Projection pair
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. 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 16384/3375 to 2.15
- 250/243: 3 729/250, 5 59049/12500 to 2.9/5
- 3125/3072: 3 3125/1024
- 20000/19683: 3 20000/6561, 5 2000000000/387420489 to 2.9/5
- 81/80: 5 81/16
- 393216/390625: 3 390625/131072
- 2109375/2097152: 3 13348388671875/4398046511104, 5 2097152/421875 to 2.75
- 15625/15552: 3 46656/15625, 5 15552/3125 to 2.5/3
- 32805/32768: 5 32768/6561
7-limit
- 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
- 33554432/33480783: 7 33554432/4782969
- 201768035/201326592: 5 201326592/40353607 to 2.3.7
- 29360128/29296875: 7 29296875/4194304
- 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
11-limit
- 33/32: 11 32/3
- 45/44: 11 45/4
- 55/54: 11 54/5
- 56/55: 11 56/5
- 245/242: 5 242/49 to 2.3.7.11
- 99/98: 11 98/9
- 100/99: 11 100/9
- 121/120: 5 121/24 to 2.3.7.11
- 1331/1323: 7 9261/1331, 11 1323/121 to 2.3.5.11/7
- 176/175: 11 175/16
- 896/891: 11 896/81
- 4375/4356: 7 4356/625 to 2.3.5.11
- 14641/14580: 5 14641/2916 to 2.3.7.11
- 243/242: 3 242/81, 11 644204/59049 to 2.5.7.11/9
- 3388/3375: 7 3375/484 to 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 4000/1331 to 2.5.7.11
- 19712/19683: 11 19683/1792
- 5632/5625: 11 5625/512
- 41503/41472: 7 41472/5929, 11 456533/41472 to 2.3.5.77
- 3025/3024: 7 3025/432 to 2.3.5.11