cpak19a

From: Gene Ward Smith (2005-10-23)
Subject: Chord pack scales

By listing chords of a particular type or types in some order, and
then testing if the chord can be added to a set of notes and preserve
an epimorphic property with respect to a particular val, one can
construct epimorphic scales with a good quantity of the desired
chords. If the list order has a good underlying logic, the scales
should be interesting.

I tried the method out and it seems to work. I listed all the tetrads
in the 11x11x11 chord cube, from [-5 -5 -5] to [5 5 5], by taking the
usual Euclidean distance from a point near [0 0 0]. If you pick the point
[-1/11 -1/13 -1/17] then it transpires that all 1331 tetrads in the
chord cub are at a unique distance, leading to a unique ordering.
Using infinitesimal elements would be less arbitary, but this was
easier and for the test I ran should lead to the same result. By going
through all six permutations of the above point, you get six different
orderings and potentially six different scales, though there is
nothing which compels the scales to be distict.

I tried it out with the standard 19 septimal val; the six different
scales boiled down to only two. Both have five otonal and five utonal
tetrads, which rises to six each upon marvel tempering. Both seven
major triads, rising to nine on marvel tempering. The first has eight
minor tetrads, rising to nine on marvel tempering, and the second has
seven minor triads, rising to nine on marvel tempering. Therefore the
inversion of the first scale is perhaps marginally the most
interesting of the two scales and two inversions. Below are the two
scales in question.

! cpak19a.scl
First 19-epimorphic ordered tetrad pack scale
19 
!
21/20
15/14
9/8
7/6
6/5
5/4
21/16
4/3
7/5
10/7
3/2
63/40
8/5
5/3
7/4
9/5
15/8
63/32
2

! cpak19b.scl
Second 19-epimorphic ordered tetrad pack scale
19
!
21/20
15/14
9/8
7/6
6/5
5/4
21/16
4/3
7/5
10/7
3/2
63/40
8/5
5/3
7/4
25/14
15/8
63/32
2

From: Gene Ward Smith (2005-10-23)
Subject: Chord pack scales

By listing chords of a particular type or types in some order, and
then testing if the chord can be added to a set of notes and preserve
an epimorphic property with respect to a particular val, one can
construct epimorphic scales with a good quantity of the desired
chords. If the list order has a good underlying logic, the scales
should be interesting.

I tried the method out and it seems to work. I listed all the tetrads
in the 11x11x11 chord cube, from [-5 -5 -5] to [5 5 5], by taking the
usual Euclidean distance from a point near [0 0 0]. If you pick the point
[-1/11 -1/13 -1/17] then it transpires that all 1331 tetrads in the
chord cub are at a unique distance, leading to a unique ordering.
Using infinitesimal elements would be less arbitary, but this was
easier and for the test I ran should lead to the same result. By going
through all six permutations of the above point, you get six different
orderings and potentially six different scales, though there is
nothing which compels the scales to be distict.

I tried it out with the standard 19 septimal val; the six different
scales boiled down to only two. Both have five otonal and five utonal
tetrads, which rises to six each upon marvel tempering. Both seven
major triads, rising to nine on marvel tempering. The first has eight
minor tetrads, rising to nine on marvel tempering, and the second has
seven minor triads, rising to nine on marvel tempering. Therefore the
inversion of the first scale is perhaps marginally the most
interesting of the two scales and two inversions. Below are the two
scales in question.

! cpak19a.scl
First 19-epimorphic ordered tetrad pack scale
19 
!
21/20
15/14
9/8
7/6
6/5
5/4
21/16
4/3
7/5
10/7
3/2
63/40
8/5
5/3
7/4
9/5
15/8
63/32
2

! cpak19b.scl
Second 19-epimorphic ordered tetrad pack scale
19
!
21/20
15/14
9/8
7/6
6/5
5/4
21/16
4/3
7/5
10/7
3/2
63/40
8/5
5/3
7/4
25/14
15/8
63/32
2