scale tree

Does anybody know how to define the tolerance on the generator at each of
these points?

> 2 3
> 5
> 7 8
> 9 12 13 11
> 11 16 19 17 18 21 19 14
> 13 20 25 23 26 31 29 22 23 31 34 29 27 30 25 17

For example, everything has an MOS at 2. As you get higher, the tolerance
becomes tighter.

Carl

Message text written by INTERNET:tuning@onelist.com
>
Does anybody know how to define the tolerance on the generator at each of
these points?
...
For example, everything has an MOS at 2. As you get higher, the tolerance
becomes tighter.<

The range of tolerance in a given scale is defined by the sub-MOS's above
it. For example, the generating interval for a 12 tone MOS scale with MOS
subsets at 5 and 7 tones must be between 2/5 and 3/7 octaves.

My horizontal presentation was a simplification, leaving out the generator
size. The trees generated by Wilson and Hanson are in this format:

...
...
...
1/2 .50000
5/11 .45454
4/9 .44444
7/16 .43750
3/7 .42857
8/19 .42105
5/12 .41666
7/17 .41176
2/5 .40000
7/18 .38888
5/13 .38461
8/21 .38095
3/8 .37500
7/19 .36842
4/11 .36363
5/14 .25714
1/3 .33333
5/16 .31250
4/13 .30769
7/23 .30434
3/10 .30000
8/27 .29629
5/17 .29411
7/24 .29166
2/7 .28571
7/25 .28000
5/18 .27777
8/29 .27586
3/11 .27272
7/26 .26923
4/15 .26666
5/19 .26315
1/4 .25000
4/17 .23529
3/13 .23076
5/22 .22727
2/9 .22222
5/23 .21739
3/14 .21428
4/19 .21052
1/5 .20000
3/16 .18750
2/11 .18181
3/17 .17647
1/6 .16666
2/13 .15384
1/7 .14285
1/8 .12500
...
...
...
0/1 .00000

Naturally, the tree can continue above 1/2 and be extended indefinitely
deep. A given ratio on the tree, 5/12, for example indicates a generator
size of 5/12 octaves and implies the complementary generator of 7/12. Over
the whole collection of 12 tones, the melodic symmetry with the cycle of
generating intervals is trivial, but the Moments of Symmetry in the
subsets, which form closed cycles only with the addition of an atypical
interval (e.g. wolfs, tritones and their analogs) to the cycle are the
points of real interest. It is at this level that the non-trivial melodic
symmetries take place.

>

STRAIGHT LINE PATTERNS Of The Scale Tree from 0/1 to 1/0
otherwise known as http://www.anaphoria.com/line.html
Here you can see the continuum of generators between 2 points on the scale tree
(the second half just repeats the first half in opposite direction

Stay tuned more is coming!

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com