The ancient Greeks called it the Lambdoma, Partch called it the Tonality

Diamond, and Novarro must have called it something.* The structure is old,

and has a funny habit of being re-discovered many times across a wide

variety of disciplines.

[* There are minor differences here. "Lambdoma" is usually reserved for

the infinite diamond of all consecutive counting numbers, while Partch's

diamond exists at some "limit" and involves adjustments and omissions

related to octave equivalence.]

But what is it, this matrix of order/limit/factor cap X?

1. A fashionable arrangement of all the ratios in a Farey series of order X?

2. The solutions to an old puzzle involving colored space-filling polytopes?

3. A list of distances between any harmonic and all other harmonics up to X?

4. A list of all the modes of a scale relative to one master key?

5. A slightly-mutilated Rothenberg interval matrix (aka difference matrix)?

6. A Cartesian cross between a set of rationals and their reciprocals?

7. The one structure with the highest complete-chords/notes ratio at limit X?

8. The coordinates in the generic tetrahedral lattice of the centers of

X-dimensional spheres in the first "shell" of their closest-packing?

#1. The number of items in a Farey series of order X and Lambdoma of cap X

is 2^X, since there's X things taken two at a time (two places for a number

in a fraction). This makes sense if you look at the diamond as a square;

as a child you learn to count objects in a grid by counting along the edges

and multiplying. The jury is still out on just how fashionable the

arrangement of Farey fractions in the diamond is...

-----------------------------------------------------------------------

#2. Some time ago, John Chalmers posted to the list about his fondness for

Martin Gardner's recreational math. Recently, when looking thru my

bookshelf for something to read on my frequent bus trips to and from New

York city, I picked out "Fractal Music Hypercards and More", which proved

to be a lot of fun.

One of the articles in it, "The Thirty Color Cubes" (it doesn't seem to say

what issue of Scientific American it was reprinted from) describes two

classic "domino tiling" puzzles involving a set of cubes. With 6 colors

and one color to a face, there are 30 ways to paint a cube so it uses each

color only once.

One puzzle is to take one of these cubes and then build a two-by-two model

of it using 8 of the remaining cubes, so that all touching interior faces

of the model match in color. A second puzzle is to find a set of 5 cubes

such that they can be placed same-color down on a table and have each of

the 5 remaining colors showing faces up.

John H. Conway found a graphic method for solving both of these problems:

an arrangement of the 30 cubes in a grid. The grid has no blocks along

this mysterious diagonal... when I saw this, my diamond warning lights went

off. With 6 factors the diamond has 36 members, 6 of which will contain

two of the same color (the 1/1's), leaving 30. [Of course, we are used to

the 11-limit diamond having 29 pitches, as we only eliminate 5 of the

1/1's, giving 31, and also eliminate the 9:3 and 3:9]

All of the solutions to the second puzzle are given by the rows and colums

in Conway's matrix. This puzzle is equivalent to selecting O or

Utonalities (minus the unity) from the diamond. The colors touching the

table are the numerary nexi.

The first puzzle is solved by finding the inversion of the prototype cube

and taking the 8 cubes for the model from the row and column that intersect

at inversion cube, minus the inversion cube itself and the 1/1's [11 minus

1 minus 2]. I do not know what musical analog this first puzzle has.

A final note is that Conway's matrix itself meets the "domino condition".

That is, all the touching faces match in color. Again I do not know the

musical significance (if any).

It seems to me that the diamond should provide solutions for similar

puzzles involving any number of things and a corresponding set of

thing-sided space-filling polytopes.

-----------------------------------------------------------------------

#3. Start with 3, go down to 2. Take 4 and go down to 3. Take 5 and go

down to 4 and to 3. Take 6, go down to 5. Take 7, go down to 6, 5, and 4.

And so on... Lambdoma city.

-----------------------------------------------------------------------

#4. This was my big breakthru with the diamond. I had come up with a

bunch of 7-limit scales (one of which was the Centaur), and I wanted to

compare them by how "low numbered" the ratios in each of the modes were.

That is, if I modulated to one of notes in the scale, what kind of

relationships would I have? In essence, I wanted to make a given note of

the scale 1/1. This is done by multiplying thru by the reciprocal of that

note. When I had done this for all the modes, I noticed inversions of the

modes running at right angles. I had myself a diamond!

-----------------------------------------------------------------------

#5. This article was inspired by John Chalmers, whose substantial work and

ready help has played no small part in my understanding of music, and who

suggested that I explain on the list my recent statement that a scale's

Rothenberg interval matrix was "just its lambdoma or tonality dimond". I

believe that he who has read my recent Rothenberg post, and up to here in

this post, should have no problems turning a diamond into an interval

matrix and back (the interval classes in the IM are the diagonals on the

diamond).

-----------------------------------------------------------------------

#6. Yessir.

-----------------------------------------------------------------------

#7. This is unproven, but I am convinced it is true.

-----------------------------------------------------------------------

#8. I believe this is also unproven (tho it works in 2 and 3 dimensions).

In fact, I have read things that seem to suggest it is not true. I wonder

what relationship this question has to question #7?

-----------------------------------------------------------------------

This is by no mean a complete list of the amazing properties of the diamond.

Carl

Speaking of diamonds ...

Does anyone have an opinion as to whether 9/3 and 3/1 are really the same

thing, or should they be viewed as an auditory pun and used in slightly

different contexts?

Generalized diamonds are much prettier, more symmetrical-looking things if

you regard them as being a bit different.

-Bram

bram wrote:

> Does anyone have an opinion as to whether 9/3 and 3/1 are really the same

> thing, or should they be viewed as an auditory pun and used in slightly

> different contexts?

Perhaps from a diamond-layout perspective they could be placed on two

bars, but obviously there's no way the ear could distinguish between the two,

except contextually.

Dear Gary

I am new to this forum but I am A Director of the South East Just Intonation

Society--

No difference just numbes are different

You should obtain Barbara Hero's Bead theory book from www.lambdoma.com or get

your hands on Kayser's Akroasis or get some graph paper and you will be able to

diagnose "equivalence w/o ambivalence.

Resinate and Extenuate

Pat Pagano, Director

South East Just intonation Society

Gary Morrison wrote:

> From: Gary Morrison <mr88cet@texas.net>

>

> bram wrote:

>

> > Does anyone have an opinion as to whether 9/3 and 3/1 are really the same

> > thing, or should they be viewed as an auditory pun and used in slightly

> > different contexts?

>

> Perhaps from a diamond-layout perspective they could be placed on two

> bars, but obviously there's no way the ear could distinguish between the two,

> except contextually.

>

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