Topic: Rectangular breed scales

4 scales

Thread (2 messages)

From: Gene Ward Smith (2005-05-21)
Subject: Rectangular breed scales

Another interesting way to construct scales for breed tempering is to
take rectangles of the generators. I checked scales of the form [a,b],
for 1 <= a,b <= 6, where by this I mean that I took (49/40)^i(10/7)^j,
with 0 <= i <= a and 0 <= j <= b, and reduced these by octaves and
2401/2400. Below I list what we get in those cases where the resulting
scale is epimorphic, listing [a,b], the scale, and the epimorph val
for the scale. Note the 3x3 square, the [2,2] scale, is epimorphic.



[2, 1]
[1, 15/14, 49/40, 10/7, 3/2, 7/4]
[6, 10, 14, 17]

[2, 2]
[1, 49/48, 15/14, 49/40, 5/4, 10/7, 3/2, 49/32, 7/4]
[9, 15, 22, 26]

[4, 1]
[1, 15/14, 9/8, 49/40, 21/16, 10/7, 3/2, 45/28, 7/4, 90/49]
[10, 16, 23, 28]

[6, 1]
[1, 15/14, 9/8, 135/112, 49/40, 21/16, 135/98, 10/7, 3/2, 45/28,
27/16, 7/4, 90/49, 63/32]
[14, 22, 32, 39]

[2, 4]
[1, 49/48, 25/24, 15/14, 35/32, 49/40, 5/4, 125/98, 10/7, 35/24, 3/2,
49/32, 25/16, 7/4, 25/14]
[15, 25, 36, 43]

[2, 6]
[1, 49/48, 25/24, 625/588, 15/14, 35/32, 125/112, 49/40, 5/4, 125/98,
125/96, 10/7, 35/24, 125/84, 3/2, 49/32, 25/16, 625/392, 7/4, 25/14,
175/96]
[21, 35, 50, 60]

[6, 2]
[1, 49/48, 15/14, 9/8, 147/128, 135/112, 49/40, 5/4, 21/16, 135/98,
45/32, 10/7, 3/2, 49/32, 45/28, 27/16, 441/256, 7/4, 90/49, 15/8, 63/32]
[21, 33, 49, 59]

From: Gene Ward Smith (2005-05-21)
Subject: Re: Rectangular breed scales

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Another interesting way to construct scales for breed tempering is to
> take rectangles of the generators. 

Looking at these further, the 7x3 rectangle strikes me as particularly
interesting, since it is epimorpic with the 21-et standard val, and is
a miracle modmos. Possibly similar results could be had for canasta
and studloco using corner clipping. 

The modmos in question is, in the mode of the scale given below, the
following:

[-22, -19, -17, -16, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, 
-4, -3, -2, 0, 1, 3, 6]

Here are some Scala scl formatted scales:

! brect73.scl
7x3 breed rectangle scale, <21 33 49 59| epimorphic
21
!
49/48
15/14
49/45
10/9
7/6
49/40
5/4
80/63
4/3
49/36
10/7
3/2
49/32
14/9
49/30
5/3
7/4
16/9
49/27
40/21
2

! brect33.scl
3x3 breed rectangle scale, <9 15 22 26| epimorphic
9
!
8/7
6/5
49/40
7/5
8/5
49/30
12/7
49/25
2

! brect35.scl
3x5 breed rectangle scale, <15 25 36 43| epimorphic
15
!
49/48
8/7
7/6
6/5
49/40
5/4
7/5
10/7
8/5
49/30
5/3
12/7
7/4
49/25
2

! brect37.scl
3x7 breed rectangle scale, <21 35 50 60| epimorphic
21
!
49/48
36/35
21/20
15/14
147/125
6/5
49/40
5/4
48/35
7/5
10/7
36/25
147/100
3/2
49/32
42/25
12/7
7/4
48/25
49/25
2