(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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