Fokker's periodicity blocks

[Paul Erlich]
>Interestingly, the same numbers which delimit ETs with good rational
>implication often also specify the number of pitch-classes in
>just-intonation systems which incorporate bridging to delineate the finity
>of the system.

Interesting, but not surprising. If we view MOS's as 1D periodicity blocks
[only those unison vectors which allow the generator to subtend a fixed
number of scale degrees count, but these are going to occur at small
intervals, relative to the number of generators in the chain, and so will
be intuitive unison vectors most of the time], then it is no more
surprising than the fact that ET's with good x's have the same number of
notes per y as MOS's of generator x and interval of equivalence y.

[Paul Hahn]
>For example, the simplest set of unison vectors for 12TET at the 5-limit
>form the matrix:
>
> | 4 -1 | = 12
> | 0 3 |
>
>By substituting (-1 2) for (0 3) (implying that the 25/24, chromatic
>semitone, "vanishes"--IOW, the 6/5 and the 5/4 are the same generic
>interval) you get
>
> | 4 -1 | = 7
> |-1 2 |
>
>implying heptatonic scales, the simplest of which is the diatonic set.
>But you can embed the diatonic set in other scales which use (4 -1) as a
>unison vector such as 19TET [(4 -1), (-1 5)] or 31TET [(4 -1), (-1 8)].

What determines if you can embed one scale in another like this? The
examples you gave share at least one unison vector. Namely, the "top" one.
Is there anything special about the top vector? Is the sharing of any one
vector sufficient?

[Paul Erlich]
>Although this number usually described in terms of the volume of a
>parallelopiped, the edges can be deformed into almost any shape one
>wishes (as long as in each of the three sets of four parallel edges,
>the four remain translationally congruent) and one gets a JI scale
>with the same number of notes.

_Three_ sets of parallel edges?

>Now of course all the properties of the block are preserved if we transpose
>one (or more) note(s) by one of the unison vectors used to create the block.
>This corresponds to distorting the edges of the block in parallel as I've
>discussed before.

I understand the transposing of notes by unison vectors, but what do you
mean about the edges?

>I've been thinking about how to answer Joe's request, and just figured it
>out. The answer: start with a sufficiently large collection of lattice
>points, postmultiply it by the inverse of the Fokker matrix, and keep the
>points whose coordinates are all greater than or equal to zero and less
than >one. Then postmultiply the resulting collection by the Fokker matrix.

It may help me to figure out what you're up to here if I knew what Joe's
request was.

>The inverse of this matrix is
<-snip->
>If we transform the set of lattice points with this matrix, we can define a
>periodicity block within any 1 X 1 square of the transformed lattice. The
>usual approach is to use the "unit square" from the origin to (1,1), but of
>course we are free to translate this square wherever we want since the
>result will still tile the plane with the unison vectors as generators.

I can't say I have a clue about this transformation. What's the inverse of
a matrix?

-C.

Carl Lumma wrote,

>[Paul Erlich]
>>Interestingly, the same numbers which delimit ETs with good rational
>>implication often also specify the number of pitch-classes in
>>just-intonation systems which incorporate bridging to delineate the finity
>>of the system.

That wasn't me, that was Joe Monzo. But recently on the list, I made a
comparison between finding good ETs and finding periodicity blocks, pointing
out the important similarities and differences.

Paul Hahn wrote,

>>But you can embed the diatonic set in other scales which use (4 -1) as a
>>unison vector such as 19TET [(4 -1), (-1 5)] or 31TET [(4 -1), (-1 8)].

Carl wrote,

>What determines if you can embed one scale in another like this? The
>examples you gave share at least one unison vector. Namely, the "top" one.
>Is there anything special about the top vector? Is the sharing of any one
>vector sufficient?

Carl, the embedding simply refers to the fact that for 5-limit purposes, the
diatonic scale is based on meantone logic, so (4 -1) must vanish (be a
unison vector). If one of the unison vectors of an ET is (4 -1), then the
diatonic scale can be embedded in that ET. That's all Paul H. was saying.

>[Paul Erlich]
>>Although this number usually described in terms of the volume of a
>>parallelopiped, the edges can be deformed into almost any shape one
>>wishes (as long as in each of the three sets of four parallel edges,
>>the four remain translationally congruent) and one gets a JI scale
>>with the same number of notes.

>_Three_ sets of parallel edges?

That's right -- I was talking about the 3D case, and in general there is one
set of parallel edges for each unison vector.

>>Now of course all the properties of the block are preserved if we
transpose
>>one (or more) note(s) by one of the unison vectors used to create the
block.
>>This corresponds to distorting the edges of the block in parallel as I've
>>discussed before.

>I understand the transposing of notes by unison vectors, but what do you
>mean about the edges?

If you distort each edge corresponding to a given unison vector in the same
way, you are not changing the volume of the periodicity block, but you may
be transposing some notes by one of the other unison vectors. If you draw a
big 5-limit lattice and split it up into the 22-tone periodicity blocks of
my sruti post, then you can see how distorting one of the two sets of edges
in a particular way will give you the desired sruti configuration.

>>I've been thinking about how to answer Joe's request, and just figured it
>>out. The answer: start with a sufficiently large collection of lattice
>>points, postmultiply it by the inverse of the Fokker matrix, and keep the
>>points whose coordinates are all greater than or equal to zero and less
>than >one. Then postmultiply the resulting collection by the Fokker
matrix.

>It may help me to figure out what you're up to here if I knew what Joe's
>request was.

He wanted to see graphs of the periodicity blocks. But the hard part was
figuring out which notes they contained!

>I can't say I have a clue about this transformation. What's the inverse of
>a matrix?

If A^-1 is the inverse of a matrix A, then A^(-1) * A = A * A^(-1) = I (I is
the identity matrix, all zeros except ones along the diagonal from upper
left to lower right). I assume you know how matrix multiplication is
defined?

>I can't say I have a clue about this transformation.

OK, let me clarify this. A matrix represents a linear transformation. It
takes the unit vectors (in the 2D case, (1 0) and (0 1)) and transforms them
to the vectors specified in the rows of the matrix. Since any conceivable
shape can be described in terms of the unit vectors, any shape can be
transformed in a well-defined way by the matrix. The inverse of a matrix
will then transform the shape back to how it started. So, to find the notes
in a periodicity block, we use the fact that the inverse of the Fokker
matrix will transform the periodicity block to a 1 X 1 square. It's much
easier to determine, especially in higher dimensions, which points are in a
unit hypercube than to determine which points are in a hyper-parallelopiped.
Then transform the points back to the lattice using the original matrix, and
voila your periodicity block.

>Carl, the embedding simply refers to the fact that for 5-limit purposes, the
>diatonic scale is based on meantone logic, so (4 -1) must vanish (be a
>unison vector). If one of the unison vectors of an ET is (4 -1), then the
>diatonic scale can be embedded in that ET. That's all Paul H. was saying.

I got the diatonic scale. But he was also embedding 9-tone scales in 31,
and your decatonic scales in 22. The diatonic and nonatonic scales each
shared one unison vector with the larger tuning, but the decatonic scale
shared two. It is simply that the diatonic and nonatonic scales can be
defined by a single vector, and any tuning that shares the vector will
work, while the decatonic scales are defined by two vectors, and any tuning
that shares those two would work? That would seem to make sense. I just
wondered if the larger tuning could have a vector in addition to the
required one(s) that would destroy the scale. Is that possible? And
exactly what properties are specified by the unison vector (which ones are
preserved across different embeddings)?

>>_Three_ sets of parallel edges?
>
>That's right -- I was talking about the 3D case

"duh."

>If you distort each edge corresponding to a given unison vector in the same
>way, you are not changing the volume of the periodicity block, but you may
>be transposing some notes by one of the other unison vectors. If you draw a
>big 5-limit lattice and split it up into the 22-tone periodicity blocks of
>my sruti post, then you can see how distorting one of the two sets of edges
>in a particular way will give you the desired sruti configuration.

Got it.

>>>I've been thinking about how to answer Joe's request, and just figured it
>>>out. The answer: start with a sufficiently large collection of lattice
>>>points, postmultiply it by the inverse of the Fokker matrix, and keep the
>>>points whose coordinates are all greater than or equal to zero and less
>>>than one. Then postmultiply the resulting collection by the Fokker
>>>matrix.
>>
>>I can't say I have a clue about this transformation. What's the inverse of
>>a matrix?
>
>If A^-1 is the inverse of a matrix A, then A^(-1) * A = A * A^(-1) = I (I is
>the identity matrix, all zeros except ones along the diagonal from upper
>left to lower right). I assume you know how matrix multiplication is
>defined?

Unfortunately, no. Nor am I familiar with the term "postmultiply". What's
the point of showing A^(-1) * A = A * A^(-1)? Isn't that just commutativity?

>OK, let me clarify this. A matrix represents a linear transformation. It
>takes the unit vectors (in the 2D case, (1 0) and (0 1)) and transforms them
>to the vectors specified in the rows of the matrix. Since any conceivable
>shape can be described in terms of the unit vectors, any shape can be
>transformed in a well-defined way by the matrix. The inverse of a matrix
>will then transform the shape back to how it started. So, to find the notes
>in a periodicity block, we use the fact that the inverse of the Fokker
>matrix will transform the periodicity block to a 1 X 1 square. It's much
>easier to determine, especially in higher dimensions, which points are in a
>unit hypercube than to determine which points are in a hyper-parallelopiped.
>Then transform the points back to the lattice using the original matrix, and
>voila your periodicity block.

Excellent thinking and explaining, Paul. Okay, I've got the idea. I guess
I just need to learn a little about matrix multiplication.

-C.

Carl wrote,

>It is simply that the diatonic and nonatonic scales can be
>defined by a single vector, and any tuning that shares the vector will
>work, while the decatonic scales are defined by two vectors, and any tuning
>that shares those two would work?

Yup.

>That would seem to make sense. I just
>wondered if the larger tuning could have a vector in addition to the
>required one(s) that would destroy the scale. Is that possible?

Only if it causes different notes in the scale to become equivalent to one
another, reducing the number of notes it contains.

>And
>exactly what properties are specified by the unison vector (which ones are
>preserved across different embeddings)?

I'm sure you know what a unison vector means, so I'm not sure what you're
asking.

>Unfortunately, no. Nor am I familiar with the term "postmultiply". What's
>the point of showing A^(-1) * A = A * A^(-1)? Isn't that just
commutativity?

Matrix multiplication is not commutative. The definition is, if A*B=C, then

C(i,j) = A(i,1)*B(1,j) + A(i,2)*B(2,j) + A(i,3)*B(3,j) . . .