Re: [tuning-math] Re: xenharmonic bridges in the 12edo comma pump

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, February 08, 2002 10:48 AM
> Subject: [tuning-math] Re: xenharmonic bridges in the 12edo comma pump
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > so our matrix for these three bridges is:
> >
> > [2 3 5] [-4 4 -1] = 81:80 syntonic comma
> > [-19/6 2 0] = 12edo==9/8 bridge = ~3.910001731 cents
> > [ -5/6 2 -1] = 12edo==10/9 bridge = ~17.59628787 cents
> >
> >
> > and note that these three bridges are linearly dependent.
>
> The second bridge is just P^(1/6), where P is the Pythagorean comma.
> The Pythagorean and syntonic commas together define the 12-et in
> the 5-limit; one way to express that is 81/80 ^ P = 81/80 X P
> (where the wedge product in this case becomes the vector cross-product)
> = h12, the [12, 19, 28] val. Your other bridge is W^(1/6), where
> W = (81/80)^6 P^(-1). Is there anything gained by taking roots
> of commas? I don't see it.
>
> > so here's the entire list of bridges which i would say
> > are in effect for the comma pump in 12edo:
> >
> > 2 3 5 ~cents
> >
> > [ -4 4 -1 ] = 81:80 syntonic comma = 21.5062896
> > [ -5/6 2 -1 ] = 12edo==10/9 bridge = 17.59628787
> > [-36/11 36/11 -9/11] = 1/11cmt==10/9 bridge = 17.59605512
> > [ -8/11 8/11 -2/11] = 1/11cmt==9/8 bridge = 3.910234472
> > [-19/6 2 0 ] = 12edo==9/8 bridge = 3.910001731
> > [161/66 -14/11 -2/11] = 12edo==1/11cmt bridge = 0.000232741
>
> It seems to me the only one on this list which is important for
> the comma pump is 81/80, and that defines it. The others define
> 12-et and 1/11 comma meantone, and have nothing to do with the
> pump so far as I can see.

thanks for all of that, Gene. i'm not sure what any of this means,
but my feeling is that all these small intervals (and many others
which i did not list) have something to do with what we're hearing
when composers intentionally mix the properties of different
tuning systems like this, so they're worth looking at.

-monz

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