Projection pair: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

By a ''projection pair'' is meant a pair of two rational intervals which can be employed by the [[Scala|Scala]] "project" command to reduce a JI scale to a scale in a [[JI_subgroup|JI subgroup]] of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing [[planar_temperaments|planar temperaments]], as the projection can then be viewed in lattice form by Scala's "lattice" or "lattice and player" command.

~~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>2011-09-01 11:34:59 UTC</tt>.<br>~~

~~: The original revision id was <tt>250056452</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">By a //projection pair// is meant a pair of two rational intervals which can be employed by the [[Scala]] "project" command to reduce a JI scale to a scale in a [[JI subgroup]] of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing [[planar temperaments]], as the projection can then be viewed in lattice form by Scala's "lattice" or "lattice and player" command.

An example of a projection pair is "7 225/32", which when applied by Scala's "project" to a 7-limit scale produces a 5-limit scale, which when tempered by marvel (225/224) temperament gives exactly the same result as the original scale does when also tempered. More than one such pair may be required to reduce to the desired subgroup; for instance "7 225/32 11 4096/375" reduces an 11-limit JI scale to a 5-limit JI scale equivalent under (unidecimal) marvel. This can happen even when only one comma is involved (codimension one temperaments.) For instance, to project a 7-limit scale in the hemimean (3136/3125) reduction to the 2.5.7 subgroup requires "5 3136/625 7 68841472/9765625".

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=List of 5-limit projection pairs=

16875/16384: 3 50625/16384 5 16384/3375 to 2.15

250/243: 3 729/250 5 59049/12500 to 2.9/5

3125/3072: 3 3125/1024

20000/19683: 3 20000/6561 5 2000000000/387420489 to 2.9/5

81/80: 5 81/16

393216/390625: 3 390625/131072

15625/15552: 3 46656/15625 5 15552/3125 to 2.5/3

32805/32768: 5 32768/6561

=List of 7-limit projection pairs=

1029/1000: 3 1000/343 to 2.5.7

36/35: 7 36/5

525/512: 7 512/75

49/48: 3 49/16 to 2.5.7

686/675: 5 3375/686 7 675/98 to 2.3.7/5

64/63: 7 64/9

854296875/843308032: 5 843308032/170859375 7 5903156224/854296875 to 2.3.7/5

64827/64000: 5 320000/64827 7 64000/9261 to 2.3.7/5

875/864: 7 864/125

3125/3087: 5 15625/3087 7 9765625/1361367 to 2.3.25/7

2430/2401: 5 2401/486 to 2.3.7

50421/50000: 3 50000/16807 to 2.5.7

245/243: 5 243/49 to 2.3.7

126/125: 7 125/18

4000/3969: 5 3969/800 7 27783/4000 to 2.3.7/5

1728/1715: 5 1728/343 to 2.3.7

1029/1024: 3 1024/343 to 2.5.7

225/224: 7 225/32

19683/19600: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7

16875/16807: 5 84375/16807 7 16875/2401 to 2.3.7/5

10976/10935: 5 10976/2187 to 2.3.7

3136/3125: 5 3136/625 7 68841472/9765625 to 2.3.25/7

5120/5103: 7 5120/729

6144/6125: 3 6125/2048 to 2.5.7

33554432/33480743: 7 33554432/4782969

201768035/201326592: 5 201326592/40353607 to 2.3.7

65625/65536: 7 65536/9375

703125/702464: 5 702464/140625 7 3454189699072/494384765625 to 2.3.25/7

420175/419904: 5 882735153125/176319369216 7 419904/60025 to 2.3.245

2401/2400: 3 2401/800 to 2.5.7

4375/4374: 7 4374/625~~</pre></div>~~

~~<h4>Original HTML content:</h4>~~

4375/4374: 7 4374/625

~~<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>Projection pair</title></head><body>By a <em>projection pair</em> is meant a pair of two rational intervals which can be employed by the <a class="wiki_link" href="/Scala">Scala</a> &quot;project&quot; command to reduce a JI scale to a scale in a <a class="wiki_link" href="/JI%20subgroup">JI subgroup</a> of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing <a class="wiki_link" href="/planar%20temperaments">planar temperaments</a>, as the projection can then be viewed in lattice form by Scala's &quot;lattice&quot; or &quot;lattice and player&quot; command. <br />

[[Category:intervals]]

~~<br />~~

[[Category:ji]]

An example of a projection pair is &quot;7 225/32&quot;, which when applied by Scala's &quot;project&quot; to a 7-limit scale produces a 5-limit scale, which when tempered by marvel (225/224) temperament gives exactly the same result as the original scale does when also tempered. More than one such pair may be required to reduce to the desired subgroup; for instance &quot;7 225/32 11 4096/375&quot; reduces an 11-limit JI scale to a 5-limit JI scale equivalent under (unidecimal) marvel. This can happen even when only one comma is involved (codimension one temperaments.) For instance, to project a 7-limit scale in the hemimean (3136/3125) reduction to the 2.5.7 subgroup requires &quot;5 3136/625 7 68841472/9765625&quot;.<br />

[[Category:method]]

~~<br />~~

[[Category:theory]]

Many projection pairs are given on the pages for various planar temperaments. When no subgroup is indicated, the default 2.3.5 5-limit subgroup is presumed. These lists of pairs can be copied and pasted into Scala and applied to any suitable JI scale.<br />

~~<br />~~

<h1 id="toc0"><a name="List of 5-limit projection pairs"></a>List of 5-limit projection pairs</h1>

~~16875/16384: 3 50625/16384 5 16384/3375 to 2.15<br />~~

~~250/243: 3 729/250 5 59049/12500 to 2.9/5<br />~~

~~3125/3072: 3 3125/1024<br />~~

~~20000/19683: 3 20000/6561 5 2000000000/387420489 to 2.9/5<br />~~

~~81/80: 5 81/16<br />~~

~~393216/390625: 3 390625/131072<br />~~

~~15625/15552: 3 46656/15625 5 15552/3125 to 2.5/3<br />~~

~~32805/32768: 5 32768/6561 <br />~~

~~<br />~~

<h1 id="toc1"><a name="List of 7-limit projection pairs"></a>List of 7-limit projection pairs</h1>

~~1029/1000: 3 1000/343 to 2.5.7<br />~~

~~36/35: 7 36/5 <br />~~

~~525/512: 7 512/75<br />~~

~~49/48: 3 49/16 to 2.5.7<br />~~

~~686/675: 5 3375/686 7 675/98 to 2.3.7/5<br />~~

~~64/63: 7 64/9<br />~~

~~854296875/843308032: 5 843308032/170859375 7 5903156224/854296875 to 2.3.7/5<br />~~

~~64827/64000: 5 320000/64827 7 64000/9261 to 2.3.7/5<br />~~

~~875/864~~: ~~7 864/125<br />~~

~~3125/3087~~: ~~5 15625/3087 7 9765625/1361367 to 2.3.25/7<br />~~

~~2430/2401~~: ~~5 2401/486 to 2.3.7<br />~~

~~50421/50000: 3 50000/16807 to 2.5.7<br />~~

~~245/243: 5 243/49 to 2.3.7<br />~~

~~126/125: 7 125/18<br />~~

~~4000/3969: 5 3969/800 7 27783/4000 to 2.3.7/5<br />~~

~~1728/1715: 5 1728/343 to 2.3.7<br />~~

~~1029/1024: 3 1024/343 to 2.5.7<br />~~

~~225/224: 7 225/32<br />~~

~~19683/19600: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7<br />~~

~~16875/16807: 5 84375/16807 7 16875/2401 to 2.3.7/5<br />~~

~~10976/10935: 5 10976/2187 to 2.3.7<br />~~

~~3136/3125: 5 3136/625 7 68841472/9765625 to 2.3.25/7<br />~~

~~5120/5103: 7 5120/729<br />~~

~~6144/6125: 3 6125/2048 to 2.5.7<br />~~

~~33554432/33480743: 7 33554432/4782969 <br />~~

~~201768035/201326592: 5 201326592/40353607 to 2.3.7<br />~~

~~65625/65536: 7 65536/9375<br />~~

~~703125/702464: 5 702464/140625 7 3454189699072/494384765625 to 2.3.25/7<br />~~

~~420175/419904: 5 882735153125/176319369216 7 419904/60025 to 2.3.245<br />~~

~~2401/2400: 3 2401/800 to 2.5.7<br />~~

~~4375/4374: 7 4374/625</body></html></pre></div>~~

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+By a ''projection pair'' is meant a pair of two rational intervals which can be employed by the [[Scala|Scala]] "project" command to reduce a JI scale to a scale in a [[JI_subgroup|JI subgroup]] of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing [[planar_temperaments|planar temperaments]], as the projection can then be viewed in lattice form by Scala's "lattice" or "lattice and player" command.
-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>2011-09-01 11:34:59 UTC</tt>.<br>
-: The original revision id was <tt>250056452</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">By a //projection pair// is meant a pair of two rational intervals which can be employed by the [[Scala]] "project" command to reduce a JI scale to a scale in a [[JI subgroup]] of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing [[planar temperaments]], as the projection can then be viewed in lattice form by Scala's "lattice" or "lattice and player" command.
 An example of a projection pair is "7 225/32", which when applied by Scala's "project" to a 7-limit scale produces a 5-limit scale, which when tempered by marvel (225/224) temperament gives exactly the same result as the original scale does when also tempered. More than one such pair may be required to reduce to the desired subgroup; for instance "7 225/32 11 4096/375" reduces an 11-limit JI scale to a 5-limit JI scale equivalent under (unidecimal) marvel. This can happen even when only one comma is involved (codimension one temperaments.) For instance, to project a 7-limit scale in the hemimean (3136/3125) reduction to the 2.5.7 subgroup requires "5 3136/625 7 68841472/9765625".
@@ Line 14: / Line 7: @@
 =List of 5-limit projection pairs=
 /16384: 3 50625/16384 5 16384/3375 to 2.15
 /243: 3 729/250 5 59049/12500 to 2.9/5
 /3072: 3 3125/1024
 /19683: 3 20000/6561 5 2000000000/387420489 to 2.9/5
 /80: 5 81/16
 /390625: 3 390625/131072
 /15552: 3 46656/15625 5 15552/3125 to 2.5/3
 /32768: 5 32768/6561
 =List of 7-limit projection pairs=
 /1000: 3 1000/343 to 2.5.7
 /35: 7 36/5
 /512: 7 512/75
 /48: 3 49/16 to 2.5.7
 /675: 5 3375/686 7 675/98 to 2.3.7/5
 /63: 7 64/9
 854296875/843308032: 5 843308032/170859375 7 5903156224/854296875 to 2.3.7/5
 /64000: 5 320000/64827 7 64000/9261 to 2.3.7/5
 /864: 7 864/125
 /3087: 5 15625/3087 7 9765625/1361367 to 2.3.25/7
 /2401: 5 2401/486 to 2.3.7
 /50000: 3 50000/16807 to 2.5.7
 /243: 5 243/49 to 2.3.7
 /125: 7 125/18
 /3969: 5 3969/800 7 27783/4000 to 2.3.7/5
 /1715: 5 1728/343 to 2.3.7
 /1024: 3 1024/343 to 2.5.7
 /224: 7 225/32
 /19600: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7
 /16807: 5 84375/16807 7 16875/2401 to 2.3.7/5
 /10935: 5 10976/2187 to 2.3.7
 /3125: 5 3136/625 7 68841472/9765625 to 2.3.25/7
 /5103: 7 5120/729
 /6125: 3 6125/2048 to 2.5.7
 33554432/33480743: 7 33554432/4782969
 201768035/201326592: 5 201326592/40353607 to 2.3.7
 /65536: 7 65536/9375
 /702464: 5 702464/140625 7 3454189699072/494384765625 to 2.3.25/7
 /419904: 5 882735153125/176319369216 7 419904/60025 to 2.3.245
 /2400: 3 2401/800 to 2.5.7
-/4374: 7 4374/625</pre></div>
-<h4>Original HTML content:</h4>
+/4374: 7 4374/625
-<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Projection pair&lt;/title&gt;&lt;/head&gt;&lt;body&gt;By a &lt;em&gt;projection pair&lt;/em&gt; is meant a pair of two rational intervals which can be employed by the &lt;a class="wiki_link" href="/Scala"&gt;Scala&lt;/a&gt; &amp;quot;project&amp;quot; command to reduce a JI scale to a scale in a &lt;a class="wiki_link" href="/JI%20subgroup"&gt;JI subgroup&lt;/a&gt; of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing &lt;a class="wiki_link" href="/planar%20temperaments"&gt;planar temperaments&lt;/a&gt;, as the projection can then be viewed in lattice form by Scala's &amp;quot;lattice&amp;quot; or &amp;quot;lattice and player&amp;quot; command. &lt;br /&gt;
+[[Category:intervals]]
-&lt;br /&gt;
+[[Category:ji]]
-An example of a projection pair is &amp;quot;7 225/32&amp;quot;, which when applied by Scala's &amp;quot;project&amp;quot; to a 7-limit scale produces a 5-limit scale, which when tempered by marvel (225/224) temperament gives exactly the same result as the original scale does when also tempered. More than one such pair may be required to reduce to the desired subgroup; for instance &amp;quot;7 225/32 11 4096/375&amp;quot; reduces an 11-limit JI scale to a 5-limit JI scale equivalent under (unidecimal) marvel. This can happen even when only one comma is involved (codimension one temperaments.) For instance, to project a 7-limit scale in the hemimean (3136/3125) reduction to the 2.5.7 subgroup requires &amp;quot;5 3136/625 7 68841472/9765625&amp;quot;.&lt;br /&gt;
+[[Category:method]]
-&lt;br /&gt;
+[[Category:theory]]
-Many projection pairs are given on the pages for various planar temperaments. When no subgroup is indicated, the default 2.3.5 5-limit subgroup is presumed. These lists of pairs can be copied and pasted into Scala and applied to any suitable JI scale.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="List of 5-limit projection pairs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;List of 5-limit projection pairs&lt;/h1&gt;
-/16384: 3 50625/16384 5 16384/3375 to 2.15&lt;br /&gt;
-/243: 3 729/250 5 59049/12500 to 2.9/5&lt;br /&gt;
-/3072: 3 3125/1024&lt;br /&gt;
-/19683: 3 20000/6561 5 2000000000/387420489 to 2.9/5&lt;br /&gt;
-/80: 5 81/16&lt;br /&gt;
-/390625: 3 390625/131072&lt;br /&gt;
-/15552: 3 46656/15625 5 15552/3125 to 2.5/3&lt;br /&gt;
-/32768: 5 32768/6561 &lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="List of 7-limit projection pairs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;List of 7-limit projection pairs&lt;/h1&gt;
-/1000: 3 1000/343 to 2.5.7&lt;br /&gt;
-/35: 7 36/5 &lt;br /&gt;
-/512: 7 512/75&lt;br /&gt;
-/48: 3 49/16 to 2.5.7&lt;br /&gt;
-/675: 5 3375/686 7 675/98 to 2.3.7/5&lt;br /&gt;
-/63: 7 64/9&lt;br /&gt;
-854296875/843308032: 5 843308032/170859375 7 5903156224/854296875 to 2.3.7/5&lt;br /&gt;
-/64000: 5 320000/64827 7 64000/9261 to 2.3.7/5&lt;br /&gt;
-/864: 7 864/125&lt;br /&gt;
-/3087: 5 15625/3087 7 9765625/1361367 to 2.3.25/7&lt;br /&gt;
-/2401: 5 2401/486 to 2.3.7&lt;br /&gt;
-/50000: 3 50000/16807 to 2.5.7&lt;br /&gt;
-/243: 5 243/49 to 2.3.7&lt;br /&gt;
-/125: 7 125/18&lt;br /&gt;
-/3969: 5 3969/800 7 27783/4000 to 2.3.7/5&lt;br /&gt;
-/1715: 5 1728/343 to 2.3.7&lt;br /&gt;
-/1024: 3 1024/343 to 2.5.7&lt;br /&gt;
-/224: 7 225/32&lt;br /&gt;
-/19600: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7&lt;br /&gt;
-/16807: 5 84375/16807 7 16875/2401 to 2.3.7/5&lt;br /&gt;
-/10935: 5 10976/2187 to 2.3.7&lt;br /&gt;
-/3125: 5 3136/625 7 68841472/9765625 to 2.3.25/7&lt;br /&gt;
-/5103: 7 5120/729&lt;br /&gt;
-/6125: 3 6125/2048 to 2.5.7&lt;br /&gt;
-33554432/33480743: 7 33554432/4782969 &lt;br /&gt;
-201768035/201326592: 5 201326592/40353607 to 2.3.7&lt;br /&gt;
-/65536: 7 65536/9375&lt;br /&gt;
-/702464: 5 702464/140625 7 3454189699072/494384765625 to 2.3.25/7&lt;br /&gt;
-/419904: 5 882735153125/176319369216 7 419904/60025 to 2.3.245&lt;br /&gt;
-/2400: 3 2401/800 to 2.5.7&lt;br /&gt;
-/4374: 7 4374/625&lt;/body&gt;&lt;/html&gt;</pre></div>