Mapping: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 255556214 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 255695308 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-19 12:33:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>255695308</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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=Temperamental Rank= | =Temperamental Rank= | ||
** | **A temperament's "rank" denotes how many independent chains of** **generators exist within the temperament.** This is a mathematical term that's borrowed from the field of group theory. It can also be viewed as the "dimensionality" of the temperament. | ||
For example: | For example: | ||
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This is, in fact, the mapping matrix for meantone temperament, which is what we wanted. | This is, in fact, the mapping matrix for meantone temperament, which is what we wanted. | ||
= | =Change of Basis= | ||
In the above example, we wrote out the meantone mapping matrix from the perspective of the two generators 2/1 and 3/2. What if we instead wanted to treat the generators as being 2/1 and 4/3? Or, what if we wanted to write it out from the perspective that the generators are 2/1 and 3/1? All of these will lead to different val lists, but will still represent the same temperament. | In the above example, we wrote out the meantone mapping matrix from the perspective of the two generators 2/1 and 3/2. What if we instead wanted to treat the generators as being 2/1 and 4/3? Or, what if we wanted to write it out from the perspective that the generators are 2/1 and 3/1? All of these will lead to different val lists, but will still represent the same temperament. | ||
In the language of mathematics, you've simply ** | In the language of mathematics, you've simply **changed the basis** for your temperament, and the resulting temperamental spaces will be **isomorphic** to one another. This is just a fancy way of stating that they're the same temperament. | ||
If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals: | If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals: | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Temperamental Rank"></a><!-- ws:end:WikiTextHeadingRule:2 -->Temperamental Rank</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Temperamental Rank"></a><!-- ws:end:WikiTextHeadingRule:2 -->Temperamental Rank</h1> | ||
<strong> | <strong>A temperament's &quot;rank&quot; denotes how many independent chains of</strong> <strong>generators exist within the temperament.</strong> This is a mathematical term that's borrowed from the field of group theory. It can also be viewed as the &quot;dimensionality&quot; of the temperament.<br /> | ||
<br /> | <br /> | ||
For example:<br /> | For example:<br /> | ||
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This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.<br /> | This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name=" | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Change of Basis"></a><!-- ws:end:WikiTextHeadingRule:6 -->Change of Basis</h1> | ||
<br /> | <br /> | ||
In the above example, we wrote out the meantone mapping matrix from the perspective of the two generators 2/1 and 3/2. What if we instead wanted to treat the generators as being 2/1 and 4/3? Or, what if we wanted to write it out from the perspective that the generators are 2/1 and 3/1? All of these will lead to different val lists, but will still represent the same temperament.<br /> | In the above example, we wrote out the meantone mapping matrix from the perspective of the two generators 2/1 and 3/2. What if we instead wanted to treat the generators as being 2/1 and 4/3? Or, what if we wanted to write it out from the perspective that the generators are 2/1 and 3/1? All of these will lead to different val lists, but will still represent the same temperament.<br /> | ||
<br /> | <br /> | ||
In the language of mathematics, you've simply <strong> | In the language of mathematics, you've simply <strong>changed the basis</strong> for your temperament, and the resulting temperamental spaces will be <strong>isomorphic</strong> to one another. This is just a fancy way of stating that they're the same temperament.<br /> | ||
<br /> | <br /> | ||
If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals:<br /> | If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals:<br /> | ||