Temperament mapping matrix: Difference between revisions

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11-limit porcupine tempers out 55/54, 64/63, and 100/99, and is supported by the patent vals for [[15-EDO]] and [[22-EDO]]. Since these two vals form a saturated submodule of the module of 11-limit vals, then the matrix formed by setting the rows to these vals is a valid mapping matrix for porcupine:

where the row vectors are still placed in bras as a notational device to signify that these are vals. If we put this in [[xenharmonic/Normal lists#x-Normal%20val%20lists|normal val list]] form, we get

or, in shorthand, [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|]. We'll call this matrix **P**.

We'll now right-multiply **P** by the following matrix **M** of two monzos, representing 2/1 and 3/2:

we can also write this matrix as

or, in shorthand, [|1 0 0 0 0&gt;, |-1 1 0 0 0&gt;], where it's understood in both cases that the kets represent columns.

**The Dual Transformation**

for which the rows are the patent vals for [[7-EDO]] and [[15-EDO]], respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the &lt;7 1| and &lt;15 2| tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix **V*P** is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|] as a result again.</pre></div>

<h4>Original HTML content:</h4>

  The multiplicative group of any set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, a &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;regular temperament&lt;/a&gt;, which is a group homomorphism &lt;strong&gt;T&lt;/strong&gt;: J -&amp;gt; K from the group J of JI rationals to a group K of tempered intervals, also has the structure of being a module homomorphism between Z-modules. This homomorphism can also be represented by a integer matrix, called a &lt;strong&gt;mapping matrix&lt;/strong&gt; or &lt;strong&gt;mapping&lt;/strong&gt; for the temperament. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a &amp;quot;monzo-map&amp;quot; or &lt;strong&gt;M-map&lt;/strong&gt; when context demands, as opposed to the &lt;a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29"&gt;V-map&lt;/a&gt; which is a mapping on vals.&lt;br /&gt;

&lt;br /&gt;

Note also that since all mapping matrices for T will have the same row module, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same &lt;a class="wiki_link" href="/Normal%20lists#x-Normal%20val%20lists"&gt;normal val list&lt;/a&gt;, or more generally if they have the same Hermite normal form.&lt;br /&gt;

&lt;br /&gt;

  Any mapping matrix can be said to represent a linear map &lt;strong&gt;M:&lt;/strong&gt; J -&amp;gt; K, where J is a module of JI intervals and K is a module of tempered intervals. There is thus an associated dual transformation &lt;strong&gt;M*:&lt;/strong&gt; K* -&amp;gt; J*, where J* and K* are the dual modules to J and K, respectively. J* is the module of vals on J, and K* is the module of &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/tmonzos%20and%20tvals"&gt;tvals&lt;/a&gt; on K, so &lt;strong&gt;M&lt;/strong&gt;* represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the submodule of vals supporting the associated temperament, and no two tvals map to the same val.&lt;br /&gt;

&lt;br /&gt;

These two transformations correspond to different types of matrix multiplication: the ordinary transformation &lt;strong&gt;M&lt;/strong&gt; corresponds to multiplication with the mapping on the left and a matrix whose columns are monzos on the right, and the dual transformation &lt;strong&gt;M&lt;/strong&gt;* corresponds to multiplication with the mapping on the right and a matrix whose rows are tvals on the left.&lt;br /&gt;

&lt;br /&gt;

  11-limit porcupine tempers out 55/54, 64/63, and 100/99, and is supported by the patent vals for &lt;a class="wiki_link" href="/15-EDO"&gt;15-EDO&lt;/a&gt; and &lt;a class="wiki_link" href="/22-EDO"&gt;22-EDO&lt;/a&gt;. Since these two vals form a saturated submodule of the module of 11-limit vals, then the matrix formed by setting the rows to these vals is a valid mapping matrix for porcupine:&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

or, in shorthand, [&amp;lt;1 2 3 2 4|, &amp;lt;0 -3 -5 6 -4|]. We'll call this matrix &lt;strong&gt;P&lt;/strong&gt;.&lt;br /&gt;

We'll now right-multiply &lt;strong&gt;P&lt;/strong&gt; by the following matrix &lt;strong&gt;M&lt;/strong&gt; of two monzos, representing 2/1 and 3/2:&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

for which the rows are the patent vals for &lt;a class="wiki_link" href="/7-EDO"&gt;7-EDO&lt;/a&gt; and &lt;a class="wiki_link" href="/15-EDO"&gt;15-EDO&lt;/a&gt;, respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the &amp;lt;7 1| and &amp;lt;15 2| tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix &lt;strong&gt;V*P&lt;/strong&gt; is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting [&amp;lt;1 2 3 2 4|, &amp;lt;0 -3 -5 6 -4|] as a result again.&lt;/body&gt;&lt;/html&gt;</pre></div>