Monzo symmetrical systems

I figured since I was on a roll with the Scala files,
I'd post some more.

These are symmetrical systems I designed about a year ago,
when I first came up with the lattice design I describe in
my website article.

When viewed on a lattice like mine, they are perfectly symmetrical, in
a way that Partch diamond-systems, which I illustrate on my
webpage, are not (those have symmetry, but not perfect symmetry).

Manuel Op de Coul says:
> 'they are point-symmetrical (don't know if that's the common
> term in English) in the origin'

and
> 'in the shape of an octahedra (sp?)'.

(They'll be available in the Scala archive too.)

The numerical guidelines for designing them were to set
an absolute limit of 2 in both directions (positive and negative)
for exponents along the prime axes, and an absolute limit
of 1 for the exponents along the composite axes. Thus they
have 'pigtails', to use Erv Wilson's term, in both directions
along all prime axes.

I wrote a pretty piece about a month ago in the 5-limit
system - an ambient thing with long sustained tones.

In practice the following pairs of notes are nearly equivalent:

in the 7-limit:

21/16 and 64/49 and their complements, 32/21 and 49/32

in the 11-limit (in addition to the above):

32/25 and 14/11 and their complements, 25/16 and 11/7
77/64 and 6/5 and their complements, 128/77 and 5/3
64/55 and 7/6 and their complements, 55/32 and 12/7
12/11 and 35/32 and their complements, 11/6 and 64/35
(11/10 and 20/11, respectively, are also close to these)
128/121 and 16/15, and their complements, 121/64 and 15/8
are also only about a comma apart.

Lattices:

5-limit system

9:8
/
15:8 /
/ '-._ /
25:16 / 3:2
'-._ / / '-._
5:4 / 6:5
/ '-._ / /
/ 1:1 /
/ / '-._ /
5:3 / 8:5
'-._ / / '-._
4:3 / 32:25
/ '-._ /
/ 16:15
/
16:9

7-limit system

9:8
/
15:8 /
12:7 ../__'-._ /
25:16 / / ''--- 3:2 ...__
10:7 ...__'-./_ / / '-._ ''-- 21:16
64:49 ..__ '-._ ''/-- 5:4 ...__ / 6:5 /
'''-- 8:7 ../__'-._ ''/- 35:32 / /
/ '-/._ '''-- 1:1 ...__ '-/._ /
/ / 64:35 ./.__'-._ ''/-- 7:4 ...__
/ 5:3 / ''-- 8:5 ../__'-._ ''-- 49:32
32:21 ..__'-._ / / '-/._ ''-- 7:5
''-- 4:3 ...__ / / 32:25
/ '-._ ''/-- 7:6
/ 16:15
/
16:9

11-limit... well, you get the idea.

That one will either have to wait till my JustMusic software
is out, or until I put a diagram on my website (unless someone
else wants to make one!)

Any further analyses are welcome.

I'd be especially interested in hearing music using
Anaphorian modulation techniques in these scales (Kraig).

!------------------- BEGIN SCALA FILE -------------
!
!\monzo-sym-5.scl
!
Monzo symmetrical system: 5-limit
13
!
16/15
9/8
6/5
5/4
32/25
4/3
3/2
8/5
5/3
16/9
15/8
25/16
2/1
!
!------------------- END SCALA FILE ---------------

!------------------- BEGIN SCALA FILE -------------
!\monzo-sym-7.scl
!
Monzo symmetrical system: 7-limit
25
!
16/15
9/8
6/5
5/4
4/3
7/5
3/2
8/5
5/3
16/9
15/8
32/21
8/7
12/7
64/49
10/7
25/16
7/6
7/4
21/16
32/25
49/32
35/32
64/35
2/1
!------------------- END SCALA FILE ---------------

!------------------- BEGIN SCALA FILE -------------
!
!\monzo-sym-11.scl
!
Monzo symmetrical system: 11-limit
41
!
16/15
9/8
6/5
5/4
4/3
7/5
3/2
8/5
5/3
16/9
15/8
121/64
33/32
11/8
11/6
55/32
11/7
11/10
77/64
32/21
8/7
12/7
64/49
10/7
25/16
7/6
7/4
21/16
32/25
49/32
35/32
64/35
128/77
20/11
64/33
16/11
12/11
64/55
14/11
128/121
2/1
!
!------------------- END SCALA FILE ---------------

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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