Projection pair: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 247766025 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 250037952 - 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:genewardsmith|genewardsmith]] and made on <tt>2011- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-01 10:39:22 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>250037952</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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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. | 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 5-limit projection pairs= | |||
250/243: 3 729/250 5 59049/12500 to 2.9/5 | |||
81/80: 5 81/16 | |||
15625/15552: 3 46656/15625 5 15552/3125 to 2.5/3 | |||
32805/32768: 5 32768/6561 | |||
=List of 7-limit projection pairs= | =List of 7-limit projection pairs= | ||
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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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="List of 7-limit projection pairs"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="List of 5-limit projection pairs"></a><!-- ws:end:WikiTextHeadingRule:0 -->List of 5-limit projection pairs</h1> | ||
250/243: 3 729/250 5 59049/12500 to 2.9/5<br /> | |||
81/80: 5 81/16<br /> | |||
15625/15552: 3 46656/15625 5 15552/3125 to 2.5/3<br /> | |||
32805/32768: 5 32768/6561 <br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="List of 7-limit projection pairs"></a><!-- ws:end:WikiTextHeadingRule:2 -->List of 7-limit projection pairs</h1> | |||
1029/1000: 3 1000/343 to 2.5.7<br /> | 1029/1000: 3 1000/343 to 2.5.7<br /> | ||
36/35: 7 36/5 <br /> | 36/35: 7 36/5 <br /> | ||
Revision as of 10:39, 1 September 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-09-01 10:39:22 UTC.
- The original revision id was 250037952.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
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 5-limit projection pairs= 250/243: 3 729/250 5 59049/12500 to 2.9/5 81/80: 5 81/16 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
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 5-limit projection pairs"></a><!-- ws:end:WikiTextHeadingRule:0 -->List of 5-limit projection pairs</h1> 250/243: 3 729/250 5 59049/12500 to 2.9/5<br /> 81/80: 5 81/16<br /> 15625/15552: 3 46656/15625 5 15552/3125 to 2.5/3<br /> 32805/32768: 5 32768/6561 <br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="List of 7-limit projection pairs"></a><!-- ws:end:WikiTextHeadingRule:2 -->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>