source file: m1495.txt
Date: Thu, 6 Aug 1998 06:39:19 -0500 (CDT)

Subject: RE: Scales and Numerology

From: Paul Hahn <Paul-Hahn@library.wustl.edu>

On Wed, 5 Aug 1998, Paul H. Erlich wrote:
> Paul Hahn wrote,
> >Interesting.  I've lately been toying with a 9-out-of-31 scale.
> 
> Care to let us in on it?

Well, I'm still working out the details, plus I lack a means at the
moment to tune it up and listen to how it sounds so my toying at the
moment just means working out theoretical quirks, but the basic scale is
4-4-2-5-3-3-4-3-3.  In the 3, 5, 7 lattice it looks like this:

	     6/5 --- 3/2 ---15/8
	     / \     / \     /
	    /   \   /   \   /
	   / 7/5 \ / 7/4 \ / 35/32
	 8/5 --- 1/1 --- 5/3


	28/15

The pivotal pitch of the scale is pitch 28 out of 31, represented twice
by 15/8 and 28/15 (separated by 225/224), this scale's equivalent of the
supertonic in that it relates the "dominant" and "subdominant" harmonic
regions to each other.  There are only two complete Otonal 4:5:6:7
tetrads, but a nice distribution of (Otonal and Utonal) 4:5:6 and 4:5:7
triads.

Eventually I plan to generalize the nonatonic framework by fixing the
1/1, 5/4, and 8/5 as the boundaries of three tetrachords, within which
the other scale degrees can vary as long as they form 8/7s, 7/6s or 6/5s
with those three "framework" pitches--but first I have to find a new
instrument, or figure out how to get one of my existing ones to work for
microtonality.  Frankly, my theorizing has never made it to this stage
before.

Oh, John and Daniel--I know full well numerology is a bunch of hooey.
My father is a mathematician, after all.  But in this case it's kind of
fun to speculate on what subconscious or cultural influences may have
been at work.  F'rinstance, cats have nine lives and I love cats, and as
I mentioned before nine is significant in Norse mythology, which I have
been interested in for a long time.  Here's a quote from _The Norse
Myths_, by Kevin Crossley-Holland:

	Nine worlds encompassed by the tree (which so becomes a symbol
	of universality known to mythologists as the World Tree); nine
	nights hanging on the tree; the number nine recurs again and
	again in Norse mythology.  Odin learns nine magic songs from a
	giant that enable him to win the mead of poetry for the gods;
	Heimdall has nine mothers; Hermod, Odin's son, journeys for nine
	nights in his attempt to win back the god Balder from Hel; the
	great religious ceremonies at the temple of Uppsala lasted for
	nine days in every ninth year, and required the sacrifice of
	nine human beings and nine animals of every kind.  Why nine was
	the most significant number in Norse mythology has not been
	satisfactorily explained, but belief in the magical properties
	of the number is not restricted to Scandinavia.  In _The Golden
	Bough_, J.G. Frazer records ceremonies involving the number nine
	in countries as widely separated as Wales, Lithuania, Siam and
	the island of Nias in the Mentawai chain.  Nine is, of course,
	the end of the series of single numbers, and this may be the
	reason why it symbolises death and rebirth in a number of
	mythologies; hence it also stands for the whole.

No luck yet on 31 though. 8-)>

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
    O
   /\        "Churchill? Can he run a hundred balls?"
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