Projection pair: Difference between revisions
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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. | 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. | ||
An example of a projection pair is | 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. | ||
More than one such pair may be required to reduce to the desired subgroup; for instance | 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>. | ||
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. | ||
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== List of projection pairs == | == List of projection pairs == | ||
=== 5-limit === | === 5-limit === | ||
* [[Negri comma|16875/16384]]: 3 50625/16384, 5 16384/3375 to 2.15 | * [[Negri comma|16875/16384]]: <code>3 50625/16384, 5 16384/3375</code> to 2.15 | ||
* [[250/243]]: 3 729/250, 5 59049/12500 to 2.9/5 | * [[250/243]]: <code>3 729/250, 5 59049/12500</code> to 2.9/5 | ||
* [[Magic comma|3125/3072]]: 3 3125/1024 | * [[Magic comma|3125/3072]]: <code>3 3125/1024</code> | ||
* [[Tetracot comma|20000/19683]]: 3 20000/6561, 5 2000000000/387420489 to 2.9/5 | * [[Tetracot comma|20000/19683]]: <code>3 20000/6561, 5 2000000000/387420489</code> to 2.9/5 | ||
* [[81/80]]: 5 81/16 | * [[81/80]]: <code>5 81/16</code> | ||
* [[Würschmidt comma|393216/390625]]: 3 390625/131072 | * [[Würschmidt comma|393216/390625]]: <code>3 390625/131072</code> | ||
* [[Semicomma|2109375/2097152]]: 3 13348388671875/4398046511104, 5 2097152/421875 to 2.75 | * [[Semicomma|2109375/2097152]]: <code>3 13348388671875/4398046511104, 5 2097152/421875</code> to 2.75 | ||
* [[15625/15552]]: 3 46656/15625, 5 15552/3125 to 2.5/3 | * [[15625/15552]]: <code>3 46656/15625, 5 15552/3125</code> to 2.5/3 | ||
* [[Schisma|32805/32768]]: 5 32768/6561 | * [[Schisma|32805/32768]]: <code>5 32768/6561</code> | ||
=== 7-limit === | === 7-limit === | ||
* [[1029/1000]]: 3 1000/343 to 2.5.7 | * [[1029/1000]]: <code>3 1000/343</code> to 2.5.7 | ||
* [[36/35]]: 7 36/5 | * [[36/35]]: <code>7 36/5</code> | ||
* [[525/512]]: 7 512/75 | * [[525/512]]: <code>7 512/75</code> | ||
* [[49/48]]: 3 49/16 to 2.5.7 | * [[49/48]]: <code>3 49/16</code> to 2.5.7 | ||
* [[686/675]]: 5 3375/686, 7 675/98 to 2.3.7/5 | * [[686/675]]: <code>5 3375/686, 7 675/98</code> to 2.3.7/5 | ||
* [[64/63]]: 7 64/9 | * [[64/63]]: <code>7 64/9</code> | ||
* [[Blackjackisma|854296875/843308032]]: 5 843308032/170859375, 7 5903156224/854296875 to 2.3.7/5 | * [[Blackjackisma|854296875/843308032]]: <code>5 843308032/170859375, 7 5903156224/854296875</code> to 2.3.7/5 | ||
* [[Squalentine comma|64827/64000]]: 5 320000/64827, 7 64000/9261 to 2.3.7/5 | * [[Squalentine comma|64827/64000]]: <code>5 320000/64827, 7 64000/9261</code> to 2.3.7/5 | ||
* [[875/864]]: 7 864/125 | * [[875/864]]: <code>7 864/125</code> | ||
* [[3125/3087]]: 5 15625/3087, 7 9765625/1361367 to 2.3.25/7 | * [[3125/3087]]: <code>5 15625/3087, 7 9765625/1361367</code> to 2.3.25/7 | ||
* [[2430/2401]]: 5 2401/486 to 2.3.7 | * [[2430/2401]]: <code>5 2401/486</code> to 2.3.7 | ||
* [[Trimyna comma|50421/50000]]: 3 50000/16807 to 2.5.7 | * [[Trimyna comma|50421/50000]]: <code>3 50000/16807</code> to 2.5.7 | ||
* [[245/243]]: 5 243/49 to 2.3.7 | * [[245/243]]: <code>5 243/49</code> to 2.3.7 | ||
* [[126/125]]: 7 125/18 | * [[126/125]]: <code>7 125/18</code> | ||
* [[4000/3969]]: 5 3969/800, 7 27783/4000 to 2.3.7/5 | * [[4000/3969]]: <code>5 3969/800, 7 27783/4000</code> to 2.3.7/5 | ||
* [[1728/1715]]: 5 1728/343 to 2.3.7 | * [[1728/1715]]: <code>5 1728/343</code> to 2.3.7 | ||
* [[1029/1024]]: 3 1024/343 to 2.5.7 | * [[1029/1024]]: <code>3 1024/343</code> to 2.5.7 | ||
* [[225/224]]: 7 225/32 | * [[225/224]]: <code>7 225/32</code> | ||
* [[Cataharry comma|19683/19600]]: 3 19600/6561, 7 1033052339200000000/150094635296999121 to 2.5.81/7 | * [[Cataharry comma|19683/19600]]: <code>3 19600/6561, 7 1033052339200000000/150094635296999121</code> to 2.5.81/7 | ||
* [[Mirkwai comma|16875/16807]]: 5 84375/16807, 7 16875/2401 to 2.3.7/5 | * [[Mirkwai comma|16875/16807]]: <code>5 84375/16807, 7 16875/2401</code> to 2.3.7/5 | ||
* [[Hemimage comma|10976/10935]]: 5 10976/2187 to 2.3.7 | * [[Hemimage comma|10976/10935]]: <code>5 10976/2187</code> to 2.3.7 | ||
* [[3136/3125]]: 5 3136/625, 7 68841472/9765625 to 2.3.25/7 | * [[3136/3125]]: <code>5 3136/625, 7 68841472/9765625</code> to 2.3.25/7 | ||
* [[5120/5103]]: 7 5120/729 | * [[5120/5103]]: <code>7 5120/729</code> | ||
* [[6144/6125]]: 3 6125/2048 to 2.5.7 | * [[6144/6125]]: <code>3 6125/2048</code> to 2.5.7 | ||
* [[Garischisma|33554432/33480783]]: 7 33554432/4782969 | * [[Garischisma|33554432/33480783]]: <code>7 33554432/4782969</code> | ||
* [[Wadisma|201768035/201326592]]: 5 201326592/40353607 to 2.3.7 | * [[Wadisma|201768035/201326592]]: <code>5 201326592/40353607</code> to 2.3.7 | ||
* [[Quasiorwellisma|29360128/29296875]]: 7 29296875/4194304 | * [[Quasiorwellisma|29360128/29296875]]: <code>7 29296875/4194304</code> | ||
* [[Horwell comma|65625/65536]]: 7 65536/9375 | * [[Horwell comma|65625/65536]]: <code>7 65536/9375</code> | ||
* [[Meter|703125/702464]]: 5 702464/140625, 7 3454189699072/494384765625 to 2.3.25/7 | * [[Meter|703125/702464]]: <code>5 702464/140625, 7 3454189699072/494384765625</code> to 2.3.25/7 | ||
* [[Wizma|420175/419904]]: 5 882735153125/176319369216, 7 419904/60025 to 2.3.245 | * [[Wizma|420175/419904]]: <code>5 882735153125/176319369216, 7 419904/60025</code> to 2.3.245 | ||
* [[2401/2400]]: 3 2401/800 to 2.5.7 | * [[2401/2400]]: <code>3 2401/800</code> to 2.5.7 | ||
* [[4375/4374]]: 7 4374/625 | * [[4375/4374]]: <code>7 4374/625</code> | ||
=== 11-limit === | === 11-limit === | ||
* [[33/32]]: 11 32/3 | * [[33/32]]: <code>11 32/3</code> | ||
* [[45/44]]: 11 45/4 | * [[45/44]]: <code>11 45/4</code> | ||
* [[55/54]]: 11 54/5 | * [[55/54]]: <code>11 54/5</code> | ||
* [[56/55]]: 11 56/5 | * [[56/55]]: <code>11 56/5</code> | ||
* [[245/242]]: 5 242/49 to 2.3.7.11 | * [[245/242]]: <code>5 242/49</code> to 2.3.7.11 | ||
* [[99/98]]: 11 98/9 | * [[99/98]]: <code>11 98/9</code> | ||
* [[100/99]]: 11 100/9 | * [[100/99]]: <code>11 100/9</code> | ||
* [[121/120]]: 5 121/24 to 2.3.7.11 | * [[121/120]]: <code>5 121/24</code> to 2.3.7.11 | ||
* [[1331/1323]]: 7 9261/1331, 11 1323/121 to 2.3.5.11/7 | * [[1331/1323]]: <code>7 9261/1331, 11 1323/121</code> to 2.3.5.11/7 | ||
* [[176/175]]: 11 175/16 | * [[176/175]]: <code>11 175/16</code> | ||
* [[896/891]]: 11 896/81 | * [[896/891]]: <code>11 896/81</code> | ||
* [[4375/4356]]: 7 4356/625 to 2.3.5.11 | * [[4375/4356]]: <code>7 4356/625</code> to 2.3.5.11 | ||
* [[Semicanousma|14641/14580]]: 5 14641/2916 to 2.3.7.11 | * [[Semicanousma|14641/14580]]: <code>5 14641/2916</code> to 2.3.7.11 | ||
* [[243/242]]: 3 242/81, 11 644204/59049 to 2.5.7.11/9 | * [[243/242]]: <code>3 242/81, 11 644204/59049</code> to 2.5.7.11/9 | ||
* [[3388/3375]]: 7 3375/484 to 2.3.5.11 | * [[3388/3375]]: <code>7 3375/484</code> to 2.3.5.11 | ||
* [[385/384]]: 11 384/35 | * [[385/384]]: <code>11 384/35</code> | ||
* [[8019/8000]]: 11 8000/729 | * [[8019/8000]]: <code>11 8000/729</code> | ||
* [[441/440]]: 11 441/40 | * [[441/440]]: <code>11 441/40</code> | ||
* [[1375/1372]]: 11 1372/125 | * [[1375/1372]]: <code>11 1372/125</code> | ||
* [[6250/6237]]: 11 6250/567 | * [[6250/6237]]: <code>11 6250/567</code> | ||
* [[540/539]]: 11 540/49 | * [[540/539]]: <code>11 540/49</code> | ||
* [[4000/3993]]: 3 4000/1331 to 2.5.7.11 | * [[4000/3993]]: <code>3 4000/1331</code> to 2.5.7.11 | ||
* [[Symbiotic comma|19712/19683]]: 11 19683/1792 | * [[Symbiotic comma|19712/19683]]: <code>11 19683/1792</code> | ||
* [[5632/5625]]: 11 5625/512 | * [[5632/5625]]: <code>11 5625/512</code> | ||
* [[Argyria|41503/41472]]: 7 41472/5929, 11 456533/41472 to 2.3.5.77 | * [[Argyria|41503/41472]]: <code>7 41472/5929, 11 456533/41472</code> to 2.3.5.77 | ||
* [[3025/3024]]: 7 3025/432 to 2.3.5.11 | * [[3025/3024]]: <code>7 3025/432</code> to 2.3.5.11 | ||
[[Category:Lists of intervals]] | [[Category:Lists of intervals]] | ||
[[Category:Just intonation]] | [[Category:Just intonation]] | ||
[[Category:Method]] | [[Category:Method]] | ||