For Joseph: Seven 19-tone 7-limit Fokker periodicity blocks translated to 72-tET

For Joseph: Seven 19-tone 7-limit Fokker periodicity blocks
translated to 72-tET

Before I begin, I note that I originally posted eight of these.
Unfortunately, the lattice diagram for the seventh one was not
correct, and the correct version shows two "unconnected" notes.
Therefore, I will leave that one out.

OK, now proceeding to translate the remaining 7 to 72-tET:

NUMBER 1

Unison vectors:
2 -3 1
4 -1 0
-3 3 1

Lattice:
45--------15
\ / \
\ / \
\ / \
\ / \
64--------34---------4
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
11--------53--------23
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
30---------0--------42
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
49--------19--------61
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
68--------38---------8
\ / \
\ / \
\ / \
\ / \
57--------27

Scale: 0,4,8,11,15,19,23,27,30,34,38,42,45,49,53,57,61,64,68(,72)
Steps: 4,4,3,4,4,4,4,3,4,4,4,3,4,4,4,4,3,4,4
(maximally even 19 out of 72)

NUMBER 2

Unison vectors:
2 2 -1
4 -1 0
-3 3 1

Lattice:
34---------4--------46--------16
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
11--------53--------23--------65
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
30---------0--------42
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
7--------49--------19--------61
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
56--------26--------68--------38

Scale: 0,4,7,11,16,19,23,26,30,34,38,42,46,49,53,56,61,65,68(,72)
Steps: 4,3,4,5,3,4,3,4,4,4,4,4,3,4,3,5,4,3,4

NUMBER 3: (the one Joseph liked in JI)

Unison vectors:
2 -3 1
-5 1 2
-3 3 1

Lattice:
34
/|\
/ | \
/ 69 \
/,'/ \`.\
11--/---\-53--------23
|\/ \/|\ / \`.
|/\ /\| \ / \ 7
46--------16 \ / \/|\
\`.\ /,'/ `.\ / /\| \
\ 30--/------0--------42 \
\ |\/ / \`. /,'/ \`.\
\|/\ / \ 56--/---\-26
65 \ / \ |\/ \/|
`.\ / \|/\ /\|
49--------19--------61
\`.\ /,'/
\ 3 /
\ | /
\|/
38

Scale: 0,3,7,11,16,19,23,26,30,34,38,42,46,49,53,56,61,65,69(,72)
Steps: 3,4,4,5,3,4,3,4,4,4,4,4,3,4,3,5,4,4,3

NUMBER 4:

Unison vectors:
2 2 -1
2 -3 1
-3 3 1

Lattice:
34---------4
/ \ / \
/ \ / \
/ \ / \
/ \ / \
11--------53--------23
|\ /|\ / \`.
| \ / | \ / \ 7
46--------16 \ / \/|\
\`.\ /,'/ `.\ / /\| \
\ 30--/------0--------42 \
\ |\/ / \`. /,'/ \`.\
\|/\ / \ 56--/---\-26
65 \ / \ | / \ |
`.\ / \|/ \|
49--------19--------61
\ / \ /
\ / \ /
\ / \ /
\ / \ /
68--------38

IN 72-tET, IT'S IDENTICAL TO NUMBER 2!

NUMBER 5:

Unison vectors:
2 2 -1
2 -3 1
-5 1 2

Lattice:
34
/ \
/ \
/ \
/ \
11--------53--------23
|\ /|\ / \`.
| \ / | \ / \ 7
4--------46--------16 \ / \/|\
\`.\ /,'/ `.\ / /\| \
\ 30--/------0--------42 \
\ |\/ / \`. /,'/ \`.\
\|/\ / \ 56--/---\-26--------68
65 \ / \ | / \ |
`.\ / \|/ \|
49--------19--------61
\ /
\ /
\ /
\ /
38

IN 72-tET, IT'S IDENTICAL TO NUMBER 2 AND NUMBER 4!

NUMBER 6:

Unison vectors:
2 2 -1
-3 3 1
-1 0 2

Lattice:
62
/
/
/
/
69--------39
/ \ `. ,'/ `.
/ \ 53--/-----23
/ \/|\/ / \`.
/ /\|/\ / \ 7
46--------16 \ / \/|\
\`. /,'/ `.\ / /\| \
\ 30--/------0--------42 \
\ |\/ / \`. /,'/ `.\
\|/\ / \ 56--/-----26
65 \ / \/|\/ /
`.\ / /\|/\ /
49--------19 \ /
`. /,' `.\ /
33---------3
/
/
/
/
10

Scale: 0,3,7,10,16,19,23,26,30,33,39,42,46,49,53,56,62,65,69(,72)
Steps: 3,4,3,6,3,4,3,4,3,6,3,4,3,4,3,6,3,4,3
THIS IS IDENTICAL TO DAVE KEENAN'S 11-LIMIT PROPOSAL!

NUMBER 8:

Unison vectors:
2 2 -1
-5 1 2
-3 -2 3

Lattice:
69
/ \`.
/ \ 53--------23
62 / \/|\ / \`.
/ `. / /\| \ / \ 7
/ 46--------16 \ / \/|\
/ \`. /,'/ `.\ / /\| \ 33
/ \ 30--/------0--------42 \ /
39 \ |\/ / \`. /,'/ `.\ /
\|/\ / \ 56--/-----26 /
65 \ / \ |\/ / `. /
`.\ / \|/\ / 10
49--------19 \ /
`.\ /
3

IN 72-tET, THIS IS IDENTICAL TO NUMBER 6, SO THIS IS ALSO IDENTICAL
TO DAVE KEENAN'S 11-LIMIT PROPOSAL!

So, after all this, I've only given you three new 19-tone scales to
try in 72-tET . . .

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_21848.html#21848

> For Joseph: Seven 19-tone 7-limit Fokker periodicity blocks
> translated to 72-tET

THIS IS AMAZING that you were able to do this so quickly!

>
> So, after all this, I've only given you three new 19-tone scales to
> try in 72-tET . . .

I guess that means that there really aren't that many that "work out"
so great...

Well, three is still quite a bit, if I decide I want to work a lot in
them...

I guess there still is a need to put the 22-note just scales into 72-
tET at your convenience, yes??

________ _____ _____ ___
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:>
>
> I guess that means that there really aren't that many that "work
out"
> so great...
>
Well, remember, Joseph, that Fokker's two conventions of

1) using a parallelopiped, rather than hexagonal prism or rhombic
dodecahedron, to delimit the periodicity block

2) centering the block exactly on a vertex of the lattice (to which
Fokker assigns the notation "D")

are arbitrary. So there may be a lot more possibilities . . .
>
> I guess there still is a need to put the 22-note just scales into
72-
> tET at your convenience, yes??
>
I'll do that eventually (the 19-note ones took about an hour,
including breaks for posting) . . . but for 22-note ones, having one
note at the exact center of the lattice is _really_ arbitrary (since
22 is an even number).

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_21848.html#21852

> --- In tuning@y..., jpehrson@r... wrote:>
> >
> > I guess that means that there really aren't that many that "work
> out" so great...
> >
> Well, remember, Joseph, that Fokker's two conventions of
>
> 1) using a parallelopiped, rather than hexagonal prism or rhombic
> dodecahedron, to delimit the periodicity block
>

On this one, Paul, I'm just going to have to take your word for it...
that is until I can seem some nice Erv Wilson 3-D graphics!

> 2) centering the block exactly on a vertex of the lattice (to which
> Fokker assigns the notation "D")
>

The notation "D." So Fokker starts on "D" and Partch starts on "G."

What ever happened to good old "C." (??)

> are arbitrary.

Sure sounds like it... And, to think that I thought YOU invented the
Periodicity Block.... I believe the people over in Russia still think
that!

So there may be a lot more possibilities . . .

Well, I've been doing quite a bit of theory, so it will be
interesting to see if my view have changed after some concrete
COMPOSING with these scales...

> >
> > I guess there still is a need to put the 22-note just scales into
> 72-tET at your convenience, yes??
> >

> I'll do that eventually (the 19-note ones took about an hour,
> including breaks for posting)

That's still pretty amazing... and while "multitasking" yet!

. . . but for 22-note ones, having one
> note at the exact center of the lattice is _really_ arbitrary
(since 22 is an even number).

Well, I guess you'll have to settle on a "quarter tone..." :)

I have PLENTY to work with now... now the job is to make it all SOUND
like something!!!

Thanks again!!

________ _____ ______ _____
Joseph Pehrson

>
> The notation "D." So Fokker starts on "D" and Partch starts on "G."

D makes sense because if the scale is mirror-symmetrical, the
notation will be too . . . I mean you'll have as many flats as
sharps, as many "ups" as "downs", etc.

>
> Sure sounds like it... And, to think that I thought YOU invented
the
> Periodicity Block.... I believe the people over in Russia still
think
> that!

OH NO! How did that ever happen? I wouldn't even be interested in
periodicity blocks had my friend Joe Monzo not asked me about finity.

Yikes!

--- In tuning@y..., paul@s... wrote:
> >
/tuning/topicId_21848.html#21861

> > The notation "D." So Fokker starts on "D" and Partch starts on
"G."
>
> D makes sense because if the scale is mirror-symmetrical, the
> notation will be too . . . I mean you'll have as many flats as
> sharps, as many "ups" as "downs", etc.
>

Got it... so it's a little like some of the other "mirrors" we were
discussing in the context of 12 out of 22-tET scales...

______ _____ _____ _
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
>
> I have PLENTY to work with now...

Yup . . . you have FOUR different 19-tone subsets of 72-tET.

1) My first one
2) My second one
3) My third one
4) Dave Keenan's 11-limit one, which is identical to some I derived as 7-limit periodicity blocks.

> now the job is to make it all SOUND
> like something!!!
>
One thing that might help would be for me to create 72-tET lattices of the four scales above . . .
these will show the new consonant intervals not present in the JI versions . . . of course these will
still only be 7-limit lattices . . . when I have some free time . . .

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_21848.html#21872

> One thing that might help would be for me to create 72-tET lattices
of the four scales above . . .
> these will show the new consonant intervals not present in the JI
versions . . . of course these will
> still only be 7-limit lattices . . . when I have some free
time . . .

Thanks, Paul! This would be very interesting...

________ _____ ______ _____
Joseph Pehrson

--- In tuning@y..., paul@s... wrote:

> D makes sense because if the scale is mirror-symmetrical, the
> notation will be too . . . I mean you'll have as many flats as
> sharps, as many "ups" as "downs", etc.

I found this comment very interesting.

The "basic" modal pattern, from the time of the Sumerians all
the way down to medieval Europe (a span of some 5000 years)
was the one represented by our notes A B C D E F G.

Last summer, when I was knee-deep in research on the Sumerians,
I realized that the most likely reason for this was because of
the symmetry in this arrangement: T S T T S T (where T = tone,
S = semitone).

The Sumerians had a love of symmetry in their art which is
displayed in nearly every surviving artifact, from the design
of bas-relief pictures to the design and stringing of their
lyres. It makes sense that their music theory would follow
suit.

> > Sure sounds like it... And, to think that I thought YOU
> > invented the Periodicity Block.... I believe the people over
> > in Russia still think that!
>
> OH NO! How did that ever happen? I wouldn't even be interested
> in periodicity blocks had my friend Joe Monzo not asked me
> about finity.
>
> Yikes!

Yikes is right! Fokker certainly deserves his due credit.
(And thanks for giving me mine, Paul.)

So why do the Russians think Paul invented this, Joe?
Is it your fault? Better start sending out some e- or
snail-mails...

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

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_21848.html#21899

>
> Yikes is right! Fokker certainly deserves his due credit.
> (And thanks for giving me mine, Paul.)
>
> So why do the Russians think Paul invented this, Joe?
> Is it your fault? Better start sending out some e- or
> snail-mails...
>

No harm done... I explained to the students in Bryansk that I was
showing Paul Erlich's "Periodicity Block" method and I showed the
scales that he generated. I didn't mention Fokker, it is true, but
the entire subject of microtonality was ENTIRELY new to them!

This was hardly a scholarly forum..., but I will be more careful the
next time!

________ ______ _______ ________
Joseph Pehrson