href="/Normal%20lists">normal val list form</a>.<br />
| | \end{bmatrix} |
| <br />
| | $$ |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:0 -->Example</h1>
| | |
| As an example, consider the mapping matrix<br />
| | This mapping represents meantone temperament. If we [[Mathematical guide/Matrix operations#Multiply matrix by vector|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. |
| [&lt;1 1 0|]<br /> | | |
| [&lt;0 1 4|]<br /> | | Now, let's use a tval {{val| 31 18 }} to figure out what the tuning of the tempered 5/4 is in 31edo. By applying this tval to the tmonzo {{monzo| -2 4 }}, we get -62 + 72 = 10 edosteps. |
| <br />
| | |
| 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.</body></html></pre></div>
| | == See also == |
| | * [[Mapped interval]] – a beginner-level introduction |
| | |
| | [[Category:Regular temperament theory]] |
| | [[Category:Math]] |
| | [[Category:Val]] |
| | [[Category:Monzo]] |