more 2-D periodicity-block math (was: new 1/6-comma meantone lattice)

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>; <tuning@yahoogroups.com>
> Sent: Thursday, December 27, 2001 4:18 AM
> Subject: [tuning-math] new 1/6-comma meantone lattice
>
>
> I've added a new lattice to my "Lattice Diagrams comparing
> rational implications of various meantone chains" webpage:
>
> http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm
>
>
> It's about 2/3 of the way down the page: a new lattice
> showing a definition of 1/6-comma meantone within a
> 55-tone periodicity-block... just under the old 1/6-comma
> lattice, below this text:
>
>
> >> And here is a more accurate lattice of the above,
> >> showing a closed 55-tone 1/6-comma meantone chain and
> >> its implied pitches, all enclosed within a complete
> >> periodicity-block defined by the two unison-vectors
> >> 81:80 = [-4 4 -1] (the syntonic comma, the shorter
> >> boundary extending from south-west to north-east on
> >> this diagram) and [-51 19 9] (the long nearly vertical
> >> boundary), portrayed here as the white area.
> >>
> >> For the bounding corners of the periodicity-block, I
> >> arbitrarily chose the lattice coordinates [-7.5 -5]
> >> for the north-west corner, [-11.5 -4] for north-east,
> >> [11.5 4] for south-west, and [7.5 5] for south-east.
> >> This produces a 55-tone system centered on n^0.

Actually, these choices turn out not to be arbitrary.
I found them intuitively, but now I've figured out how
to formalize it.

There's a very simple formula which finds the corners of
a 2-dimensional periodicity-block from the unison-vectors.

For matrix M:

M = [a b]
[c d]

Each of the four corners NW, NE, SW, and SE are:

NW = [ ( a-c)/2 , ( b-d)/2 ]
NE = [ (-a-c)/2 , (-b-d)/2 ]
SW = [ ( a+c)/2 , ( b+d)/2 ]
SE = [ (-a+c)/2 , (-b+d)/2 ]

(I invite canditates for better terms.)

In simple generalized terms:

[ (+/-a +/-c) (+/-b +/-d) ]
---------------------------
2

Plugging our example in, we get:

M = [ 4 -1]
[19 9]

NW = [ -7.5 -5]
NE = [-11.5 -4]
SW = [ 11.5 4]
SE = [ 7.5 5]

Then, to get the corners of the tiling periodicity-blocks
which fill the rest of the lattice, simply add or subtract
either of the unison-vectors separately to any of these
coordinates, and iterate the process as long as necessary
to find as many tiles as desired.

One of the unison-vectors will tile the plane along
either of two parallel sides of the parallelogram, and the
other unison-vector will tile the plane along either
of the other two parallel sides.

Has this ever been explained explicitly like this before?
If so, please give references, tuning-math list URLs, etc.

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> One of the unison-vectors will tile the plane along
> either of two parallel sides of the parallelogram, and the
> other unison-vector will tile the plane along either
> of the other two parallel sides.

This is exactly what the "Gentle Introduction" shows.