--- 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:

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:

> 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