Topic: Some tetrad pack scales

5 scales

Thread (1 messages)

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

I've investigated further the idea of finding scales by packing
tetrads in order, and the results are interesting. In the case of 12,
I got some
of the cv family--scales with five tetrads--discussed here:

http://groups.yahoo.com/group/tuning-math/message/11451

This invesitation was very promising, but could not be pushed to
larger scales because the computations became too heavy. The tetrad
pack scales are far easier to find, and probably zero in on some of
the same maximal tetrad scales. 

Below I give my pick of the best of the lot for 12, 15, 19, 22 and 31
notes to the octave. In the case of 22, something happened which had
concerned me as a possibility, namely that a scale was started but
there was no way to finish it correctly. Nonetheless, some good tetrad
pack scales emerged.

! cpak12.scl
optimal tetrad pack scale = cv1
12
!
21/20
9/8
6/5
5/4
21/16
7/5
3/2
8/5
12/7
7/4
15/8
2

! cpak15.scl
optimal tetrad pack scale
15
!
21/20
35/32
9/8
49/40
5/4
21/16
7/5
35/24
3/2
63/40
105/64
7/4
147/80
15/8
2

! cpak19.scl
optimal tetrad pack scale
19
!
21/20
35/32
9/8
7/6
49/40
5/4
21/16
4/3
7/5
35/24
3/2
63/40
105/64
5/3
7/4
147/80
15/8
63/32
2

! cpak22.scl
optimal tetrad pack scale
22
!
25/24
15/14
35/32
8/7
7/6
6/5
5/4
9/7
4/3
48/35
10/7
35/24
3/2
25/16
8/5
5/3
12/7
7/4
64/35
15/8
35/18
2

! cpak31.scl
optimal tetrad pack scale
31
!
36/35
21/20
15/14
35/32
9/8
8/7
7/6
6/5
60/49
5/4
9/7
21/16
4/3
48/35
7/5
10/7
72/49
3/2
54/35
25/16
8/5
105/64
5/3
12/7
7/4
9/5
64/35
15/8
48/25
96/49
2