Projection pair
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Original Wikitext content:
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". 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. =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 3376/686 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
Original HTML content:
<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> "project" 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 "lattice" or "lattice and player" command. <br /> <br /> 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".<br /> <br /> 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 /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="List of 7-limit projection pairs"></a><!-- ws:end:WikiTextHeadingRule:0 -->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 3376/686 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</body></html>