Intro to Mappings: Difference between revisions
Wikispaces>keenanpepper **Imported revision 442486098 - Original comment: ** |
Wikispaces>keenanpepper **Imported revision 442486128 - 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:keenanpepper|keenanpepper]] and made on <tt>2013-07-26 21: | : This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2013-07-26 21:27:30 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>442486128</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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As an example, let's consider the familiar [[12edo]] considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the [[3-limit]], that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. (For people familiar with mathematical notation, this can be written as | As an example, let's consider the familiar [[12edo]] considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the [[3-limit]], that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. (For people familiar with mathematical notation, this can be written as | ||
[[math]] | [[math]] | ||
\left\{440\right\} | \left\{440\cdot 2^a\cdot 3^b\center|a,b\in\mathbb Z\right\} | ||
[[math]] | [[math]] | ||
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<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left\{440\right\}&lt;br/&gt;[[math]] | \left\{440\cdot 2^a\cdot 3^b\center|a,b\in\mathbb Z\right\}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\left\{440\right\}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | --><script type="math/tex">\left\{440\cdot 2^a\cdot 3^b\center|a,b\in\mathbb Z\right\}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | ||
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A val maps JI onto one chain of generators, or relates that generator chain back to JI. However, many temperaments incorporate more than one of chain of generators. The familiar meantone temperament is an example, as it requires two: the fifth and the octave. However, vals only relate a single generator chain to JI. If we want to evolve out of the realm of isolated EDOs and consider higher-dimensional temperaments in their full glory, we're going to have to raise the bar on what vals can do for us. Luckily, the mathematics to do so is simple enough.<br /> | A val maps JI onto one chain of generators, or relates that generator chain back to JI. However, many temperaments incorporate more than one of chain of generators. The familiar meantone temperament is an example, as it requires two: the fifth and the octave. However, vals only relate a single generator chain to JI. If we want to evolve out of the realm of isolated EDOs and consider higher-dimensional temperaments in their full glory, we're going to have to raise the bar on what vals can do for us. Luckily, the mathematics to do so is simple enough.<br /> | ||