Yahoo Tuning Groups Ultimate Backup tuning Moments of Symmetry at 12

I just noticed something I found very interesting
(given my interest in the importance of primes
in music):

In both the positive and negative methods of
linear mapping, the only significant MOS which
does *not* have a number of elements which
is prime is that with 12.

Positive mapping produces MOS at systems
with 2, 3, 5, 7, 12, 17, 29, and 41 notes.
Negative mapping produces MOS at systems
with 2, 3, 5, 7, 12, 19, and 31 notes.

Has this ever been pointed out before?
Any ideas on its importance, or on why
12 pops up among all those primes?

---------
references (by Erv Wilson):

letter to Chalmers pertaining to Moments of
Symmetry/ Tanabe Cycle
http://www.anaphoria.com/mos.html

On the Development of Intonational Systems
by Extended Linear Mapping
(from Xenharmonikon 3)
http://www.anaphoria.com/xen3b.html

- Monzo
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🔗D. Stearns <stearns@capecod.net>

-----Original Message-----
From: Joseph L Monzo

>I just noticed something I found very interesting(given my interest in the
importance of primes in music): In both the positive and negative methods of
linear mapping, the only significant MOS which does *not* have a number of
elements which is prime is that with 12.

I think the question should be: When does(n*O)+f,+F [where "O" is 12, "f" is
5, and "F" is 7]* result in two numbers that are neither prime nor a product
of primes?

Dan

* F f
\ /
--------------
O
/ | \
O+f | O+F
/ | \
...2*O...

🔗Joseph L Monzo <monz@xxxx.xxxx>

I wrote:
> In both the positive and negative
> methods of linear mapping, the only
> significant MOS which does *not* have
> a number of elements which is prime
> is that with 12.

Dan Stearns replied:
> I think the question should be:
>
> When does(n*O)+/-f,+/-F
> [where "O" is 12, "f" is 5, and "F" is 7]
> result in two numbers that are neither
> prime nor a product of primes >/= f and F?
>
> * F f
> \ /
> --------------
> O
> / | \
> O+f | O+F
> / | \
> ...2*O...

I've redrawn Dan's "Navaho eagle"
(my name for it) in a standard font for
those who saw the garbled version of it
(it's just a fleshing out of the above
diagram, but I think it looks beautiful):

12
/ \
17 19
__/ \__
e (12*2) e
___/ / \ \___
e 29 31 e
/ ___/ \___ \
e (17*2) (12*3) (19*2) e
/ / /\ \ \
e e 41 43 e e
/ / ___/ \___ \ \
e e e (12*4) e e e
/ / \ \
...(17*3) 53 5*11 (19*3)...

And yes, it is basically the same as Erv
Wilson's "Scale Tree", and I've already
drawn stuff like this years ago, when I
first got interested in JI, so obviously,
I understand intuitivly what's happening,
but I still don't quite get it.

The answer to my question appears to
be this: simply because the generating
interval in the scales demonstrated by
Wilson is the "5th" or its complement "4th",
which have step sizes of 7 and 5 semitones
in the 12-tone scale, then the above
mathematical progression results from the
combinations of 5 and 7 with 12 and its
higher multiples.

But that still seems like a circular
explanation to me. The "5th" and "4th"
*don't* contain 7 and 5 steps in the
higher-degree scales such as 19, 31, etc.
So why does the progression still work?
Or am I just ignorant of some basic
mathematical process here? (a distinct
possiblity). My math is telling me I
understand it, but my logic still says
I don't.

Does it have something to do with the
importance of the fact that there are
Moments of Symmetry at 5 and 7 themselves?
(5+7=12) . . . ???

🔗Kraig Grady <kraiggrady@xxxxxxxxx.xxxx>

🔗Daniel Wolf <DJWOLF_MATERIAL@xxxxxxxxxx.xxxx>

(Use fixed-width font for the following)

As I remember it, the center of Wilson's scale tree is like this:

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
etc.

Each number represents the number of tones in a scale. Each number is equal
to the sum of the two scales immediately above. (For our immediate
purposes, think of these as ETs.) Each of the scales contributing to the
sum represents an MOS subset. As an example, consider the 13 tone scale
formed by the sum of 5 and 8. The 5-toned MOS subset is:

0 738 277 1015 554--------1200
277 554 1015 1200 554 738 1200 277 738 1015
L L S L S L L S L S L L S L S

Explaination: the generating interval is 8/13 of an octave, ca. 738 cents.
A series of five of these intervals falls short of an octave multiple by
ca. 646 cents, but this interval functions melodically like the typical
generating intervals in that it subtends the same number of scale degrees.
Moreover, the sequence of subtended scale-steps (indicated here as L's and
S's) form a symmetrical pattern.

🔗Kraig Grady <kraiggrady@xxxxxxxxx.xxxx>

Daniel Wolf wrote:

> From: Daniel Wolf <DJWOLF_MATERIAL@compuserve.com>
>
> (Use fixed-width font for the following)
>
> As I remember it, the center of Wilson's scale tree is like this:
>
> 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
> etc.
>
> Each number represents the number of tones in a scale. Each number is equal
> to the sum of the two scales immediately above. (For our immediate
> purposes, think of these as ETs.) Each of the scales contributing to the
> sum represents an MOS subset. As an example, consider the 13 tone scale
> formed by the sum of 5 and 8. The 5-toned MOS subset is:
>
> 0 738 277 1015 554--------1200
> 277 554 1015 1200 554 738 1200 277 738 1015
> L L S L S L L S L S L L S L S
>
> Explaination: the generating interval is 8/13 of an octave, ca. 738 cents.
> A series of five of these intervals falls short of an octave multiple by
> ca. 646 cents, but this interval functions melodically like the typical
> generating intervals in that it subtends the same number of scale degrees.
> Moreover, the sequence of subtended scale-steps (indicated here as L's and
> S's) form a symmetrical pattern.

All can see the Scale Tree for themselves!

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

🔗Kraig Grady <kraiggrady@xxxxxxxxx.xxxx>

Kraig Grady wrote:

> nded scale-steps (indicated here as L's and
> > S's) form a symmetrical pattern.
>
> All can see the Scale Tree for themselves!

For some reason my link didnt work the scale tree can be seen at;
http://www.anaphoria.com/scaletree.html

>
>
> -- Kraig Grady
> North American Embassy of Anaphoria Island
> www.anaphoria.com