Re: [tuning-math] Digest Number 125

Thanks to Gene and Graham for your explanations, and Paul for your offer
(I emailed you back). I'm still catching up, and I tried to come up with
an example to start working through for myself. Let's say I wanted a scale
where four times 13:1 was once 7:1, that is I want the "comma" 13^4 /
7*2^12, or 28561/28672, to vanish. And I further want two 13s to be three
7s, that is I want the comma 338/343 to disappear. So I get the following
matrix:

[4 -1]
[2 -3]

whose determinant is 10, so I'm looking at a periodicity block of 10
notes. And indeed 10-tet has good approximations of 13:8 and 7:4--7/10ths
and 8/10ths of an octave respectively--and the congruences in the matrix
are true, 4* 7/10 == 8/10 and 2* 7/10 == 3*8/10.

If I understand correctly I can choose from many possible 10-note subsets
of the Z^2 lattice to construct my just scale, as long as I don't pick any
pairs of notes separated by my unison vectors (4, -1) or (2, -3). But this
can still lead to different just scales, so I imagine I can't say that the
commas 28561/28672 and 338/343 "define" whichever region of the lattice I
choose as my just scale.

Right so far? --Jon

I wrote:

> I want the "comma" 13^4 /
> 7*2^12, or 28561/28672, to vanish.
[snip]
> I want the comma 338/343 to disappear. So I get the following
> matrix:
>
> [4 -1]
> [2 -3]

(where the first column is powers of 13, and the second powers of 7, and
for my toy example I don't care about the others)

Inverting this matrix, if I remember how to do it properly, gives me this:

[ 3/10 -1/10 ]
[ 1/5 -2/5 ]

How do I obtain a useful matrix, i.e. one I could read off different ets
from, from this?

--- In tuning-math@y..., jon wild <wild@f...> wrote:

> If I understand correctly I can choose from many possible 10-note
subsets
> of the Z^2 lattice to construct my just scale, as long as I don't
pick any
> pairs of notes separated by my unison vectors (4, -1) or (2, -3).
But this
> can still lead to different just scales, so I imagine I can't say
that the
> commas 28561/28672 and 338/343 "define" whichever region of the
lattice I
> choose as my just scale.

> Right so far? --Jon

It's much more restrictive than that--the notes can't be spread all
over the lattice, but must be bunched together. Let's construct your
example using my definition of block.

(1) First check that we have a valid basis;
(28672/28561 * 343/338)^10 = 1.204 < 2, so we are in very valid
territory.

(2) Form the matrix <2, 282672/28561, 343/338> and invert it:

[1 0 0]^(-1) [1.0 0.0 0.0]
[12 1 -4] = [2.8 -0.2 0.4]
[1 3 -2] [3.7 -0.3 0.1]

(3) Define a norm on q = 2^a 7^b 13^c by setting

||q|| = max(|a + 2.8*b + 3.7*c|, |-.2*b-.3*c|, |.4*b+.1*c|

This makes the numbers 2^a 7^b 13^c into a three-dimensional lattice;
we can measure the distance between lattice points p and q by

d(p, q) = ||p/q||.

(4) Find sets such that the diameter is less than 1. This means for
s1, s2 in S, we have d(s1, d2) < 1.

Without loss of generality, we can assume that the points are
contained in ||q|| <= 1/2. We may also use integer arithmetic if we
choose, by rescaling by a factor of 10; this would mean ||q||<=5,
where now v1=10*a+28*b+378c, v2=-2*b-3*c and v3=4*b+c are integers
less than or equal to 5 in absolute value. If we look at triples
[v1,v2,v3] less than 5 in absolute and which correspond to integer
[a,b,c], we find nine candidates:

169/224 [-4 -4 -2]
13/16 [-3 -3 1]
7/8 [-2 -2 4]
13/14 [-1 -1 -3]
1 [ 0 0 0]
14/13 [ 1 1 3]
8/7 [ 2 2 -4]
16/13 [ 3 3 -1]
224/169 [ 4 4 2]

We now have eight possibilities to complete the scale, [+-5 +-5 +-5];
however modulo transposition this becomes four, and modulo inversion
as well, two; hence there really aren't vast numbers of possibilities.
Our eight choices are 91/64,64/91,91/128,128/91,2197/1568,1568/2197,
3136/2197, 2197/3168. If we pick 91/64, [5 -5 5], we have a block
corresponding to your choice of commas--enjoy!

Gene wrote (quoting me):

>> If I understand correctly I can choose from many possible 10-note
>> subsets of the Z^2 lattice to construct my just scale, as long as I
>> don't pick any pairs of notes separated by my unison vectors (4, -1)
>> or (2, -3). But this can still lead to different just scales, so I
>> imagine I can't say that the commas 28561/28672 and 338/343 "define"
>> whichever region of the lattice I choose as my just scale.
>>
>> Right so far? --Jon
>
> It's much more restrictive than that--the notes can't be spread all
> over the lattice, but must be bunched together.

right - I first wrote ->connected<- subset but then thought that was
redundant, so deleted the word. I understand from the rest of what you
wrote that the requirement is stronger still than connected: that the span
of the set is also limited. Thanks --Jon

p.s. sorry for the lame subject fields, I kept forgetting to edit them
from the digest I'm replying to.

--- In tuning-math@y..., jon wild <wild@f...> wrote:

> I understand from the rest of what you
> wrote that the requirement is stronger still than connected: that
the span
> of the set is also limited. Thanks --Jon

It is at least for classic blocks, which is what I just defined.

The question is what to do about what I was calling "semiblocks";
simply requiring them to come from a convex region with a span of
less than 1 on the first coordinate (which I think is in effect what
Paul suggests in his Gentle Introduction) is pretty weak, and it
allows for more extreme examples than he gave, but my definition may
still be too restrictive--it doesn't include his example of the
Indian diatonic, with sixth degree raised a comma, for instance. That
makes for a distance of 8/7 to 4/3, which is a little far. Possibly a
semiblock should just be convex and the diameter would serve as a
measure of how extreme it is, so the above example would be an 8/7-
semiblock.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., jon wild <wild@f...> wrote:
>
> > If I understand correctly I can choose from many possible 10-note
> subsets
> > of the Z^2 lattice to construct my just scale, as long as I don't
> pick any
> > pairs of notes separated by my unison vectors (4, -1) or (2, -3).
> But this
> > can still lead to different just scales, so I imagine I can't say
> that the
> > commas 28561/28672 and 338/343 "define" whichever region of the
> lattice I
> > choose as my just scale.
>
> > Right so far? --Jon
>
> It's much more restrictive than that--the notes can't be spread all
> over the lattice, but must be bunched together.

Well, in the diatonic case, you're free to transpose any note by any
number of 81:80s, since this is the commatic unison vector and is
ignored in diatonic notation (I object to the idea of there being a
JI diatonic scale of fixed specification). As for the 25:24, this is
the chromatic unison vector, so transposing notes by it will lead to
chromatically altered diatonic scales (still diatonic, as no
chromatic semitones will be present between notes of the scale). My
construction for the unaltered diatonic scale is as follows: divide
the 5-limit lattice into "bands" or "strips" which run parallel to
the commatic UV (81:80), and which are exactly wide enough to allow
one chromatic UV (25:24) to reach from one edge of the strip to the
other. Then, if you stay within one strip, simply choose any 7 notes
so that no pair is seperated by 81:80 or a power thereof (i.e. a
multiple of the syntonic comma).