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]] | ||
Revision as of 14:06, 25 November 2025
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 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. 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 7 225/32, 11 4096/375 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 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 projection pairs
5-limit
- 16875/16384:
3 50625/16384, 5 16384/3375to 2.15 - 250/243:
3 729/250, 5 59049/12500to 2.9/5 - 3125/3072:
3 3125/1024 - 20000/19683:
3 20000/6561, 5 2000000000/387420489to 2.9/5 - 81/80:
5 81/16 - 393216/390625:
3 390625/131072 - 2109375/2097152:
3 13348388671875/4398046511104, 5 2097152/421875to 2.75 - 15625/15552:
3 46656/15625, 5 15552/3125to 2.5/3 - 32805/32768:
5 32768/6561
7-limit
- 1029/1000:
3 1000/343to 2.5.7 - 36/35:
7 36/5 - 525/512:
7 512/75 - 49/48:
3 49/16to 2.5.7 - 686/675:
5 3375/686, 7 675/98to 2.3.7/5 - 64/63:
7 64/9 - 854296875/843308032:
5 843308032/170859375, 7 5903156224/854296875to 2.3.7/5 - 64827/64000:
5 320000/64827, 7 64000/9261to 2.3.7/5 - 875/864:
7 864/125 - 3125/3087:
5 15625/3087, 7 9765625/1361367to 2.3.25/7 - 2430/2401:
5 2401/486to 2.3.7 - 50421/50000:
3 50000/16807to 2.5.7 - 245/243:
5 243/49to 2.3.7 - 126/125:
7 125/18 - 4000/3969:
5 3969/800, 7 27783/4000to 2.3.7/5 - 1728/1715:
5 1728/343to 2.3.7 - 1029/1024:
3 1024/343to 2.5.7 - 225/224:
7 225/32 - 19683/19600:
3 19600/6561, 7 1033052339200000000/150094635296999121to 2.5.81/7 - 16875/16807:
5 84375/16807, 7 16875/2401to 2.3.7/5 - 10976/10935:
5 10976/2187to 2.3.7 - 3136/3125:
5 3136/625, 7 68841472/9765625to 2.3.25/7 - 5120/5103:
7 5120/729 - 6144/6125:
3 6125/2048to 2.5.7 - 33554432/33480783:
7 33554432/4782969 - 201768035/201326592:
5 201326592/40353607to 2.3.7 - 29360128/29296875:
7 29296875/4194304 - 65625/65536:
7 65536/9375 - 703125/702464:
5 702464/140625, 7 3454189699072/494384765625to 2.3.25/7 - 420175/419904:
5 882735153125/176319369216, 7 419904/60025to 2.3.245 - 2401/2400:
3 2401/800to 2.5.7 - 4375/4374:
7 4374/625
11-limit
- 33/32:
11 32/3 - 45/44:
11 45/4 - 55/54:
11 54/5 - 56/55:
11 56/5 - 245/242:
5 242/49to 2.3.7.11 - 99/98:
11 98/9 - 100/99:
11 100/9 - 121/120:
5 121/24to 2.3.7.11 - 1331/1323:
7 9261/1331, 11 1323/121to 2.3.5.11/7 - 176/175:
11 175/16 - 896/891:
11 896/81 - 4375/4356:
7 4356/625to 2.3.5.11 - 14641/14580:
5 14641/2916to 2.3.7.11 - 243/242:
3 242/81, 11 644204/59049to 2.5.7.11/9 - 3388/3375:
7 3375/484to 2.3.5.11 - 385/384:
11 384/35 - 8019/8000:
11 8000/729 - 441/440:
11 441/40 - 1375/1372:
11 1372/125 - 6250/6237:
11 6250/567 - 540/539:
11 540/49 - 4000/3993:
3 4000/1331to 2.5.7.11 - 19712/19683:
11 19683/1792 - 5632/5625:
11 5625/512 - 41503/41472:
7 41472/5929, 11 456533/41472to 2.3.5.77 - 3025/3024:
7 3025/432to 2.3.5.11