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

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?

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

Yeah.

> 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 . . .

Ah. But you're only thinking 7-limit, not 9 or 11-limit.

> Always using epimoric ratios for the unison vectors has one
> advantage . . .

Why do we need this word "epimoric"? What's wrong with
"superparticular"? They both mean numerator is one more than
denominator. Correct?

> the size of the numbers in the ratio immediately
> tells you both the melodic smallness of the interval,

Yes.

> 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).

I don't see how it "immediately" tells you that. Surely you have to
prime factorise it first.

> 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?

I don't understand. Maybe an example?

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> Ah. But you're only thinking 7-limit, not 9 or 11-limit.

True (thus the subject line).
>
> Why do we need this word "epimoric"? What's wrong with
> "superparticular"? They both mean numerator is one more than
> denominator. Correct?

Yeah -- just trying to tie in to the current discussion.
>
> > the size of the numbers in the ratio immediately
> > tells you both the melodic smallness of the interval,
>
> Yes.
>
> > 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).
>
> I don't see how it "immediately" tells you that. Surely you have to
> prime factorise it first.

You don't! That the beauty of this metric. Try it!

> > 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?
>
> I don't understand. Maybe an example?
>
The amount each consonance has to be tempered is inversely
proportional to the total number of consonances involved in the
unison vector, yes? The taxicab lattice distance is a measure of the
number of consonances involved. So lattice distance contributes a
factor of N . . . and the smallness of the unison vector contributes
another factor of N. . . . still want an example?