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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
A '''projection pair''' is 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 temperament]]s, 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>2011-08-22 12:52:27 UTC</tt>.<br>
: The original revision id was <tt>247703671</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".
An example of a projection pair is <code>7 225/32</code>, 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. This can be thought of as marvel temperament replacing 7 by 225/32.


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.</pre></div>
More than one such pair may be required to reduce to the desired subgroup; for instance <code>7 225/32, 11 4096/375</code> reduces an 11-limit JI scale to a 5-limit JI scale equivalent under (undecimal) 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 <code>5 3136/625, 7 68841472/9765625</code>.
<h4>Original HTML 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;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;
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;
 
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;
== List of projection pairs ==
&lt;br /&gt;
=== 5-limit ===
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;/body&gt;&lt;/html&gt;</pre></div>
{| class="wikitable"
|+ Projection pairs for 5-limit temperaments
|-
! Temperament !! Associated comma !! Projection pair !! Target subgroup
|-
| [[Negri]] || [[16875/16384]] || <code>3 50625/16384, 5 16384/3375</code> || 2.15
|-
| [[Porcupine]] || [[250/243]] || <code>3 729/250, 5 59049/12500</code> || 2.9/5
|-
| [[Magic]] || [[3125/3072]] || <code>3 3125/1024</code> || 2.5
|-
| [[Tetracot]] || [[20000/19683]] || <code>3 20000/6561, 5 2000000000/387420489</code> || 2.9/5
|-
| [[Meantone]] || [[81/80]] || <code>5 81/16</code> || 2.3
|-
| [[Würschmidt]] || [[393216/390625]] || <code>3 390625/131072</code> || 2.5
|-
| [[Orson]] || [[2109375/2097152]] || <code>3 13348388671875/4398046511104, 5 2097152/421875</code> || 2.75
|-
| [[Hanson]] || [[15625/15552]] || <code>3 46656/15625, 5 15552/3125</code> || 2.5/3
|-
| [[Helmholtz]] || [[32805/32768]] || <code>5 32768/6561</code> || 2.3
|}
 
=== 7-limit ===
{| class="wikitable"
|+ Projection pairs for 7-limit temperaments
|-
! Temperament !! Associated comma !! Projection pair !! Target subgroup
|-
| [[Keegic]] || [[1029/1000]] || <code>3 1000/343</code> || 2.5.7
|-
| [[Mint]] || [[36/35]] || <code>7 36/5</code> || 2.3.5
|-
| [[Avicennmic]] || [[525/512]] || <code>7 512/75</code> || 2.3.5
|-
| [[Semaphoresmic]] || [[49/48]] || <code>3 49/16</code> || 2.5.7
|-
| [[Sengic]] || [[686/675]] || <code>5 3375/686, 7 675/98</code> || 2.3.7/5
|-
| [[Archytas]] || [[64/63]] || <code>7 64/9</code> || 2.3.5
|-
| [[Blackjackismic]] || [[Blackjackisma|854296875/843308032]] || <code>5 843308032/170859375, 7 5903156224/854296875</code> || 2.3.7/5
|-
| [[Squalentine]] || [[64827/64000]] || <code>5 320000/64827, 7 64000/9261</code> || 2.3.7/5
|-
| [[Keemic]] || [[875/864]] || <code>7 864/125</code> || 2.3.5
|-
| [[Gariboh]] || [[3125/3087]] || <code>5 15625/3087, 7 9765625/1361367</code> || 2.3.25/7
|-
| [[Nuwell]] || [[2430/2401]] || <code>5 2401/486</code> || 2.3.7
|-
| [[Trimyna]] || [[50421/50000]] || <code>3 50000/16807</code> || 2.5.7
|-
| [[Sensamagic]] || [[245/243]] || <code>5 243/49</code> || 2.3.7
|-
| [[Starling]] || [[126/125]] || <code>7 125/18</code> || 2.3.5
|-
| [[Octagar]] || [[4000/3969]] || <code>5 3969/800, 7 27783/4000</code> || 2.3.7/5
|-
| [[Orwellismic]] || [[1728/1715]] || <code>5 1728/343</code> || 2.3.7
|-
| [[Gamelismic]] || [[1029/1024]] || <code>3 1024/343</code> || 2.5.7
|-
| [[Marvel]] || [[225/224]] || <code>7 225/32</code> || 2.3.5
|-
| [[Cataharry]] || [[19683/19600]] || <code>3 19600/6561, 7 1033052339200000000/150094635296999121</code> || 2.5.81/7
|-
| [[Mirkwai]] || [[16875/16807]] || <code>5 84375/16807, 7 16875/2401</code> || 2.3.7/5
|-
| [[Hemimage]] || [[10976/10935]] || <code>5 10976/2187</code> || 2.3.7
|-
| [[Hemimean]] || [[3136/3125]] || <code>5 3136/625, 7 68841472/9765625</code> || 2.3.25/7
|-
| [[Hemifamity]] || [[5120/5103]] || <code>7 5120/729</code> || 2.3.5
|-
| [[Porwell]] || [[6144/6125]] || <code>3 6125/2048</code> || 2.5.7
|-
| [[Garischismic]] || [[33554432/33480783]] || <code>7 33554432/4782969</code> || 2.3.5
|-
| [[Wadismic]] || [[201768035/201326592]] || <code>5 201326592/40353607</code> || 2.3.7
|-
| [[Quasiorwellismic]] || [[29360128/29296875]] || <code>7 29296875/4194304</code> || 2.3.5
|-
| [[Horwell]] || [[65625/65536]] || <code>7 65536/9375</code> || 2.3.5
|-
| [[Metric]] || [[703125/702464]] || <code>5 702464/140625, 7 3454189699072/494384765625</code> || 2.3.25/7
|-
| [[Wizmic]] || [[420175/419904]] || <code>5 882735153125/176319369216, 7 419904/60025</code> || 2.3.245
|-
| [[Breed]] || [[2401/2400]] || <code>3 2401/800</code> || 2.5.7
|-
| [[Ragismic]] || [[4375/4374]] || <code>7 4374/625</code> || 2.3.5
|}
 
=== 11-limit ===
{| class="wikitable"
|+ Projection pairs for 11-limit temperaments
|-
! Temperament !! Associated comma !! Projection pair !! Target subgroup
|-
|  || [[33/32]] || <code>11 32/3</code> || 2.3.5.7
|-
| [[Cake]] || [[45/44]] || <code>11 45/4</code> || 2.3.5.7
|-
|  || [[55/54]] || <code>11 54/5</code> || 2.3.5.7
|-
|  || [[56/55]] || <code>11 56/5</code> || 2.3.5.7
|-
| [[Frostmic]] || [[245/242]] || <code>5 242/49</code> || 2.3.7.11
|-
| [[Mothwellsmic]] || [[99/98]] || <code>11 98/9</code> || 2.3.5.7
|-
| [[Ptolemismic]] || [[100/99]] || <code>11 100/9</code> || 2.3.5.7
|-
| [[Biyatismic]] || [[121/120]] || <code>5 121/24</code> || 2.3.7.11
|-
|  || [[1331/1323]] || <code>7 9261/1331, 11 1323/121</code> || 2.3.5.11/7
|-
| [[Valinorsmic]] || [[176/175]] || <code>11 175/16</code> || 2.3.5.7
|-
| [[Pentacircle]] || [[896/891]] || <code>11 896/81</code> || 2.3.5.7
|-
|  || [[4375/4356]] || <code>7 4356/625</code> || 2.3.5.11
|-
| [[Semicanousmic]] || [[14641/14580]] || <code>5 14641/2916</code> || 2.3.7.11
|-
| [[Rastmic]] || [[243/242]] || <code>3 242/81, 11 644204/59049</code> || 2.5.7.11/9
|-
|  || [[3388/3375]] || <code>7 3375/484</code> || 2.3.5.11
|-
| [[Keenanismic]] || [[385/384]] || <code>11 384/35</code> || 2.3.5.7
|-
| [[Trimitone]] || [[8019/8000]] || <code>11 8000/729</code> || 2.3.5.7
|-
| [[Werckismic]] || [[441/440]] || <code>11 441/40</code> || 2.3.5.7
|-
|  || [[1375/1372]] || <code>11 1372/125</code> || 2.3.5.7
|-
|  || [[6250/6237]] || <code>11 6250/567</code> || 2.3.5.7
|-
| [[Swetismic]] || [[540/539]] || <code>11 540/49</code> || 2.3.5.7
|-
|  || [[4000/3993]] || <code>3 4000/1331</code> || 2.5.7.11
|-
| [[Symbiotic]] || [[19712/19683]] || <code>11 19683/1792</code> || 2.3.5.7
|-
|  || [[5632/5625]] || <code>11 5625/512</code> || 2.3.5.7
|-
| [[Argyric]] || [[41503/41472]] || <code>7 41472/5929, 11 456533/41472</code> || 2.3.5.77
|-
| [[Lehmerismic]] || [[3025/3024]] || <code>7 3025/432</code> || 2.3.5.11
|}
 
[[Category:Lists of intervals]]
[[Category:Just intonation]]
[[Category:Method]]
Retrieved from "https://en.xen.wiki/w/Projection_pair"