> 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

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--- 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.