Diamonds are Forever Retool

Dave Keenan's recent post on the tonality diamond inspired me to dig up and
revise an article I posted here on 12/31/98. Enjoy!

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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/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 all the modes of a scale relative to one master key?
4. A Cartesian cross between a set of rationals and their reciprocals?
5. A slightly-mutilated Rothenberg interval matrix (aka difference matrix)?
6. The one structure with the highest chord/note ratio at limit X?
7. The solution to a sphere-packing problem in X dimensions?

[#1.] The number of items in a Farey series of order X, and the number of
items in a Lambdoma of cap-X, is X^2. A Farey series of order X is a list
of all the fractions that you can make using whole numbers up to X. So X^2
makes sense, because you're taking X things two at a time; there are two
places for a number in a fraction. X^2 makes sense for the Lambdoma when
you realize the Lambdoma is a square; as a child you learn to count objects
in a grid by counting along the edges and multiplying.

[#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 puzzles --
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 then eliminate the 9/3 and 3/9).

All of the solutions to the second puzzle are given by the rows and columns
in Conway's matrix. This puzzle is equivalent to selecting o- or
utonalities (minus the unity) from the diamond. The color touching the
table is the numerary nexus.

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 contain
the 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.

[#3-4.] This was my breakthru with the diamond. I had come up with a
bunch of 7-limit scales (one of which was identical to Kraig Grady's
"centaur" scale), 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 the
other notes in the scale, what kind of relationships would I have? In
essence, I wanted to make a given note in the scale the "1/1". This can be
done by multiplying the entire scale by the reciprocal of the given note.
Doing this for each note of the scale in turn, I wound up with the
cross-set of the original scale and its subharmonic inversion --- 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 --
when he suggested I explain on the list my recent statement that a scale's
Rothenberg interval matrix was "just its Lambdoma or tonality diamond". I
believe that anyone who read that recent Rothenberg post, and up to here in
this article, should have no problems turning a diamond into an interval
matrix and back (hint: the interval classes in the interval matrix are the
diagonals on the diamond). Of course, Partch used the diamond to generate
a pitch set, while Rothenberg used it to generate an interval set (it would
make sense to take the interval matrix of a diamond, but not the diamond of
an interval matrix), but the structure is the same.

[#6.] I am not aware of a proof, but there can be no doubt.

[#7.] A famous problem asks for the most compact arrangement of generic
spheres of unit diameter in N dimensions. In 2-D, the generic sphere is
the circle, and the closest packing has been proven to be "hexagonal". If
we tile a plane with circles in this way, and mark the centers of the
circles on the plane, we get the 2-D triangular lattice. If we then take
any given circle, and all the circles whose centers lie one unit away, and
erase their centers' marks, we will have a hexagonal hole in our lattice.
Interestingly, the 3-factor diamond is this hexagon when mapped to the 2-D
(3-factor) triangular lattice.

A similar situation occurs in 3-D. It was conjectured long ago, but only
recently proven, that the closest packing of 3-D spheres in 3-space is the
so-called "face-centered cubic" packing. In this arrangement, the centers
of a given sphere and all those 1 unit away form the vertices of a
cuboctahedron. Again, when the 4-factor diamond is mapped to the
tetrahedral (3-D triangular) lattice, the diamond represents this "unit
packing" of spheres (the cuboctahedron).

So it is tempting to say that the closest packing of N-dimensional spheres
is achieved by centering them on the vertices of the N-dimensional
triangular lattice, and that the (N+1) factor diamond represents the unit
packing on this lattice. I am not aware of any proof of this, however. It
is interesting to speculate on the relationship between this statement and
the one of #6. . .

-Carl

Carl Lumma wrote,

>So it is tempting to say that the closest packing of N-dimensional spheres
>is achieved by centering them on the vertices of the N-dimensional
>triangular lattice,

This doesn't work in 8 dimensions or higher. In 8 dimensions, there's so
much space between the spheres in this packing that you can insert a whole
other copy of the packing into the spaces, creating what I believe is called
the Leech lattice. Paul Hahn probably knows more.

Carl Lumma wrote,

6. The one structure with the highest chord/note ratio at limit X?

[#6.] I am not aware of a proof, but there can be no doubt.

Sorry, it ain't so.

5-limit diamond:

5/3---5/4
/ \ / \
/ \ / \
4/3---1/1---3/2
\ / \ /
\ / \ /
8/5---6/5

6 consonant chords, 7 notes

Weird 12-tone scale:

25/24-25/16
/ \ / \
/ \ / \
5/3---5/4--15/8
/ \ / \ / \
/ \ / \ / \
4/3---1/1---3/2---9/8
\ / \ / \ /
\ / \ / \ /
8/5---6/5---9/5

13 consonant chords, 12 notes

What would be an entry level book covering this subject of Diamonds?

Thanks,
Micah

> -----Original Message-----
> From: Paul H. Erlich [SMTP:PErlich@Acadian-Asset.com]
> Sent: Tuesday, January 11, 2000 10:56 AM
> To: 'tuning@onelist.com'
> Subject: RE: [tuning] Diamonds are Forever Retool
>
> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> Carl Lumma wrote,
>
> 6. The one structure with the highest chord/note ratio at limit X?
>
> [#6.] I am not aware of a proof, but there can be no doubt.
>
> Sorry, it ain't so.
>
>
> 5-limit diamond:
>
> 5/3---5/4
> / \ / \
> / \ / \
> 4/3---1/1---3/2
> \ / \ /
> \ / \ /
> 8/5---6/5
>
> 6 consonant chords, 7 notes
>
>
> Weird 12-tone scale:
>
> 25/24-25/16
> / \ / \
> / \ / \
> 5/3---5/4--15/8
> / \ / \ / \
> / \ / \ / \
> 4/3---1/1---3/2---9/8
> \ / \ / \ /
> \ / \ / \ /
> 8/5---6/5---9/5
>
> 13 consonant chords, 12 notes
>
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>What would be an entry level book covering this subject of Diamonds?

They are introduced in Partch's _Genesis of a Music_; I'm not aware of any
other books that go into them.

Carl Lumma wrote,

>A similar situation occurs in 3-D. It was conjectured long ago, but only
>recently proven, that the closest packing of 3-D spheres in 3-space is the
>so-called "face-centered cubic" packing. In this arrangement, the centers
>of a given sphere and all those 1 unit away form the vertices of a
>cuboctahedron. Again, when the 4-factor diamond is mapped to the
>tetrahedral (3-D triangular) lattice, the diamond represents this "unit
>packing" of spheres (the cuboctahedron).

>So it is tempting to say that the closest packing of N-dimensional spheres
>is achieved by centering them on the vertices of the N-dimensional
>triangular lattice, and that the (N+1) factor diamond represents the unit
>packing on this lattice.

I wrote,

>This doesn't work in 8 dimensions or higher. In 8 dimensions, there's so
>much space between the spheres in this packing that you can insert a whole
>other copy of the packing into the spaces

From http://mathworld.wolfram.com/HyperspherePacking.html:

"The analog of face-centered cubic packing is the densest lattice packing in
4- and 5-D. In 8-D, the densest lattice packing is made up of two copies of
face-centered cubic. In 6- and 7-D, the densest lattice packings are Cross
Sections of the 8-D case. In 24-D, the densest packing appears to be the
Leech Lattice."

Also of possible interest:

The largest number of unit Circles which can touch another is six. For
Spheres, the maximum number is 12. Newton considered this question long
before a proof was published in 1874. The maximum number of hyperspheres
that can touch another in n-D is the so-called Kissing Number.

Go to http://mathworld.wolfram.com/KissingNumber.html for more on Kissing
Numbers.

[Daniel Wolf wrote...]
>>6. The one structure with the highest chord/note ratio at limit X?
>
>Depends on how you define "chords", but if we accept any triad from the
>given limit, then combination-product sets beat diamons. I.e. a hexany,
>with six tones, has eight triads.

Really? What's the triad/note ratio of the 4-factor diamond?

[responding to Paul Erlich, I wrote...]
>Hmm. Maybe the rule even holds for scales of cardinality up to 2x, or
>something. . .

Obviously not, since the scale you gave has less than 14 tones, and in
general you can always take two (or more) diamonds and overlap them, when
tones at the intersection will serve double duty (you can cut the third
diamond out of the scale you give and get 10 triads with 10 notes, or add
one and get a ratio of 8/7). But I think that at each layer of sphere
packing you get the highest ratio for up to that cardinality.

-Carl