Re: 3&7 based scales (was: Re: [MMM] lost in appalachia)

I'd like to follow up to something Gene Ward Smith wrote last spring
(I wasn't here then):

> One thing you might want to look at are the {3,7} Fokker blocks you
> get by taking a pair of 12-et compatible, {3,7}-commas. I've listed
> some of these below. The TM basis is {64/63, 729/686}, another
> important comma is the Pythagorean comma.

Questions: (1) What is meant by "12-et compatible"? (2) How does one
arrive at a list of such commas?

- Rich Holmes

--- In tuning@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>
> I'd like to follow up to something Gene Ward Smith wrote last spring
> (I wasn't here then):
>
> > One thing you might want to look at are the {3,7} Fokker blocks
you
> > get by taking a pair of 12-et compatible, {3,7}-commas. I've
listed
> > some of these below. The TM basis is {64/63, 729/686}, another
> > important comma is the Pythagorean comma.
>
> Questions: (1) What is meant by "12-et compatible"?

Gene means that these commas vanish in 12-equal. If you try to
construct these intervals in 12-equal, using 12-equal's best
approximations to the consonant intervals that make them up in JI,
you get a unison in each case.

Let's make this precise, since I think you're a physicist. The below
is the sort of material you see on tuning-math, but since I think
you're new, I'll break the "rule" and post it here.

Firstly, 12-equal is consistent in the 7-limit. Knowing that, it's
pretty unobjectionable to write the covector for the 12-equal pitch
contours as

<12 19 28 34]

(If anyone hasn't read my new paper -- which I'm snail-mailing out to
everyone for free -- just note that this is how 12-equal approximates
primes 2, 3, 5, and 7, in steps of 12-equa).

Now 64/63 can be written as 2^6 * 3^(-2) * 5^0 * 7^(-1), or as a
vector in the JI lattice,

[6 -2 0 -1>

To calculate its represenation in 12-equal, just take the bracket
product of this with the covector above:

<12 19 28 34|6 -2 0 -1>

= 12*6 - 19*2 - 34

= 0.

So 64/63 vanishes in 12-equal.

729/686 can be written as 2^(-1) * 3^6 * 5^0 * 7^(-3), or

[-1 6 0 -3>.

So, as above, we compute

<12 19 28 34|-1 6 0 -3>

= -12 + 19*6 - 3*34

= 0.

So 729/686 vanishes in 12-equal too.

> (2) How does one
> arrive at a list of such commas?

Brute force works -- create an array containing a large chunk of the
lattice, and compute the bracket product of each vector in this array
with the 12-equal covector. Look for where the result is zero -- the
corresponding vectors are commas that vanish in 12-equal. The set of
positions where a zero occurs -- called the kernel -- will form a
sublattice of your original lattice.

So another, quicker, way is to construct a lattice basis for the
kernel of 12-equal, and then flesh out the resulting lattice. The key
here is finding a valid lattice basis for the kernel. Any old
linearly independent set of vectors representing commas that vanish
in 12-equal won't necessarily do the trick, because it may only
generate a sublattice of the kernel lattice. In that case, the wedge
product of the linearly independent set of vectors will result in an
integer multiple of the 12-equal covector. What you need is a set
whose wedge product is exactly the 12-equal covector.

Let's ignore prime 5 and just consider primes 2, 3, and 7. These
generate a three-dimensional tuning system. So we will need two
independent vanishing commas to end up with the one-dimensional
system that is 12-equal. We already know that 64/63 works as one,
since

<12 19 28 34|6 -2 0 -1> = 0.

Since we're ignoring prime 5, we can just write this as

<12 19 34|6 -2 -1> = 0.

Similarly, 729/686 works as one, since

<12 19 34|-1 6 -3> = 0

To test whether these two serve as a basis for the full 12-equal
kernel lattice, and not just a subset, calculate the wedge product
and take its complement. In this, three-dimensional, case, another
word for this is "taking the cross-product", which operation should
be familiar to physicists. The result is

[6 -2 -1> X [-1 6 -3> = <12 19 34]

So indeed we've found a basis for 12-equal's kernel lattice.
Therefore, the entire infinite set of commas in {2,3,7}-JI -- that
is, JI built from the primes 2, 3, and 7 -- which vanish in 12-equal
can be constructed using two integer parameters, a and b, which both
vary across all integers, and calculating

a*[6 -2 -1> + b*[-1 6 -3>

In practice, running each a and b independently from, say, -50 to 50
will more than suffice. This will give you the list you were looking
for.

More later -- and should we stay at this level of mathematics,
perhaps we should move it to tuning-math (sorry for the repeated
moving requests :) ).