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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | Suppose v is a [[Vals_and_Tuning_Space|val]] <n ... | whose first coordinate is a positive integer n, and suppose the coordinates of the val, reduced [http://en.wikipedia.org/wiki/Modulo_operation modulo] n, are distinct. An example would be <12 19 28 34|; reduced mod 12 this is <0 7 4 10| and 0, 7, 4, and 10 are all distinct. Starting from 1, take the odd positive integers in order of increasing size, 1, 3, 5, 7, ... and map them by the val v, reducing the result mod n. If this number (from 0 to n-1) has not appeared before, add the odd positive integer to a set. When n values have been obtained and no further additions are possible, take the resulting set and reduce its elements to an octave. The result is Dwarf(v), the dwarf scale resulting from the val v. Examples may be found on the [[Scalesmith|Scalesmith]] page. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-25 16:17:23 UTC</tt>.<br>
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| : The original revision id was <tt>242784343</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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| <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">Suppose v is a [[Vals and Tuning Space|val]] <n ... | whose first coordinate is a positive integer n, and suppose the coordinates of the val, reduced [[http://en.wikipedia.org/wiki/Modulo_operation|modulo]] n, are distinct. An example would be <12 19 28 34|; reduced mod 12 this is <0 7 4 10| and 0, 7, 4, and 10 are all distinct. Starting from 1, take the odd positive integers in order of increasing size, 1, 3, 5, 7, ... and map them by the val v, reducing the result mod n. If this number (from 0 to n-1) has not appeared before, add the odd positive integer to a set. When n values have been obtained and no further additions are possible, take the resulting set and reduce its elements to an octave. The result is Dwarf(v), the dwarf scale resulting from the val v. Examples may be found on the [[Scalesmith]] page.
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| Of particular interest are dwarf scales resulting from equal temperament vals which are [[http://xenharmonic.wikispaces.com/Periodic+scale|epimorphic]] for the val v, but even vals far removed from an equal temperament will produce a scale. Dwarf scales often produce results which are rich harmonically, with a tendency to favor [[otonal]]ities over [[utonal]]ities. The name "dwarf" refers to the fact that you are choosing for each degree the smallest [[Benedetti height]]. | | Of particular interest are dwarf scales resulting from equal temperament vals which are [[Periodic_scale|epimorphic]] for the val v, but even vals far removed from an equal temperament will produce a scale. Dwarf scales often produce results which are rich harmonically, with a tendency to favor [[otonal|otonal]]ities over [[utonal|utonal]]ities. The name "dwarf" refers to the fact that you are choosing for each degree the smallest [[Benedetti_height|Benedetti height]]. |
| | | [[Category:benedetti]] |
| | | [[Category:dwarf]] |
| </pre></div>
| | [[Category:math]] |
| <h4>Original HTML content:</h4>
| | [[Category:scale]] |
| <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>Dwarves</title></head><body>Suppose v is a <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> &lt;n ... | whose first coordinate is a positive integer n, and suppose the coordinates of the val, reduced <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Modulo_operation" rel="nofollow">modulo</a> n, are distinct. An example would be &lt;12 19 28 34|; reduced mod 12 this is &lt;0 7 4 10| and 0, 7, 4, and 10 are all distinct. Starting from 1, take the odd positive integers in order of increasing size, 1, 3, 5, 7, ... and map them by the val v, reducing the result mod n. If this number (from 0 to n-1) has not appeared before, add the odd positive integer to a set. When n values have been obtained and no further additions are possible, take the resulting set and reduce its elements to an octave. The result is Dwarf(v), the dwarf scale resulting from the val v. Examples may be found on the <a class="wiki_link" href="/Scalesmith">Scalesmith</a> page.<br />
| | [[Category:theory]] |
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| | [[Category:todo:reduce_mathslang]] |
| Of particular interest are dwarf scales resulting from equal temperament vals which are <a href="http://xenharmonic.wikispaces.com/Periodic+scale">epimorphic</a> for the val v, but even vals far removed from an equal temperament will produce a scale. Dwarf scales often produce results which are rich harmonically, with a tendency to favor <a class="wiki_link" href="/otonal">otonal</a>ities over <a class="wiki_link" href="/utonal">utonal</a>ities. The name &quot;dwarf&quot; refers to the fact that you are choosing for each degree the smallest <a class="wiki_link" href="/Benedetti%20height">Benedetti height</a>.</body></html></pre></div>
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Suppose v is a val <n ... | whose first coordinate is a positive integer n, and suppose the coordinates of the val, reduced modulo n, are distinct. An example would be <12 19 28 34|; reduced mod 12 this is <0 7 4 10| and 0, 7, 4, and 10 are all distinct. Starting from 1, take the odd positive integers in order of increasing size, 1, 3, 5, 7, ... and map them by the val v, reducing the result mod n. If this number (from 0 to n-1) has not appeared before, add the odd positive integer to a set. When n values have been obtained and no further additions are possible, take the resulting set and reduce its elements to an octave. The result is Dwarf(v), the dwarf scale resulting from the val v. Examples may be found on the Scalesmith page.
Of particular interest are dwarf scales resulting from equal temperament vals which are epimorphic for the val v, but even vals far removed from an equal temperament will produce a scale. Dwarf scales often produce results which are rich harmonically, with a tendency to favor otonalities over utonalities. The name "dwarf" refers to the fact that you are choosing for each degree the smallest Benedetti height.