Re: Eikosany as ball in Wilson lattice

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

Sorry, this was supposed to go on tuning-math.

I thought i would respond in part even though this was sent to the wrong list.
One can always tell the consonances within an Eikosany from any one tone to another in that it will have 2 factors in common.
Since the eikosany is a structure and not limited to any particular ratios , on could use 3 squared as one of the factors, and i know i have seen Erv write it out this way. Obviously one could pick factors that could detroy the overall purpose of the structure. Regardless other consonances will appear from time to time that, but the way in which these tones work can be more on the lines of a dissonance if veiwed and heard within the overall context and useage of the tones involved. Each tone will function as a unique 'role' in functioning as three factors harmonically and 3 subharmonically and one becomes quite use to this overall meaning within the structure. In a sense it creates a noncentered functional harmony matrix. each chord implying the others.
As far as the center, the structure can be generated by and of the factors squared and it recipocal. The 3-5-7-9-11 being the recipocal of 1 squared.

>Message: 5 > Date: Wed, 16 Mar 2005 21:23:40 -0000
> From: "Gene Ward Smith" <gwsmith@svpal.org>
>Subject: Eikosany as ball in Wilson lattice
>
>
>If L is an odd number, we may call a product of odd numbers up through
>L, 3^e3 5^e5 ... L^el, a Wilson product. These do not represent pitch
>classes when L>7, because of the lack of unique factorization.
>However, you can put a symmetrical A_n lattice structure on them,
>which makes the Wilson products into a lattice, where the L-limit
>consonaces are the closest products to the unison, and are all at the
>same distance. This means, of course, we would count 9/7 as a
>consonance, but not 3^2/7, and would regard 9/3^2 as a comma.
>
>This, according to Paul, is Wilson's point of view, and if we adopt
>it, the Eikosany is a deep hole, whose center is 3^(1/2) 5^(1/2)
>7^(1/2) 9^(1/2) 11^(1/2). This does not extend far enough to involve
>the 9/3^2 comma, for ball scales of larger radius we will have
>multiple Wilson products corresponding to a single pitch class, and
>will have "scales" where some of the scale steps are 9/3^2 or 3^2/9.
>
>
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
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