Full thread (2 messages)
From: Gene Ward Smith (2004-07-13)
Subject: Hahn reduced scales
By a Hahn reduced scale I mean a 5 or 7 limit scale reduced according
to a comma set which defines a val for the number of notes in the
scale to the correct prime limit. The smallest Hahn distance is
picked, and if there is a tie, Tenney distance is used to break the tie.
Since Hahn distance and symmetrical Euclidean distance are so similar
this will be similar if not the same to the Euclidean reduction I've
mentioned already, but Hahn distance might be preferred; in any case,
I'm giving it; I give Scala files for the scales, but before that I
list some pertinant data. The numbers on the second line are major
tetrads and major tetrads tempered by marvel, and the same for the
other pairs, which are minor tetrads, supermajor tetrads, subminor
tetrads and 9-limit quintads in otonal and utonal flavors. Since
225/224 is a scale step for Hahn16, the count is a little confused there.
The third line gives the scale by giving its major/minor tetrads in
lattice format, union all the notes left over.
Hahn12 is noteable for being one of the
(15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 = 2 scales.
7-limit Hahn reduced scales
12: epimorphic, strictly proper, CS, superparticular
2/3, 2/2, 1/1, 0/1; 0/1, 0/1; smallest = 25/24
{[-1, 0, -1], [0, 0, 0], [0, 0, 1], [-1, -1, -1]} U {5/3}
15: epimorphic, improper, CS, superparticular
3/4, 2/3; 1/2, 2/3; 2/3, 1/2; smallest = 50/49
{[-1, 0, -1], [0, 0, 0], [0, -1, 0], [0, -1, -1], [0, 0, -1]} U
{16/15,10/9,9/5,15/8}
16: epimorphic, improper, not CS, not superparticular
2/4, 1/4; 1/2, 2/2; 1/2, 1/2; smallest = 225/224 (vanishes in marvel)
{[-1, 0, -1], [0, 1, 1], [0, 0, 0], [0, 0, 1], [-1, -1, -1]} U
{25/16,5/3,28/15}
19: epimorphic, improper, CS, superparticular
4/5, 4/6; 4/5, 3/4; 3/4, 3/4; smallest = 50/49
{[-1, 0, -1], [0, 0, 0], [-1, 1, 0], [-1, 1, 1], [0, -1, 0],
[0, -1, -1], [0, 0, -1], [0, 0, 1]} U {14/9,35/18}
22: epimorphic, improper, CS, superparticular
6/7, 5/6; 5/6, 5/6; 4/5, 3/4; smallest = 126/125
{[-1, 0, -1], [1, -2, -1], [1, -1, -1], [0, 0, 0],
[1, -1, 0], [0, -1, -1],
[0, 0, -1], [0, 0, 1], [-1, -1, -1], [-1, 0, 1],
[-1, 0, 0]} U {14/9,9/5}
! hahn12.scl
Hahn-reduced 12 note scale
12
!
15/14
8/7
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2
! hahn15.scl
Hahn-reduced 15 note scale
15
!
16/15
10/9
7/6
6/5
5/4
4/3
7/5
10/7
3/2
8/5
5/3
7/4
9/5
15/8
2
! hahn16.scl
Hahn-reduced 16 note scale
16
!
15/14
9/8
8/7
6/5
5/4
21/16
4/3
7/5
3/2
25/16
8/5
5/3
7/4
28/15
15/8
2
! hahn19.scl
Hahn-reduced 19 note scale
19
!
21/20
15/14
9/8
7/6
6/5
5/4
9/7
4/3
7/5
10/7
3/2
14/9
8/5
5/3
7/4
9/5
15/8
35/18
2
! hahn22.scl
Hahn-reduced 22 note scale
22
!
25/24
15/14
10/9
8/7
7/6
6/5
5/4
9/7
4/3
25/18
7/5
35/24
3/2
14/9
8/5
5/3
12/7
7/4
9/5
15/8
35/18
2
From: Gene Ward Smith (2004-07-14)
Subject: Re: Hahn reduced scales
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
Some of these scales can be adjusted, retaining epimorphicity, so that
they have more complete tetrads and all of the notes can be given in
terms of tetrads. If we look for triads of the 11 notes of the 12 note
scale below, minus 5/3, we find that adjusting 5/3 to 12/7 allows it
to be harmonized by the tonic minor, 1-6/5-3/2-12/7, which is given by
[-1,0,0], so we could add that to the set of triads for 12 notes.
Similarly, for 22 we can adjust 14/9 up a 50/40 to 100/63, and 9/5 up
a 250/243 to 50/27, and add the [1,-3,-1] tetrad to the list. This
sort of thing improves things for 7-limit tetrads but not always in
general. I give the adjusted scales below.
! hen12.scl
Adjusted Hahn12
12
!
15/14
8/7
6/5
5/4
4/3
7/5
3/2
8/5
12/7
7/4
15/8
2
! hen22.scl
Adjusted Hahn22
22
!
25/24
15/14
10/9
8/7
7/6
6/5
5/4
9/7
4/3
25/18
7/5
35/24
3/2
100/63
8/5
5/3
12/7
7/4
50/27
15/8
35/18
2