catakleismic34trans

From: genewardsmith (2011-08-15)
Subject: Three views of Catakleismic[34]

When studying chord relationships in a MOS, it can be useful to look at a transversal. If you have a temperament like miracle or myna where 5 is relatively complex, you can take a 5-limit transversal, stick it into Scala and look at the lattice diagram, and use the triads as a guide to other chords, such as in particular the 7-limit tetrads. This works because the triads extend to tetrads.

However, consider for example catakleismic. This is a higher limit extension of kleismic which hasn't gained much traction since the 7 and 11 are so much more complex than the 5-limit, while 13, not so complex, is not as well in tune. But it's interesting at least in theory; for one thing, there's not much difference in tuning between marvel, tempering out 225/224, and catakleismic, which adds 4375/4374 to the mix. So it's one way of organizing anything in marvel temperament. But just looking at the 5-limit transversal for a catakleismic MOS is exactly the same as looking at kleismic; it's not much help for the more complex 7-limit. Below I give a kleismic transversal, but also a 2.5.7 transversal, and a 17-note 2.3.7 transversal; the latter because catakleismic is contorted as a 2.2.7 temperament. By sticking these various transversals into Scala you can get different views of Catakleismic[34].


! kleismic34trans.scl
!
Kleismic[34] transversal (detempering)
 34
!
 128/125
 25/24
 16/15
 27/25
 10/9
 9/8
 144/125
 75/64
 6/5
 100/81
 5/4
 32/25
 162/125
 4/3
 27/20
 25/18
 45/32
 36/25
 40/27
 3/2
 125/81
 25/16
 8/5
 81/50
 5/3
 128/75
 125/72
 16/9
 9/5
 50/27
 15/8
 48/25
 125/64
 2/1


! catakleismic34semitransversal.scl
!
17 note 2.3.7 semitransversal of Catakleismic[34]
 17
!
 28/27
 243/224
 9/8
 7/6
 243/196
 9/7
 4/3
 112/81
 81/56
 3/2
 14/9
 392/243
 12/7
 16/9
 448/243
 27/14
 2/1


! catakleismic34trans.scl
!
Catakleismic[34] 2.5.7 transversal
 34
!
 128/125
 401408/390625
 48828125/44957696
 15625/14336
 125/112
 28/25
 3584/3125
 11239424/9765625
 1953125/1605632
 15625/12544
 5/4
 32/25
 100352/78125
 12845056/9765625
 78125/57344
 625/448
 7/5
 896/625
 114688/78125
 9765625/6422528
 78125/50176
 25/16
 8/5
 25088/15625
 3211264/1953125
 9765625/5619712
 3125/1792
 25/14
 224/125
 28672/15625
 89915392/48828125
 390625/200704
 125/64
 2/1

From: genewardsmith (2011-08-15)
Subject: Three views of Catakleismic[34]

When studying chord relationships in a MOS, it can be useful to look at a transversal. If you have a temperament like miracle or myna where 5 is relatively complex, you can take a 5-limit transversal, stick it into Scala and look at the lattice diagram, and use the triads as a guide to other chords, such as in particular the 7-limit tetrads. This works because the triads extend to tetrads.

However, consider for example catakleismic. This is a higher limit extension of kleismic which hasn't gained much traction since the 7 and 11 are so much more complex than the 5-limit, while 13, not so complex, is not as well in tune. But it's interesting at least in theory; for one thing, there's not much difference in tuning between marvel, tempering out 225/224, and catakleismic, which adds 4375/4374 to the mix. So it's one way of organizing anything in marvel temperament. But just looking at the 5-limit transversal for a catakleismic MOS is exactly the same as looking at kleismic; it's not much help for the more complex 7-limit. Below I give a kleismic transversal, but also a 2.5.7 transversal, and a 17-note 2.3.7 transversal; the latter because catakleismic is contorted as a 2.2.7 temperament. By sticking these various transversals into Scala you can get different views of Catakleismic[34].


! kleismic34trans.scl
!
Kleismic[34] transversal (detempering)
 34
!
 128/125
 25/24
 16/15
 27/25
 10/9
 9/8
 144/125
 75/64
 6/5
 100/81
 5/4
 32/25
 162/125
 4/3
 27/20
 25/18
 45/32
 36/25
 40/27
 3/2
 125/81
 25/16
 8/5
 81/50
 5/3
 128/75
 125/72
 16/9
 9/5
 50/27
 15/8
 48/25
 125/64
 2/1


! catakleismic34semitransversal.scl
!
17 note 2.3.7 semitransversal of Catakleismic[34]
 17
!
 28/27
 243/224
 9/8
 7/6
 243/196
 9/7
 4/3
 112/81
 81/56
 3/2
 14/9
 392/243
 12/7
 16/9
 448/243
 27/14
 2/1


! catakleismic34trans.scl
!
Catakleismic[34] 2.5.7 transversal
 34
!
 128/125
 401408/390625
 48828125/44957696
 15625/14336
 125/112
 28/25
 3584/3125
 11239424/9765625
 1953125/1605632
 15625/12544
 5/4
 32/25
 100352/78125
 12845056/9765625
 78125/57344
 625/448
 7/5
 896/625
 114688/78125
 9765625/6422528
 78125/50176
 25/16
 8/5
 25088/15625
 3211264/1953125
 9765625/5619712
 3125/1792
 25/14
 224/125
 28672/15625
 89915392/48828125
 390625/200704
 125/64
 2/1