2-D periodicity-block math

(Please indulge my cross-posting. Thanks.)

This is an expansion of a post I sent to tuning-math.

> 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.
>>
>> The grey area represents the part of the JI lattice
>> outside the defined periodicity-block (and thus, with
>> each of those pitch-classes in its own periodicity-block),
>> and the lattice should be imagined as extending infinitely
>> in all four directions. The other periodicity-blocks,
>> all identical to this one, can be tiled against it to
>> cover the entire space.

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.

Here's something I'd like to be able to do to enhance this:

Enable the user to find unison-vectors by:

- click-and-drag on a pitch-height graph to specify an
approximate size for a unison-vector

- click on various lattice-points to create vectors directly

- use mouse-click and the Ctrl or Shift button together, to
drag a periodicity-block shape across the lattice

JustMusic would then calculate appropriate candidates for
the unison-vectors, displaying parameters of them in various ways.

Then, the user could also explore the graphing of various
linear or planar temperaments on the lattice, *within* the
rational periodicity-block already defined.

Of course, I'd also like to be able to work the other way
around: specify parameters of a temperament first, then find
the unison-vectors and periodicity-blocks from it.

Manuel, at this point, if you'd like to incorporate *any*
(or even all) of my JustMusic ideas into Scala, I'd be happy
to collaborate.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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