Yahoo Tuning Groups Ultimate Backup tuning RE: RE: Fokker periodicity blocks from the 3-5-7-harmoni c lattice

Joe Monzo wrote,

>I'm really tired right now and don't remember
>if Paul posted lattices to describe any of what he discussed,
>but I'm very interested in seeing the graphs

I didn't. By the way, I had a few typos in that post; the unison vector (3
-1 -3) I had as (3 -1 3) in the second matrix, making a periodicity block of
22 notes (coincidentally), though it should only have had 2 notes. 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. For
example, for the matrix

2 2 -1
-8 -6 2
1 -3 -2

the inverse is

-0.81818 -0.31818 0.090909
0.63636 0.13636 -0.18182
-1.3636 -0.36364 -0.18182

the points in the unit cube after the inverse transformation are

0.45455 0.95455 0.72727
0.36364 0.86364 0.18182
0.27273 0.77273 0.63636
0.90909 0.90909 0.45455
0.18182 0.68182 0.090909
0.81818 0.81818 0.90909
0.090909 0.59091 0.54545
0.72727 0.72727 0.36364
0 0.5 0
0.63636 0.63636 0.81818
0.54545 0.54545 0.27273
0.45455 0.45455 0.72727
0.36364 0.36364 0.18182
0.27273 0.27273 0.63636
0.90909 0.40909 0.45455
0.18182 0.18182 0.090909
0.81818 0.31818 0.90909
0.090909 0.090909 0.54545
0.72727 0.22727 0.36364
0 0 0
0.63636 0.13636 0.81818
0.54545 0.045455 0.27273

and transformed back to lattice points they are

-6 -7 0
-6 -5 1
-5 -6 0
-5 -5 0
-5 -4 1
-4 -6 -1
-4 -5 0
-4 -4 0
-4 -3 1
-3 -5 -1
-3 -3 0
-2 -4 -1
-2 -2 0
-1 -3 -1
-1 -2 -1
-1 -1 0
0 -3 -2
0 -2 -1
0 -1 -1
0 0 0
1 -2 -2
1 0 -1

Examining the step sizes in this scale, the largest one is 60.6 cents and
the smallest one is 43.4 cents. Thus, this is a periodicity block of the
type I've been looking for since the unison vectors are 7.7, 4.1, and 5.4
cents.

Let's choose [-2 -4 -1] as the 1/1; then it becomes (with cents values)

-4 -3 1 602
-4 -1 2 1144
-3 -2 1 490
-3 -1 1 877
-3 0 2 1032
-2 -2 0 223
-2 -1 1 379
-2 0 1 765
-2 1 2 920
-1 -1 0 112
-1 1 1 653
0 0 0 0
0 2 1 541
1 1 0 1088
1 2 0 275
1 3 1 430
2 1 -1 821
2 2 0 977
2 3 0 163
2 4 1 318
3 2 -1 710
3 4 0 51

Joe, it would be easy enough to graph this in terms of any of the
lattice-representations you've mentioned. I'd use the
Wilson/Chalmers/Erlich/Hahn version, but that wouldn't fit on an
80-character screen. So I'll use a rotated version, defined like this:

1
/|\
/ | \
/ 7 \
/,' `.\
3---------5

OK, here goes:

602
\
\
\
\
( )
/
/
/
/
490
,' \
223 \
\ 1144
\ ,'
877
/
/
/
/
379
,' \
112 \
\ 1032
\ ,'
765
/
/
/
/
( )
,' \
0 \
\ \ 920
\ \ ,'
\ 653
\ ,'
( )
/
/
/
/
1088
,' \
821 \
\ 541
\ .'
275
/
/
/
/
977
,' \
710 \
\ 430
\ ,'
163
/
/
/
/
( )
\
\
\ 318
\ ,'
51

Note the two identical halves, 602 cents apart.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

>Now we're talkin'. That's exactly why I like the
>lattice diagrams: you can see interesting properties
>like this at a glance! This is much easier for me
>to digest than a table of numbers (altho it's important
>to provide the actual datatoo). It also makes what one
>hears in music that uses a particular (JI) tuning that
>much easier to grasp.

It took me a while to draw that diagram. Wouldn't Canright's program be able
to do that automatically? (Last time I tried, I wasn't able to download the
necessary components).

Anyway, Joe, hopefully you now have the machinery to investigate these
periodicity blocks yourself, and plot them in your own lattice framework if
you like. Suffice it to say that I won't be attempting ASCII-diagrams of
58-, 99-, or 171-tone periodicity blocks anytime soon. But I'd be happy to
provide you with the coordinates if you like (or if someone wants to plug
them into Canright's program). For example, the 27-tone periodicity block
formed from the matrix

2 -3 1
3 -1 -3
-4 3 -2

-3 2 -2 329
-2 1 -2 645
-2 1 -1 414
-2 2 -3 62
-1 0 -2 960
-1 0 -1 729
-1 1 -3 378
-1 1 -2 147
0 -1 -1 1045
0 -1 0 814
0 0 -3 694
0 0 -2 462
0 0 -1 231
0 0 0 0
0 1 -4 111
1 -2 0 1129
1 -1 -3 1009
1 -1 -2 778
1 -1 -1 547
1 0 -3 195
2 -2 -2 1094
2 -2 -1 862
2 -1 -3 511
2 -1 -2 280
3 -3 -1 1178
3 -2 -2 596
4 -3 -2 911

The smallest step is 22 cents, while the unison vectors are all between 13
and 14 cents, so this is another "good" one.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Let me try graphing the 27-tone periodicity block that I posted on Thu
9/23/99 5:26 PM.

The matrix of unison vectors is

2 -3 1
3 -1 -3
-4 3 -2

and the coordinates are

I think I'll go back to the usual orientation for this one:

5
/:\
/ : \
/ 7 \
/,' `.\
1---------3

And we're off:

329
\`.
\ 62
\ : 414
\: ,' `.
645-------147
\`. ,'/:\ 0
\ 378 / : \ ,'/
\ : /.729-------231 /
\:/,'**\`.\ ,'/:\/
960-------462 / :/\
`. : ,' : / 814 \
694--\:/,'195`.\
\ 1045-/:\--547
\ `/ : \' : `.
\ / 778-\-----280
\ /.' \`.\@@,'/:\
1009------511 / : \
\ : / 862 \
\:/,' `.\
1094------596
:\
: \
1178\
`.\
911

**111 goes here, but there was not enough room to write it.
@@1129 goes here, but there was not enough room to write it.

This one is amazingly rich in 7-limit triads and tetrads, so much so that
it's straining my meager ASCII symbology to show all the connections.

🔗Jonathan M. Szanto <jszanto@xxxx.xxxx>

Paul,

{you wrote...}
>This one is amazingly rich in 7-limit triads and tetrads, so much so that
>it's straining my meager ASCII symbology to show all the connections.

Then how about this:

1. Either compose (and perform) or improvise a piece that demonstrates the
richness.
2. Record the results.
3. Post it on the web somewhere.

It would be worth so much, maybe more than a _paper_.

Musically, all yours,
Jon
`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`
Jonathan M. Szanto : Corporeal Meadows - Harry Partch, online.
jszanto@adnc.com : http://www.corporeal.com/
`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`