blackjack lattice question

Why is it that the *otonal* tetrads proceed OUTWARD, toward the
observer on the blackjack lattice and the *utonal* proceed INWARD
toward the distance away from the observer.

There must be some easy explanation for this, but it isn't coming to
me right away...

Thanks!

_______ _______ _____
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> Why is it that the *otonal* tetrads proceed OUTWARD, toward the
> observer on the blackjack lattice and the *utonal* proceed INWARD
> toward the distance away from the observer.
>
> There must be some easy explanation for this, but it isn't coming to
> me right away...

This is not specific to blackjack. It's a general lattice thing. A
utonality is an otonality turned inside-out. I'll try to demonstrate
for the 2D case in slow-motion-ASCII. You'll have to try to imagine
what this means when generalised to 3D or higher.

Start with a 5-limit otonality (aka major triad). Check that there's
nothing up my sleeves.

5
/ \
/ \
/ \
4-------6

Now break it up into its component dyads and make them all head
towards the point at the center of the triad

5 5
\/
/\
4--/--\--6
4 6

and then pass thru it and out the other side.

5 5
4--\--/--6
\/
/\
4 6

Now we pause to re-label the vertices, but notice that this doesn't
change the intervals in any way. e.g. 4:6 = 1/6 : 1/4

1/6 1/4
1/6-\--/-1/4
\/
/\
1/5 1/5

Now continue the movement until they join up again.

1/6-----1/4
\ /
\ /
\ /
1/5

Viola! 5-limit utonality (aka minor triad).

Now generalise that to the 7-limit tetrads.

Regards,
-- Dave Keena

On 6/30/01 3:16 AM, "Dave Keenan" <D.KEENAN@UQ.NET.AU> wrote:

>1/6-----1/4
> \ /
> \ /
> \ /
> 1/5
>
> Viola! 5-limit utonality (aka minor triad).

How about for a 'Cello...?

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_25849.html#25854

>
> This is not specific to blackjack. It's a general lattice thing.

Thanks a lot, Dave... this was quite helpful...

______ ______ ____
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> Why is it that the *otonal* tetrads proceed OUTWARD, toward the
> observer on the blackjack lattice and the *utonal* proceed INWARD
> toward the distance away from the observer.
>
> There must be some easy explanation for this, but it isn't coming to
> me right away...
>
> Thanks!

It's just the way I drew the ASCII lattice. But you should have thrown that away by now, and be
using the color lattice I put in the Files section. In that lattice, both types of tetrads are viewed
roughly "edge on" . . . though they're still mirror-images of one another, as they always must be .
. .

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., jpehrson@r... wrote:
> > Why is it that the *otonal* tetrads proceed OUTWARD, toward the
> > observer on the blackjack lattice and the *utonal* proceed INWARD
> > toward the distance away from the observer.
> >
> > There must be some easy explanation for this, but it isn't coming
to
> > me right away...
> >
> > Thanks!
>
> It's just the way I drew the ASCII lattice. But you should have
thrown that away by now, and be
> using the color lattice I put in the Files section. In that
lattice, both types of tetrads are viewed
> roughly "edge on" . . . though they're still mirror-images of one
another, as they always must be .
> . .

But, in that new blackjack 3D color lattice I have, the otonal still
seem to come out toward the observer, at least the center point, and
the utonal _away_ from the observer...

At least that's the way *I'm* looking at it... (??)

________ ______ _______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., jpehrson@r... wrote:
> > > Why is it that the *otonal* tetrads proceed OUTWARD, toward the
> > > observer on the blackjack lattice and the *utonal* proceed
INWARD
> > > toward the distance away from the observer.
> > >
> > > There must be some easy explanation for this, but it isn't
coming
> to
> > > me right away...
> > >
> > > Thanks!
> >
> > It's just the way I drew the ASCII lattice. But you should have
> thrown that away by now, and be
> > using the color lattice I put in the Files section. In that
> lattice, both types of tetrads are viewed
> > roughly "edge on" . . . though they're still mirror-images of one
> another, as they always must be .
> > . .
>
> But, in that new blackjack 3D color lattice I have, the otonal
still
> seem to come out toward the observer, at least the center point,
and
> the utonal _away_ from the observer...
>
> At least that's the way *I'm* looking at it... (??)
>
Sorry, Joseph, I got a little confused. I actually made the color
lattice correspond as closely as possible with the ASCII
lattice . . . so yes, the otonal tetrads point toward the observer,
and the utonal tetrads point away . . . again, that's just an
arbitrary convention, though whatever convention you choose, otonal
and utonal will always be mirror-images of one another, as Dave
Keenan beautifully showed . . .