For Joseph: Eight 19-tone 7-limit Fokker periodicity blocks

Well Joseph, it may be too late, since a it sound like adaptive JI based on
19-tET scores might be what you're really going for (I would). But in case
anyone cares, here's a little "zoo":

Unison vectors:
2 -3 1
4 -1 0
-3 3 1
Scale:
0 1 1
70.672 25 24
133.24 27 25
182.4 10 9
253.08 125 108
315.64 6 5
386.31 5 4
448.88 162 125
498.04 4 3
568.72 25 18
631.28 36 25
701.96 3 2
751.12 125 81
813.69 8 5
884.36 5 3
946.92 216 125
1017.6 9 5
1066.8 50 27
1129.3 48 25
Lattice:
125/81----125/108
\ / \
\ / \
\ / \
\ / \
50/27-----25/18-----25/24
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
10/9-------5/3-------5/4
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
4/3-------1/1-------3/2
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
8/5-------6/5-------9/5
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
48/25-----36/25-----27/25
\ / \
\ / \
\ / \
\ / \
216/125---162/125

Unison vectors:
2 2 -1
4 -1 0
-3 3 1
Scale:
cents numerator denominator
0 1 1
70.672 25 24
111.73 16 15
182.4 10 9
274.58 75 64
315.64 6 5
386.31 5 4
427.37 32 25
498.04 4 3
568.72 25 18
631.28 36 25
701.96 3 2
772.63 25 16
813.69 8 5
884.36 5 3
925.42 128 75
1017.6 9 5
1088.3 15 8
1129.3 48 25
Lattice:
25/18-----25/24-----25/16-----75/64
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
10/9-------5/3-------5/4------15/8
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
4/3-------1/1-------3/2
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
16/15------8/5-------6/5-------9/5
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
128/75-----32/25-----48/25-----36/25

Unison vectors:
2 -3 1
-5 1 2
-3 3 1
Scale:
cents numerator denominator
0 1 1
48.77 36 35
119.44 15 14
182.4 10 9
266.87 7 6
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
568.72 25 18
631.28 36 25
701.96 3 2
764.92 14 9
813.69 8 5
884.36 5 3
933.13 12 7
1017.6 9 5
1080.6 28 15
1151.2 35 18
Lattice:
25/18
/|\
/ | \
/35/18\
/,'/ \`.\
10/9-/---\-5/3-------5/4
|\/ \/|\ / \`.
|/\ /\| \ / \15/14
14/9-------7/6 \ / \/|\
\`.\ /,'/ `.\ / /\| \
\ 4/3-/-----1/1-------3/2 \
\ |\/ / \`. /,'/ \`.\
\|/\ / \12/7-/---\-9/7
28/15\ / \ |\/ \/|
`.\ / \|/\ /\|
8/5-------6/5-------9/5
\`.\ /,'/
\36/35/
\ | /
\|/
36/25

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

Scale:
cents numerator denominator
0 1 1
70.672 25 24
119.44 15 14
182.4 10 9
266.87 7 6
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
568.72 25 18
631.28 36 25
701.96 3 2
764.92 14 9
813.69 8 5
884.36 5 3
933.13 12 7
1017.6 9 5
1080.6 28 15
1129.3 48 25
Lattice:
25/18-----25/24
/ \ / \
/ \ / \
/ \ / \
/ \ / \
10/9-------5/3-------5/4
|\ /|\ / \`.
| \ / | \ / \15/14
14/9-------7/6 \ / \/|\
\`.\ /,'/ `.\ / /\| \
\ 4/3-/-----1/1-------3/2 \
\ |\/ / \`. /,'/ \`.\
\|/\ / \12/7-/---\-9/7
28/15\ / \ | / \ |
`.\ / \|/ \|
8/5-------6/5-------9/5
\ / \ /
\ / \ /
\ / \ /
\ / \ /
48/25-----36/25

Unison vectors:
2 2 -1
2 -3 1
-5 1 2
Scale:
cents numerator denominator
0 1 1
62.961 28 27
119.44 15 14
182.4 10 9
266.87 7 6
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
568.72 25 18
631.28 36 25
701.96 3 2
764.92 14 9
813.69 8 5
884.36 5 3
933.13 12 7
1017.6 9 5
1080.6 28 15
1137 27 14
Lattice:
25/18
/ \
/ \
/ \
/ \
10/9-------5/3-------5/4
|\ /|\ / \`.
| \ / | \ / \15/14
28/27-----14/9-------7/6 \ / \/|\
\`.\ /,'/ `.\ / /\| \
\ 4/3-/-----1/1-------3/2 \
\ |\/ / \`. /,'/ \`.\
\|/\ / \12/7-/---\-9/7------27/14
28/15\ / \ | / \ |
`.\ / \|/ \|
8/5-------6/5-------9/5
\ /
\ /
\ /
\ /
36/25

Unison vectors:
2 2 -1
-3 3 1
-1 0 2
Scale:
cents numerator denominator
0 1 1
48.77 36 35
119.44 15 14
160.5 192 175
266.87 7 6
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
546.82 48 35
653.18 35 24
701.96 3 2
764.92 14 9
813.69 8 5
884.36 5 3
933.13 12 7
1039.5 175 96
1080.6 28 15
1151.2 35 18
Lattice:
175/96
/
/
/
/
35/18-----35/24
/ \ `. ,'/ `.
/ \ 5/3-/-----5/4
/ \/|\/ / \`.
/ /\|/\ / \15/14
14/9-------7/6 \ / \/|\
\`. /,'/ `.\ / /\| \
\ 4/3-/-----1/1-------3/2 \
\ |\/ / \`. /,'/ `.\
\|/\ / \12/7-/-----9/7
28/15\ / \/|\/ /
`.\ / /\|/\ /
8/5-------6/5 \ /
`. /,' `.\ /
48/35-----36/35
/
/
/
/
192/175

Unison vectors:
2 2 -1
-5 1 2
1 2 1
Scale:
cents numerator denominator
0 1 1
48.77 36 35
111.73 16 15
160.5 192 175
266.87 7 6
323.35 135 112
386.31 5 4
435.08 9 7
498.04 4 3
546.82 48 35
653.18 35 24
701.96 3 2
764.92 14 9
813.69 8 5
876.65 224 135
933.13 12 7
1039.5 175 96
1088.3 15 8
1151.2 35 18
Lattice:
175/96
/
/
/
/
35/18-----35/24
/ \ / `.
/ \ / 5/4------15/8
/ \ / / \`. ,'/
/ \ / / \15/14/
14/9-------7/6 / \/|\/
\`. ,'/ `. / /\|/\
\ 4/3-/-----1/1-------3/2 \
\/|\/ / `. /,' `.\
/\|/\ / 12/7-------9/7
/28/15\ / / \ /
/,' `.\ / / \ /
16/15------8/5 / \ /
`. / \ /
48/35-----36/35
/
/
/
/
192/175

Unison vectors:
2 2 -1
-5 1 2
-3 -2 3
Scale:
cents numerator denominator
0 1 1
48.77 36 35
119.44 15 14
168.21 54 49
266.87 7 6
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
554.53 135 98
645.47 196 135
701.96 3 2
764.92 14 9
813.69 8 5
884.36 5 3
933.13 12 7
1031.8 49 27
1080.6 28 15
1151.2 35 18
Lattice:
35/18
/ \ `.
/ \ 5/3-------5/4
49/27 / \/|\ / \`.
/ `. / /\| \ / \15/14
/ 14/9-------7/6 \ / \/|\
/ \`. /,'/ `.\ / /\| \ 135/98
/ \ 4/3-/-----1/1-------3/2 \ /
196/135 \ |\/ / \`. /,'/ `.\ /
\|/\ / \12/7-/-----9/7 /
28/15\ / \/|\/ / `. /
`.\ / /\|/\ / 54/49
8/5-------6/5 \ /
`.\ /
36/35

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> Well Joseph, it may be too late, since a it sound like adaptive JI
based on
> 19-tET scores might be what you're really going for (I would). But
in case
> anyone cares, here's a little "zoo":

Waa! Paul, that's awesome. How long did that take you?

Do you have an algorithm to generate sets of unison vectors having a
given periodicity?

Regards,
-- Dave Keenan

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_19050.html#19050

> Well Joseph, it may be too late, since a it sound like adaptive JI
based on 19-tET scores might be what you're really going for (I
would). But in case anyone cares, here's a little "zoo":

THESE ARE AMAZING!

TERRIFIC GRAPHICS TOO!

Certainly, whether I end up using them or not in a piece, I will be
"tuning up" SCALA to listen to all these scales, probably with Graham
Breed's Midi Relay.

Thanks so VERY much again...

INCREDIBLY impressive!!!

________ _____ _____ ___
Joseph Pehrson

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_19050.html#19050

CALLING ROBERT WALKER!!!

Robert!

Did you see the post above??!

Is there some way we can make some kind of beautiful 2-D or, even
better, 3-D graphics of these wonderful lattices?

They're beautiful as it is, but can you imagine if they had colorful
"enhancement??" Even the way Pierre Lamothe does them would make
them incredible!!!

_______ _____ _____ _
Joseph Pehrson

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_19050.html#19050

> Well Joseph, it may be too late, since a it sound like adaptive JI
based on 19-tET scores might be what you're really going for (I
would). But in case anyone cares, here's a little "zoo":
>
> Unison vectors:
> 2 -3 1
> 4 -1 0
> -3 3 1

Hi Paul...

Actually, unless these scales sound really "baaaaad," most probably I
will be working with them, wolves and all. (Of course in an AUDITORY
way, so only when I NEED a "wolf bite" will I use one...).

Also, I think it's just great to use original scale material for a
composition. It seems to me like a "new frontier" and I'm not sure,
given these scales, that I will want to "retreat" to 19-tET...[I
know,
it DOES have some advantages that we have been over] Besides, I
wouldn't want to "insult" some of my JI friends, and they
sometimes seem a little "sensitive" about such matters.

HOWEVER, I have a couple of questions for you. Could you please
briefly explain in LAYMAN'S terms what you did???

I see some "unison vectors" and then the resulting ratios and cents.
How do you get that again? I think I understand a bit the PRINCIPLE
of the periodicity block... when close pitches cycle around and
around again from some generating intervals, you just choose a
"limitation" somewhat like focusing the lens on a camera.... I think.

What are the generating intervals??

Also, I was wondering how you can apparently so quickly make these
lattices in ascii? Do you do it right in your e-mail program?? It
would take me YEARS, and that's besides figuring it out.

I have to repeat again, and I WILL NOT USE CAPS this time that this
post is really one of the most incredible things I have ever seen on
the Tuning List, and there have been some pretty amazing things over
the last year.

I think I'm going to use these scales... Besides, Kraig Grady is
going to put some kind of "bad karma" on me if I use 19-tET. I'll
probably wake up to see some "big puppets" in my room...

______ ___ ____ ____ _
Joseph Pehrson

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_19050.html#19050
>
>
> Unison vectors:
> 2 -3 1
> -5 1 2
> -3 3 1
> Scale:
> cents numerator denominator
> 0 1 1
> 48.77 36 35
> 119.44 15 14
> 182.4 10 9
> 266.87 7 6
> 315.64 6 5
> 386.31 5 4
> 435.08 9 7
> 498.04 4 3
> 568.72 25 18
> 631.28 36 25
> 701.96 3 2
> 764.92 14 9
> 813.69 8 5
> 884.36 5 3
> 933.13 12 7
> 1017.6 9 5
> 1080.6 28 15
> 1151.2 35 18
> Lattice:
> 25/18
> /|\
> / | \
> /35/18\
> /,'/ \`.\
> 10/9-/---\-5/3-------5/4
> |\/ \/|\ / \`.
> |/\ /\| \ / \15/14
> 14/9-------7/6 \ / \/|\
> \`.\ /,'/ `.\ / /\| \
> \ 4/3-/-----1/1-------3/2 \
> \ |\/ / \`. /,'/ \`.\
> \|/\ / \12/7-/---\-9/7
> 28/15\ / \ |\/ \/|
> `.\ / \|/\ /\|
> 8/5-------6/5-------9/5
> \`.\ /,'/
> \36/35/
> \ | /
> \|/
> 36/25
>

Purely from the "symmetrical" point of view, this seems like about
the most beautiful of the 19-tone 7-limit JI "animals."

Are those hexanies I see there?? Shore looks like them...

Paul, you may have, inadventently, given me composing materials for
the next few years... since I like the scale-step size of AROUND
19-tET and also the idea of composing in Just...

(31-tET is a bit too teeny-tiny for me... at least in conjunction
with traditional keyboards...)

I haven't HEARD any of these yet, so more on that later...

THANKS!

________ ____ ____ _____
Joseph Pehrson

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_19050.html#19050

Paul, on the following latticing of one of your Just scales:

>>
> Unison vectors:
> 2 2 -1
> -5 1 2
> 1 2 1
> Scale:
> cents numerator denominator
> 0 1 1
> 48.77 36 35
> 111.73 16 15
> 160.5 192 175
> 266.87 7 6
> 323.35 135 112
> 386.31 5 4
> 435.08 9 7
> 498.04 4 3
> 546.82 48 35
> 653.18 35 24
> 701.96 3 2
> 764.92 14 9
> 813.69 8 5
> 876.65 224 135
> 933.13 12 7
> 1039.5 175 96
> 1088.3 15 8
> 1151.2 35 18
> Lattice:
> 175/96
> /
> /
> /
> /
> 35/18-----35/24
> / \ / `.
> / \ / 5/4------15/8
> / \ / / \`. ,'/
> / \ / / \15/14/
> 14/9-------7/6 / \/|\/
> \`. ,'/ `. / /\|/\
> \ 4/3-/-----1/1-------3/2 \
> \/|\/ / `. /,' `.\
> /\|/\ / 12/7-------9/7
> /28/15\ / / \ /
> /,' `.\ / / \ /
> 16/15------8/5 / \ /
> `. / \ /
> 48/35-----36/35
> /
> /
> /
> /
> 192/175
>

I'm not seeing the 135/112 or the 224/135.

Those were the only two ratios in the entire production of 8 scales
that were not shown...

Was there a reason for this, or were they just inadvertently
omitted...?

I'm particularly interested, of course, in the "big" ratios and how
they fit in with the "connectedness" of the total picture...

Thanks!

_______ _____ _____ _
Joseph Pehrson

Oops -- Thanks for spotting that, Joseph . . .

What you didn't notice was that I had 28/15 and 15/14 in the lattice, even
though they aren't in the scale.

It turns out, coincidentally, that the ratios I left out, 224/135 and
135/112, are not connected to any other notes in the lattice.

Here is the corrected lattice:

Unison vectors:
2 2 -1
-5 1 2
1 2 1
Scale:
cents numerator denominator
0 1 1
48.77 36 35
111.73 16 15
160.5 192 175
266.87 7 6
323.35 135 112
386.31 5 4
435.08 9 7
498.04 4 3
546.82 48 35
653.18 35 24
701.96 3 2
764.92 14 9
813.69 8 5
876.65 224 135
933.13 12 7
1039.5 175 96
1088.3 15 8
1151.2 35 18
Lattice:
175/96
/
/
/
/
35/18-----35/24
/ \ / `.
/ \ / 5/4------15/8
/ \ / / \ /
/ \ / / \ / 112/135
14/9-------7/6 / \ /
`. ,' `. / \ /
4/3-------1/1-------3/2
/ \ / `. ,' `.
/ \ / 12/7-------9/7
224/135 / \ / / \ /
/ \ / / \ /
16/15------8/5 / \ /
`. / \ /
48/35-----36/35
/
/
/
/
192/175

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_19050.html#19179

> Oops -- Thanks for spotting that, Joseph . . .
>
> What you didn't notice was that I had 28/15 and 15/14 in the
lattice, even though they aren't in the scale.
>

You are so right, Paul! I hadn't noticed that yet!

> It turns out, coincidentally, that the ratios I left out, 224/135
and 135/112, are not connected to any other notes in the lattice.
>

Hmmm. Well that's not so "heartening." Even my "Sparky the Robot"
scale had connections...

It might make me "think twice" before using this particular one.

Of course, with all these scales, the "virtues" of having Just
Intonation intervals are offset by the different sized scale steps...

Something to think about. They're a little "funky," but maybe I like
that...

________ ______ _____ _
Joseph Pehrson