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).