For Joseph: Eight 22-tone 7-limit Fokker periodicity blocks

Joseph wrote,

>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.

Actually I'm embarassed about the fact that these all use the parallelopiped
method, while, as I've indicated, the hexagonal prism and the rhombic
dodecadron can sometimes be preferable as the boundaries for a 3-d
periodicity block . . .

Anyway, on to 22-tone (again, restricted to parallelopipeds):

Unison vectors:
-5 1 2
-4 -2 0
-2 0 -1
Scale:
cents numerator denominator
0 1 1
62.961 28 27
111.73 16 15
155.14 35 32
203.91 9 8
266.87 7 6
342.91 128 105
386.31 5 4
435.08 9 7
498.04 4 3
546.82 48 35
590.22 45 32
653.18 35 24
701.96 3 2
764.92 14 9
813.69 8 5
857.09 105 64
933.13 12 7
996.09 16 9
1044.9 64 35
1088.3 15 8
1137 27 14
Lattice:

35/24-----35/32----105/64
/ `. ,' `. ,' `.
/ 5/4------15/8------45/32
/ / \ / \ /
/ / \ / \ /
28/27-------14/9-------7/6 / \ / \ /
`. ,' `. ,' `. / \ / \ /
16/9-------4/3-------1/1-------3/2-------9/8
\ / \ / `. ,' `. ,' `.
\ / \ / 12/7-------9/7-------27/14
\ / \ / /
\ / \ / /
16/15------8/5 /
,' `. ,' `. /
128/105----64/35-----48/35

Unison vectors:
3 -1 -3
-2 0 -1
0 2 -2
Scale:
cents numerator denominator
0 1 1
35.697 49 48
84.467 21 20
182.4 10 9
231.17 8 7
266.87 7 6
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
546.82 48 35
582.51 7 5
653.18 35 24
701.96 3 2
764.92 14 9
813.69 8 5
884.36 5 3
933.13 12 7
968.83 7 4
1017.6 9 5
1115.5 40 21
1164.3 96 49
Lattice:

35/24
,'/|\`.
10/9-------5/3-/-|-\-5/4
|\`. ,'/|\/49/48\/|\
| \40/21/ |/,' `.\| \
14/9-------7/6-------7/4 \
`.\|/,' \`.\ /,'/ \`.\
4/3-----\-1/1-/---\-3/2
\`.\ ,\/|\/. ,\/|\`.
\ 8/7-/\|/\12/7--\|---9/7
\ |\/.7/5------21/20\ |
\|/,96/49.\|/,' `.\|
8/5-------6/5-------9/5
`.\|/ ,'
48/35

Unison vectors:
3 -1 -3
-3 3 1
-2 0 -1
Scale:
cents numerator denominator
0 1 1
48.77 36 35
84.467 21 20
182.4 10 9
231.17 8 7
266.87 7 6
315.64 6 5
386.31 5 4
422.01 245 192
498.04 4 3
546.82 48 35
582.51 7 5
653.18 35 24
701.96 3 2
777.99 384 245
813.69 8 5
884.36 5 3
933.13 12 7
968.83 7 4
1017.6 9 5
1115.5 40 21
1151.2 35 18
Lattice:
245/192
,'
35/18-----35/24
,' \`. ,'/ \`.
10/9-----\-5/3-/---\-5/4
\`. ,\/|\/ \/|\
\40/21/\|/\ /\| \
\ | / 7/6-------7/4 \
\|/,' \`.\ /,'/ \`.\
4/3-----\-1/1-/---\-3/2
\`.\ ,\/|\/. ,\/|\
\ 8/7-/\|/\12/7 /\| \
\ |\/.7/5------21/20\
\|/,' `.\|/,' `.\
8/5-------6/5-------9/5
`.\ /,' `.\ ,'
48/35-----36/35
,'
384/245

Unison vectors:
3 -1 -3
-5 1 2
0 2 -2
Scale:
cents numerator denominator
0 1 1
48.77 36 35
84.467 21 20
182.4 10 9
231.17 8 7
266.87 7 6
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
533.74 49 36
582.51 7 5
666.26 72 49
701.96 3 2
764.92 14 9
813.69 8 5
884.36 5 3
933.13 12 7
968.83 7 4
1017.6 9 5
1115.5 40 21
1151.2 35 18
Lattice:
35/18
,'/|\`.
10/9-/-|-\-5/3-------5/4
|\/49/36\/|\ /|\
|,'40/21`.| \ / | \
14/9-------7/6-------7/4 \
`.\|/,' \`.\ /,'/ \`.\
4/3-----\-1/1-/---\-3/2
\`.\ ,\/|\/. ,\/|\`.
\ 8/7-/\|/\12/7-/\|-\-9/7
\ | /.7/5------21/20\/|
\|/,' `.\|/,' * `.\|
8/5-------6/5-------9/5
`.\|/ ,'
36/35
* 72/49 goes here

Unison vectors:
3 -1 -3
-5 1 2
-3 3 1
Scale:
cents numerator denominator
0 1 1
48.77 36 35
97.541 1296 1225
182.4 10 9
218.1 245 216
266.87 7 6
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
546.82 48 35
582.51 7 5
653.18 35 24
701.96 3 2
764.92 14 9
813.69 8 5
884.36 5 3
933.13 12 7
981.9 432 245
1017.6 9 5
1102.5 1225 648
1151.2 35 18
Lattice:
1225/648
\
\
\
\
245/216
`.
35/18-----35/24
,' \`. ,'/ `.
10/9-----\-5/3-/-----5/4
\ \/|\/ / \
\ /\|/\ / \
\ / 7/6 \ / \
\ /,' \`.\ / \
4/3-----\-1/1-------3/2
\ \/|\`. ,'/ \
\ /\| \12/7 / \
\ / 7/5 \/|\/ \
\ /,' `.\|/\ \
8/5-------6/5-------9/5
`. /,' `.\ ,'
48/35-----36/35
`.
432/245
\
\
\
\
1296/1225

Unison vectors:
2 2 -1
3 -1 -3
-2 0 -1
Scale:
cents numerator denominator
0 1 1
35.697 49 48
111.73 16 15
155.14 35 32
231.17 8 7
266.87 7 6
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
546.82 48 35
582.51 7 5
653.18 35 24
701.96 3 2
764.92 14 9
813.69 8 5
884.36 5 3
933.13 12 7
968.83 7 4
1044.9 64 35
1088.3 15 8
1164.3 96 49
Lattice:
35/24-----35/32
,'/|\`. ,'/ `.
5/3-/-|-\-5/4-/-----15/8
/|\/49/48\/|\/ /
/ |/,' `.\|/\ /
14/9-------7/6-------7/4 \ /
`. /,' \`.\ /,'/ `.\ /
4/3-----\-1/1-/-----3/2
/ \`. ,\/|\/. ,'/ `.
/ \ 8/7-/\|/\12/7-------9/7
/ \/|\/.7/5 \ | /
/ /\|/,96/49.\|/
16/15------8/5-------6/5
`. / `.\|/ ,'
64/35-----48/35

Unison vectors:
2 2 -1
3 -1 -3
-3 3 1
Scale:
cents numerator denominator
0 1 1
48.77 36 35
119.44 15 14
160.5 192 175
218.1 245 216
266.87 7 6
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
546.82 48 35
582.51 7 5
653.18 35 24
701.96 3 2
764.92 14 9
813.69 8 5
884.36 5 3
933.13 12 7
981.9 432 245
1039.5 175 96
1080.6 28 15
1151.2 35 18
Lattice:
175/96
/
/
245/216 /
`. /
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------7/5 \/|\/ /
`.\ /,' `.\|/\ /
8/5-------6/5 \ /
`. /,' `.\ /
48/35-----36/35
/ `.
/ 432/245
/
/
192/175

Unison vectors:
2 2 -1
3 -1 -3
-5 1 2
Scale:
cents numerator denominator
0 1 1
48.77 36 35
111.73 16 15
168.21 54 49
218.1 245 216
266.87 7 6
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
546.82 48 35
582.51 7 5
653.18 35 24
701.96 3 2
764.92 14 9
813.69 8 5
884.36 5 3
933.13 12 7
981.9 432 245
1031.8 49 27
1088.3 15 8
1151.2 35 18
Lattice:
245/216
/ `.
/ 35/18-----35/24
/ / \`. ,'/ `.
/ / \ 5/3-/-----5/4------15/8
49/27 / \/|\/ / \ /
`. / /\|/\ / \ /
14/9-------7/6 \ / \ /
`. /,' \`.\ / \ /
4/3-----\-1/1-------3/2
/ \ \/|\`. ,'/ `.
/ \ /\| \12/7-/-----9/7
/ \ / 7/5 \/|\/ / `.
/ \ /,' `.\|/\ / 54/49
16/15------8/5-------6/5 \ / /
`. /,' `.\ / /
48/35-----36/35 /
`. /
432/245

Cheers!

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

/tuning/topicId_19127.html#19127

>
> Actually I'm embarassed about the fact that these all use the
parallelopiped method, while, as I've indicated, the hexagonal prism
and the rhombic dodecadron can sometimes be preferable as the
boundaries for a 3-d periodicity block . . .
>

Yeah... me too... Wazzat??

> Anyway, on to 22-tone (again, restricted to parallelopipeds):
>

Well, these are pretty cool... Lots of "hexanies" in them and they
seem, for whatever reason, to be a bit more "compressed." (??) Maybe
it's just the effect by having three more notes... Dunno.

Anyway, perhaps I'll move on to 22 as well...

THANKS!

________ _____ ______ ____
Joseph Pehrson

Hello Paul,

These are interesting Paul especially the 22 tone scales, I'm still playing
around with scales for my guitar and and I have found that 22 tones is
probably the most realistic amount of tones per octave. Would you be able
to post some 22 tone 11 and 13 limit Fokker periodicity blocks ?

< Actually I'm embarassed about the fact that these all use the
parallelopiped
method, while, as I've indicated, the hexagonal prism and the rhombic
dodecadron can sometimes be preferable as the boundaries for a 3-d
periodicity block . . .

Anyway, on to 22-tone (again, restricted to parallelopipeds):>

BTW good to see you using letter names not ratios on your tempered lattice
for Joseph. But lattices refer to prime numbers and different dimension for
each prime. Lattices are a good tool but perhaps a different sort of
lattice is required for tempered scales ? I'm not sure ascii is the
greatest tool for this though !

Justin White

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

/tuning/topicId_19127.html#19127

Hi Paul...

I know you are the "big 22 man" and I was wondering if you could
explain why there would be advantages to using a just scale in 22
over a just scale in 19.

Certainly I can understand that 22 is more "unusual" and might be
more distinctive from that aspect, but otherwise??

I can think of one obvious DISADVANTAGE for users of a "standard"
keyboard, and that is simply the fact that the interval of as 12th on
a standard keyboard is quite a nice "boundary" for 19 tones... quite
recognizable and memorable.

A span from a low C (below "middle C" let's say) to a Bb above middle
"C" for 22 tones would be slightly less "natural," no??

_________ _____ ____ ___
Joseph Pehrson

Justin wrote,

>Lattices are a good tool but perhaps a different sort of
>lattice is required for tempered scales ?

The only difference could be that you could "wrap" the lattice in a higher
dimension to lower the number of dimensions in which it is infinite. For
example, If you think of 12-tET as an approximation of Pythagorean tuning,
it takes the infinite straight-line chain of fifths and bends in into a
finite circle. Similarly, I've spoken of how meantone temperament can be
thought of taking the 5-limit plane, infinite in 2 dimensions, and wrapping
it into an infinite cylinder, infinite in 1 dimension, with four perfect
fifths making one "turn" around the cylinder to land one a single step
further along the infinite dimension . . .

On Tuesday, Justin White wrote,

>These are interesting Paul especially the 22 tone scales, I'm still playing
>around with scales for my guitar and and I have found that 22 tones is
>probably the most realistic amount of tones per octave. Would you be able
>to post some 22 tone 11 and 13 limit Fokker periodicity blocks ?

Let me try to go through the same process I went through for 41-tone
13-limit, now for 22-tone 11-limit.

I get the following unison vectors:

63:64
175:176
224:225
384:385

The Fokker periodicity block that results from these unison vectors at a
particular position in the lattice is:

0 1 1
43.408 525 512
119.44 15 14
155.14 35 32
231.17 8 7
274.58 75 64
315.64 6 5
386.31 5 4
435.08 9 7
498.04 4 3
551.32 11 8
590.22 45 32
653.18 35 24
701.96 3 2
772.63 25 16
813.69 8 5
857.09 105 64
933.13 12 7
968.83 7 4
1044.9 64 35
1088.3 15 8
1133.8 77 40

You are of course free to move any note by one, two, or three of the unison
vectors, in any direction, to increase connectivity and/or even out the step
sizes . . .

Please e-mail me privately if you have any questions.

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> Let me try to go through the same process I went through for 41-tone
> 13-limit, now for 22-tone 11-limit.
>
> I get the following unison vectors:
>
> 63:64
> 175:176
> 224:225
> 384:385
>
> The Fokker periodicity block that results from these unison vectors
at a
> particular position in the lattice is:
...
> You are of course free to move any note by one, two, or three of the
unison
> vectors, in any direction, to increase connectivity and/or even out
the step
> sizes . . .

And of course this one is amenable to microtempering by distributing
the two smallest unison vectors or commas (224:225 and 384:385). If
you made such a fretboard you would probably be tempted to use those
intervals which differ from Just by those commas, and so you might
want to distribute them.

My favourite distribution of these is achieved by narrowing all the
2:3's by 1.05 c, the 4:5's by 3.18 c, the 4:7's by 0.47 c, and the
8:11's by 1.63 c. The changes in the other intervals follow from
these. Apart from the 4:5, all other deviations are less than 2.5 c,
which I understand is typical of the tuning errors of (even expertly
tuned) acoustic stringed instruments.

Regards,
-- Dave Keenan
Regards,

Here's an old post of mine from 2/22 (pretty spooky!):

> On Tuesday, Justin White wrote,
>
> >These are interesting Paul especially the 22 tone scales, I'm
still playing
> >around with scales for my guitar and and I have found that 22
tones is
> >probably the most realistic amount of tones per octave. Would you
be able
> >to post some 22 tone 11 and 13 limit Fokker periodicity blocks ?
>
> Let me try to go through the same process I went through for 41-tone
> 13-limit, now for 22-tone 11-limit.
>
> I get the following unison vectors:
>
> 63:64
> 175:176
> 224:225
> 384:385
>
> The Fokker periodicity block that results from these unison vectors
at a
> particular position in the lattice is:
>
> 0 1 1
> 43.408 525 512
> 119.44 15 14
> 155.14 35 32
> 231.17 8 7
> 274.58 75 64
> 315.64 6 5
> 386.31 5 4
> 435.08 9 7
> 498.04 4 3
> 551.32 11 8
> 590.22 45 32
> 653.18 35 24
> 701.96 3 2
> 772.63 25 16
> 813.69 8 5
> 857.09 105 64
> 933.13 12 7
> 968.83 7 4
> 1044.9 64 35
> 1088.3 15 8
> 1133.8 77 40
>
> You are of course free to move any note by one, two, or three of
the unison
> vectors, in any direction, to increase connectivity and/or even out
the step
> sizes . . .
>
> Please e-mail me privately if you have any questions.

Now Joseph, I think Dave Keenan was saying that 225:224 and 385:384
are unison vectors that vanish in 72-tET . . . hence any periodicity
block constructed from these unison vectors is going to have a whole
lot of extra consonances in 72-tET . . . good news!

In 72-tET, this scale comes out to

0
2
7
9
14
16
19
23
26
30
33
35
39
42
46
49
51
56
58
63
65
68
(72)

Of course, moving notes by a 64:63 would result in a shift of 2 steps
in 72-tET . . . so this is far from "crystallized". Anyway, just
thought I'd throw this out there . . . lots more possibilities to
explore . . .

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

/tuning/topicId_19127.html#21792

> Of course, moving notes by a 64:63 would result in a shift of 2
steps
> in 72-tET . . . so this is far from "crystallized". Anyway, just
> thought I'd throw this out there . . . lots more possibilities to
> explore . . .

Quite frankly, I think when I get "tired" of working with 19-tone
scales, I, most probably, will be working with 22 as well...

It seems about in the "practical" range in terms of step size and
keyboard mapping that I want to pursue at the moment...

I know you also did the generation of scales from the periodicity
blocks for 22-note scales, but I don't seem to have the correct
archive number at the moment...

Do you happen to have this off hand??

Thanks!

_________ _____ _______ ___
Joseph Pehrson