Fwd: True nature of the blackjack scale (in 7-limit) . . . and more epimores

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
Dave Keenan wrote (privately),

>What's the
>set of 7-limit UV's for Blackjack again?

The blackjack scale is the result of forming a periodicity block with
the unison vectors 2401:2400, 225:224, and 36:35; treating the
2401:2400 and 225:224 as commatic unison vectors and tempering them
out; and treating 36:35 as a chromatic unison vector and not
tempering it out.

Confused? Maybe it would help to add that the good old diatonic scale
in 5-limit is the result of forming a periodicity block with the
unison vectors 81:80 and 25:24; treating 81:80 as a commatic unison
vector and tempering it out; and treating 25:24 as a chromatic unison
vector and not tempering it out.

In other words, the diatonic scale is an infinite 'band' of the
infinite 2D 5-limit lattice, and the thickness of the band is given
by the 25:24 interval. This is clearly explained and depicted in my
paper, _The Forms Of Tonality_.

The blackjack lattice that I posted (and need to correct) shows that
the blackjack scale is an infinite 'slice' of the infinite 3D 7-limit
lattice, and the thickness of the slice is given by the 36:35
interval. You can see that there is no 36:35 interval, which would be
formed by moving two red connectors to the right, one green connector
to the lower-left, and one blue connector to the lower left, within
the blackjack scale. If you were to transpose the blackjack scale by
this interval, it would fit, with no gaps, on top of its transposed
self . . . and an infinite number of such 'layers' would fill the
infinite 3D 7-limit lattice.

So the blackjack scale is a "Form Of Tonality" (perhaps "Form of
Modality" would be better since no 'tonal center' is necessarily
implicated) with commatic and chromatic unison vectors very much in
accordance with the sizes of commatic and chromatic unison vectors in
the scales I've already described in that paper. Funny how things
that appear unrelated at first seem to 'fit together'!

Canasta seems less interesting from this point of view because its
chromatic unison vector is 81:80 . . . which is more like a commatic
unison vector in size, begging to be tempered out . . . and if you do
that, you get the wonderful 31-tET . . .

Always using epimoric ratios for the unison vectors has one
advantage . . . the size of the numbers in the ratio immediately
tells you both the melodic smallness of the interval, and its taxicab
distance in the triangular lattice (suitably constructed, as in the
second-to-last lattice on Kees van Prooijen's page
http://www.kees.cc/tuning/lat_perbl.html). So tempering out an
epimoric unison vector that uses numbers N times smaller than another
one means that the constituent consonances will have to be tempered
N^2 times as much . . . am I on to something?
--- End forwarded message ---

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

/tuning-math/message/133

Thanks for posting and cross-posting this message, Paul. I never
even understood that blackjack is an infinite 7-limit lattice as
compared with our traditional diatonic being an infinite 5-limit
lattice... Somehow I never "got" that before from the previous
dialogues... No wonder there are so many "just tetrads" in it!

__________ ___________ _______
Joseph Pehrson

--- In tuning-math@y..., jpehrson@r... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning-math/message/133
>
> Thanks for posting and cross-posting this message, Paul. I never
> even understood that blackjack is an infinite 7-limit lattice as
> compared with our traditional diatonic being an infinite 5-limit
> lattice... Somehow I never "got" that before from the previous
> dialogues... No wonder there are so many "just tetrads" in it!
>
Hey Joseph . . . look at blackjack3.bmp in the new bjlatt.ZIP . . .
then look at Figure 8 in my paper _The Forms of Tonality_ . . . same
idea, different scale (and different lattice orientation) . . .

You can think of them as infinite, as we are here, or you can think
of them simply as 'wrapping around' to meet themselves. Remember
the 'donut'?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

/tuning-math/message/137

> Hey Joseph . . . look at blackjack3.bmp in the new bjlatt.ZIP . . .
> then look at Figure 8 in my paper _The Forms of Tonality_ . . .
same
> idea, different scale (and different lattice orientation) . . .
>
> You can think of them as infinite, as we are here, or you can think
> of them simply as 'wrapping around' to meet themselves. Remember
> the 'donut'?

Yes, indeed! It became patently obvious once I was SHOWN it! :)

Regarding the list, I don't know. I'm reluctant to post over to
the "big" list right now, just because it's so "big!" and there have
been so many complaints.

Maybe it is right to keep a list of more numerical or math items over
here. I was even reluctant to post the recent question Monz had
about fractional remainders after division... And probably that
would have been of interest to more people than just Monz and
myself....

Dunno. Maybe it's best to keep things running like they are until
there are "complaints" even though the vote indicated otherwise.
After all, there were lots of abstentions...

For myself, I'm pretty much getting used to having so many lists
now... and, since I'm still trying to read most all of them,
MATHEMATICALLY, it really doesn't make any difference.

See... I *was* going to get to the "math" part...

_________ _________ _______
Joseph Pehrson