[tuning] questions about Graham's matrices (was: 13-limit mappings)

Thanks for the explanation, Graham. A lot of it is now
much clearer.

I started this post as a series of questions about what
I still didn't understand, but by working thru it I've
gotten most of it. I decided to post my working-out here
in the hope that it will help others to understand.

A few questions remain below...

> From: <graham@microtonal.co.uk>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, June 27, 2001 4:35 AM
> Subject: [tuning] Re: 13-limit mappings
>
>
> > > mapping by period and generator:
> > > ([1, 0], ([0, 2, -1], [5, 1, 12]))
>
> The first two-element list shows the mapping of the octave. The second
> element is always zero for both my scripts, as the period is always a
> fraction of an octave.

So in other words you always use the nearest integer here?
I'm still confused about that "0".

> So the first number tells you how many equal
> parts the octave is being divided into. Here it's 1 which is the
> simplest case.

Confused about this too... I thought this example divided the
octave into 41 parts? Again, is this pair of numbers expressing
the nearest integer fraction of an octave?

Also, I think it's confusing the way you give the "octave correction"
first and the "number of generators" second in this line, but
it's reversed in all the following lines, generator first and
octave second.

>
> In more familiar terms, the generator is a 5:4 major third. 5 major
> thirds are a 3:1 perfect twelfth.

(2^(380.391/1200))^5 does indeed equal exactly 3.

Following you so far...

> An octave less two major thirds is a 9:7 supermajor third.

OK... using the regular ratios (which I understand
are only approximated by your generator)...

In regular math:

2 / ((5/4)^2) = 2 * (16/25) = 32/25

In vector addition:

2/1 = [ 1 0 0]
(5/4)^2 = [-2 0 1] * 2 = [-4 0 2]

and

[ 1 0 0]
-[-4 0 2]
----------
[ 5 0 -2] = 32/25

32/25 is ~7.7 cents narrower than 9:7, but both of these are
approximated well by your actual result of 2 / ((2^(380.391/1200))^2).

So I follow this too. Now comes the tricky part...

> (2*(5,0) - (12,-1) = (-2, 1))

OK, so as I said above, ((2^(380.391/1200))^5) * (2^0) = 3 .
The "2*" means that we square that, and so the first group
stands for 3^2 = 9 .

And ((2^(380.391/1200))^12) * (2^-1) = ~6.983305074 ,
which agrees with your definition above as ~7.

The minus sign means we divide the terms, and...
Voil�! ... ~9/7 .

And checking the answer:
((2^(380.391/1200))^-2) * (2^1) does indeed equal the ~9/7.

So you're putting an equivalence relationship in here.

That was confusing... I had a hard time understanding how
~9/7 = "An octave less two major thirds". Now it's clear.

> > > mapping by steps:
> > > [(22, 19), (35, 30), (51, 44), (62, 53)]
>
> Each pair shows the size of a prime interval in terms of scale steps.
> Call the steps x and y. An octave is 22x+19y. For the case where x=y,
> you have 41-equal. Where x=0, you have 19-equal. Where y=0, you have
> 22-equal. So 19, 22 and 41-equal are all members of this temperament
> family.
>
> 3:1 is 35x+30y, 5:1 is 51x+44y and 7:1 is 62x+53y. You can get any
> 7-prime limit interval in terms of x and y by combining these.

OK, I understand all the math here, but I'm not quite following the
logic which deterimines that they are "all members of this temperament
family". How does your program find the 19, 22 and 41 in this example?

>
> For visualising the scale, it can be simpler to reduce each prime
> interval to be within the octave.
>
> > > [(22, 19), (35, 30), (51, 44), (62, 53)]
>
> 3:2 is 13x + 11y

Because

[ 35 30]
-[ 22 19]
----------
[ 13 11]

> 5:4 is 7x + 6y

[ 51 44]
-[ 22 19] * 2

=

[ 51 44]
-[ 44 38]
----------
[ 7 6]

> 8:7 is 4x + 4y

[ 62 53]
-[ 22 19] * 3

=

[ 62 53]
-[ 66 57]
-----------
[- 4 - 4]

>
> and use simpler coordinates. Here, q=x+y and p=x
>
> 3:2 is 11q + 2p
> 5:4 is 6q + p
> 8:7 is 4q

Oops... now you lost me.

> So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal
> and p is 1 step in 41-or 22-equal, and no steps in 19-equal.

Getting foggier...

>
>
> > > unison vectors:
> > > [[-10, -1, 5, 0], [5, -12, 0, 5]]

So these are the ratios 3125/3072 and 537824/531441 ?

-monz
http://www.monz.org
"All roads lead to n^0"

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