Topic: Octonys, breed balls and miraclized breed balls

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Thread (1 messages)

From: Gene Ward Smith (2005-05-21)
Subject: Octonys, breed balls and miraclized breed balls

If we require the hexagon projection of the hexany using the breed
temperament to produce regular hexagons, we are in effect putting a
norm on note classes with notes written 2^a (49/40)^b (10/7)^c of
sqrt(2b^2+c^2). We can take the midpoint of an interval of 10/7 and
draw circles around it, obtaining scales. Calling these breed balls,
the first breed ball around the interval 49/40-7/4 is of course just
this interval, but the second is an interesting scale we might call
the octony. It is an eight-note scale containing a hexany, plus the
two intervals the hexany hexagon contains. Despite having only eight
notes, already 2401/2400 relationships put in an appearence. Below I
give an octony in 7-limit JI form, but really it should be considered
as a tempered object--tempering by 441 or 612 would be excellent.
Despite the simplicity of this scale, Scala knows not of it.

! octony.scl
octony around 49/40-7/4 interval
8
!
15/14
49/40
5/4
10/7
3/2
12/7
7/4
2

The breed balls thus far discussed are microtemperings of the 7-limit
down to the 5-limit. However, the 11-limit makes an appearence in a
natural way, as the approximations of miracle are all over the place
in breed balls. As a scale of miracle, the octony is
[-7,-5,-2,0,1,3,6,8]. We see therefore that from -7 to +8 secors we
get an 11/8, and we also have some 11/9 (3 secors.)

Here are other breed balls:

! bree3.scl
Third breed ball around 49/40-7/4
12
!
49/48
21/20
15/14
49/40
5/4
7/5
10/7
3/2
49/32
12/7
7/4
2

Miracle form: [-10,-7,-5,-4,-2,0,1,3,5,6,8,11]

! bree4.scl
fourth breed ball around 49/40-7/4
14
!
1
49/48
21/20
15/14
6/5
49/40
5/4
7/5
10/7
3/2
49/32
12/7
7/4
25/14
2

Miracle form: [-12,-10,-7,-5,-4,-2,0,1,3,5,6,8,11,13]