RE: [tuning] Re: For Joseph: Eight 19-tone 7-limit Fokker periodi city blocks

Dave Keenan wrote,

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

About an hour, mainly drawing the lattices.

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

Not really -- I have a program that will search a given set of unison
vectors (for this exercise I included all 7-limit unison vectors smaller
than a quartertone and with numerator <= 2048) for n-tuplets that yield
periodicity blocks with a given cardinality; then I have another program
which takes a given n-tuplet and constructs the block, outputting cents,
numerator, and denominator. These programs are very short.

Hi Joseph.

>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...

Certainly I don't like 7-limit harmony in 19-tET . . . but don't forget
there are a lot of possibilities besides JI and ET, that will get you much
of the purity of the former along with much of the flexibility of the latter
. . . adaptive JI is clearly one answer, but even among fixed tunings
there's much more to consider . . . for example, my favorite 19-tone tuning
for 7-limit harmony is approximately 19 consecutive notes of 1/4-comma
meantone (which Margo indicates may have been rather common in one part of
Europe at one point in the 16th century, but probably without higher than
5-limit harmonic usage). A portion of the infinite lattice for this tuning
would look like this:

F---------C---------G---------D---------A---------E---------B---------F#
|\/ \/|\/ \/|\/ \/|\`. ,'/|\`. ,'/|\`. ,'/|\`. ,'/
|/\ /\|/\ /\|/\ /\| \ Cb-/-|-\--Gb-/-|-\--Db-/-|-\--Ab-/-
B---------F#--------C#--------G#--------D#--------A#--------E# \ | /
|\`.\ /,'/ \`.\ /,'/ \`.\ /,'/ \`.\|/,'/ \`.\|/,'/ \`.\|/,'/ \`.\|/
|-\--Ab-/---\--Eb-/---\--Bb-/---\--F--/---\--C--/---\--G--/---\--D----
E# \ |\/ \/|\/ \/|\/ \/|\/ \/|\/ \/|\/ \ |\`.
`.\|/\ /\|/\ /\|/\ /\|/\ /\|/\ /\|/\ \| \ Cb
-----D---------A---------E---------B---------F#--------C#--------G# \ |
/|\`.\ /.'/|\`.\ /,'/|\`.\ /,'/|\`.\ /.'/ \`.\ /,'/ \`.\ ,'/ \`.\|
/ | \ Cb-/-|-\--Gb-/-|-\--Db-/-|-\--Ab-/---\--Eb-/---\--Bb--------F
-----G#--------D#--------A#--------E# \ |\/ \/|\/ \/|\/ \/|
/,'/ \`.\|/,'/ \`.\|/,'/ \`.\ /,'/ \`.\|/\ /\|/\ /\|/\ /\|
Bb--------F---------C---------G---------D---------A---------E---------B
|\/ \/|\/ \/|\/ \/|\/ \/|\`.\ /,'/|\`.\ /,'/|\`.\ /,'/|
|/\ /\|/\ /\|/\ /\|/\ /\| \ Cb-/-|-\--Gb-/-|-\--Db / |
E---------B---------F#--------C#--------G#--------D#--------A#--------E#
|\`.\ /,'/|\`.\ /,'/ \`.\ /,'/ \`.\ /,'/ \`.\|/,'/ \`.\|/,'/ \`.\|/,'/
|-\--Db-/-|-\--Ab-/---\--Eb-/---\--Bb-/---\--F--/---\--C--/---\--G--/--
A#--------E# \ |\/ \/|\/ \/|\/ \/|\/ \/|\/ \/|\/
`.\|/,' `.\|/\ /\|/\ /\|/\ /\|/\ /\|/\ /\|/\
-----G---------D---------A---------E---------B---------F#--------C#----
/|\ /|\`.\ /.'/|\`.\ /,'/|\`.\ /,'/|\`.\ /.'/ \`.\ /,'/ \`.\
/ | \ / | \ Cb-/-|-\--Gb-/-|-\--Db-/-|-\--Ab-/---\--Eb-/---\--Bb
-----C#--------G#--------D#--------A#--------E# \ | / \ | / \ |
/,' `.\ /,' `.\|/,' `.\|/.' `.\|/,' `.\|/ \|/ \|
Eb--------Bb--------F---------C---------G---------D---------A---------E

>Besides, I
>wouldn't want to "insult" some of my JI friends, and they
>sometimes seem a little "sensitive" about such matters.

You have friends who could be insulted by how you chose to compose?

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

Could I ask you to go over the Gentle Introdiction to Fokker Periodicity
Blocks again, particularly part 3?

>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??

Yes, with a lot of cutting and pasting.

>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.

It's no big deal. Here, I'll post a bunch of 22-tone 7-limit periodicity
blocks next . . .

Joseph wrote,

>> 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...

There are two hexanies: 4/3-14/9-10/9-35/18-5/3-7/6 and
36/35-6/5-12/7-3/2-9/7-9/5.

Note that 19 notes of approximately 1/4-comma meantone has _six_ hexanies,
only slightly out-of-tune. And, of course, if you accept the 7-limit
approximations of 19-tET (I don't) or you use adaptive JI, you will have
_nineteen_ hexanies.

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

/tuning/topicId_19113.html#19120

> Hi Joseph.
>
> >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...
>
> Certainly I don't like 7-limit harmony in 19-tET . . . but don't
forget there are a lot of possibilities besides JI and ET, that will
get you much of the purity of the former along with much of the
flexibility of the latter . . . adaptive JI is clearly one answer,
but even among fixed tunings there's much more to consider . . . for
example, my favorite 19-tone tuning for 7-limit harmony is
approximately 19 consecutive notes of 1/4-comma meantone

Hi Paul...

That's the same tuning we were discussing some time ago, yes?:

0: 1/1 0.000 unison, perfect prime
1: 76.049 cents 76.049
2: 117.108 cents 117.108
3: 193.157 cents 193.157
4: 269.206 cents 269.206
5: 310.265 cents 310.265
6: 5/4 386.314 major third
7: 462.363 cents 462.363
8: 503.422 cents 503.422
9: 579.471 cents 579.471
10: 620.529 cents 620.529
11: 696.578 cents 696.578
12: 25/16 772.627 classic augmented fifth
13: 8/5 813.686 minor sixth
14: 889.735 cents 889.735
15: 965.784 cents 965.784
16: 1006.843 cents 1006.843
17: 1082.892 cents 1082.892
18: 125/64 1158.941 classic augmented seventh,
octave - minor diesis
19: 2/1 1200.000 octave

That's a beautiful section of lattice you posted...

>
> >Besides, I
> >wouldn't want to "insult" some of my JI friends, and they
> >sometimes seem a little "sensitive" about such matters.
>
> You have friends who could be insulted by how you chose to compose?
>

Well, I'm being a little facetious. However, it works the "other
way" as well... I have some former teachers who view my work in
alternate tunings as "out of tune" so, to a degree there really *ARE*
some prejudices out there!

> >HOWEVER, I have a couple of questions for you. Could you please
> >briefly explain in LAYMAN'S terms what you did???
>
> Could I ask you to go over the Gentle Introdiction to Fokker
Periodicity Blocks again, particularly part 3?
>

Willdoo... with questions later...

JP

______ ____ ____ ____ _
Joseph Pehrson

Joseph wrote,

>That's the same tuning we were discussing some time ago, yes?

Glad you remember! As you can see, the 7:4, 7:6, 3:2, 7:5, and 6:5 are not
too far off in this tuning, while the 5:4 is just.

On Tue, 20 Feb 2001 15:15:35 -0500, "Paul H. Erlich"
<PERLICH@ACADIAN-ASSET.COM> wrote:

>Hi Joseph.
>
>>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...
>
>Certainly I don't like 7-limit harmony in 19-tET . . . but don't forget
>there are a lot of possibilities besides JI and ET, that will get you much
>of the purity of the former along with much of the flexibility of the latter
>. . . adaptive JI is clearly one answer, but even among fixed tunings
>there's much more to consider . . .

One of the hand-tuned scales that I was playing around with before I knew
much about tuning is a 19-note scale with a few good 7-limit approximations
-- which pretty much came about unintentionally, as I was tuning the scale
for its particular "character" without regard for how close it came to JI.
It looks pretty haphazard. But I actually have an old sample of this scale
on my Netscape site, so you can judge the results for yourself:

http://sites.netscape.net/thryomanes/rrilad.ra

The actual pitches of the scale in Hz are:

469 489 507 524 547 565 586 612 627 654 681 702 731 757 782 819 842 875 912

Here are some fairly good 7/4's in this scale:

912/524 = 959.4 cents
2 * 681/782 = 960.6
2 * 612/702 = 962.5
2 * 547/627 = 963.7
819/469 = 965.1
(7/4 = 968.8)

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning@y..., Herman Miller <hmiller@I...> wrote:

/tuning/topicId_19113.html#19148

>
> One of the hand-tuned scales that I was playing around with before
I knew much about tuning is a 19-note scale with a few good 7-limit
approximations -- which pretty much came about unintentionally, as I
was tuning the scale for its particular "character" without regard
for how close it came to JI. It looks pretty haphazard.

Hello Herman

Gee. I wonder if this isn't a good endorsement of the virtues of
"not knowing much about tuning...." :)

This tuning in the way you are using it is one of the most beautiful
of your efforts, in my humble opinion... It really sounds great.

I guess tuning a scale by ear to find its "character" has some
certain advantages!

Again, it's the use of "sound logic," and it's hard to argue with
that...

This was somewhat of an "ear opener..."
________ _____ ______ _
Joseph Pehrson