Yahoo Tuning Groups Ultimate Backup tuning ASCII lattices

[Paul Erlich:]
>Nice work, Monz! Glad you're comfortable with "our" way of drawing
lattices.

When Paul says this, I'm of course assuming he means the triangular,
4:5:6 and 1/(4:5:6) ASCII lattices that are a mainstay of the TD. The
way I understand this, or at least have been going about it lately, is
to let all prime members of the harmonic series collapse into the
triangle formed on the 1/1... so in other words a 16-32 harmonic
series would be -- theoretically, as this would be impossibly crowded
if all the primes of the 16-32 were to be simultaneously used and
fleshed out used -- something on the order of:

25
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
5----------------15
/ \ / \
19 \ / \
/ \ / \
/ 11 / \
/ 7----\-------/---21 \
17 29` `. 23 / .` `. \
/ .` 13 \ / .` `. \
/31` `. \ /.` `. \
1-----------------3-----------------9---------------27

So with inversional symmetry, this would mean that a series reduced to
primes would collapse into an otonal identity as say h...h*2, and an
utonal identity as h*2...h, or to put it another way, an 8-16 series
would collapse into:

8:(9):10:11:12:13:14:(15):16

and:

16:(15):14:13:12:11:10:(9):8

as:

5/4
/ \
/ \
/ \
/ \
/ \
/ 11/8
/ \
/ . 7/4 . \
/ .` `. \
/ .` 13/8 \
/ .` `. \
4/3---------------------1/1---------------------3/2
\ `. .`/
\ 16/13 .` /
\ `. .` /
\ ` 8/7 ` /
\ /
16/11 /
\ /
\ /
\ /
\ /
\ /
8/5

I've never really bothered to ask these questions as I often prefer to
work things out for myself, so this may not be the way the conventions
are working at all... but as there certainly are some ("our" way)
lattice conventions at work here at the TD, maybe someone could take
the time to post some basic 'lattice etiquette' for the sake of others
who might wonder some of these same things (like how to best
illustrate commatic differences that would create non-inversional
symmetry, like the 121/120, etc., etc.).

thanks,
Dan

🔗D.Stearns <stearns@capecod.net>

This is a slight rewrite of a bit from a post previous post (though
this was probably clear from the text that preceded it and the example
that followed it):

So with inversional symmetry, this would mean that a series reduced to
primes would collapse into an otonal identity as say h...2h, and an
utonal identity as 2h...h (where h is a harmonic series power of two).
Or to put it another way, an 8-16 series would map:

8:(9):10:11:12:13:14:(15):16

and

16:(15):14:13:12:11:10:(9):8

onto the collapsed O and U triangles (assuming h is =/> 2^2) as only
the reducible primes of h...2h and 2h...h:

Dan

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

I don't think there's a good way to portray 11- and higher limit lattices in
2 dimensions, let alone in ASCII, but the best attempts have probably been
Erv Wilson's: for the chord 1 3 5 7 9 11, or sometimes a different hexad, he
uses a (sometimes slightly distorted to add "depth") regular pentagon with
one note at the center and the others at the vertices, and often leaves many
of the connections out. See

http://www.anaphoria.com/dal12.html

http://www.anaphoria.com/dal13.html

http://www.anaphoria.com/dal14.html

http://www.anaphoria.com/dal15.html

http://www.anaphoria.com/dal16.html

http://www.anaphoria.com/dal17.html

and the accompanying text at

http://www.anaphoria.com/dal01.html.

🔗D.Stearns <stearns@capecod.net>

[Paul H. Erlich:]
>I don't think there's a good way to portray 11- and higher limit
lattices in 2 dimensions, let alone in ASCII, but the best attempts
have probably been Erv Wilson's:

Thanks for the Wilson links Paul, they are to my sensibilities,
*extremely* beautiful and very inspiring.

What I mostly wanted to do with 11 and higher limits in this ASCII and
two-dimensional format was to have a way to plot and look at different
scales (and this has certainly been done here at the TD with 11-limit
scales on more than a few occasions). So I'm basically attempting to
take this 16-32 template:

5/
/ \
19/ \
/ \
/ \11
/ 7, \
17/ 29, ` \23
/ , `13\
1/31, ` \3

and on a scale by scale, example by example basis, make it fit as
reasonably, and aesthetically pleasingly as I can on a smaller scale
(whenever a smaller scale is indeed possible).

So as I said before, I have seen 11-limit scales illustrated this way
here at the TD in the past, and as such I wouldn't imagine that
something like this 1/1, 12/11, 6/5, 4/3, 3/2, 18/11, 9/5, 2/1, two
1/(9:10:11:12) tetrachord scale, would be much of a cause for
confusion if it were illustrated as:

4/3-------1/1-------3/2
\ / \
\ / \
12/11-+---18/11
\ / \
6/5-------9/5

However, if I were to illustrate say a 1/1, 9/8, 24/19, 4/3, 3/2,
32/19, 36/19, 2/1, 19-limit diatonic as a:

4/3-------1/1-------3/2-------9/8
/ \ / \ / \ /
/ \ / \ / \ /
64/57-----32/19-----24/19-----36/19
/ \ / \ / \ /

then I would expect that this would perhaps be somewhat confusing
without the aid of the:

O = 16:17:(18):19:20:(21):22:23:24:(25):26:(27):28:29:(30):31:32

U = 32:31:(30):29:28:(27):26:(25):24:23:22:(21):20:19:(18):17:16

16-32 prime template,

5/
/ \
19/ \
/ \
/ \11
/ 7, \
17/ 29, ` \23
/ , `13\
1/31, ` \3

and an understanding, or an acceptance of the compromises that are
made for sake of size reduction.

So at least to this type of a limited extent (and as I actually use
quite a few scales that are fairly easily shown with this type of a
mapping), I find it satisfying (if not ideal) to plot in this way,
higher limits in the (standard) ASCII two-dimensional format.

Dan

🔗Joe Monzo <monz@juno.com>

> [Paul Erlich, TD 533.12]
> I don't think there's a good way to portray 11- and higher
> limit lattices in 2 dimensions, let alone in ASCII, but the
> best attempts have probably been Erv Wilson's: <snip>

So by omission it seems that you're saying my lattices don't
do the job at least as well as Erv's?

I know the copy of my book that you have is very sloppy-looking,
Paul, and I also know that there are some deep disagreements
in the way the two of us look at the subject, but I do wish
you'd really read it from start to finish and try to get what
usefulness you can out of it. If you get nothing else from
it, you (and any other reader) should at least gain an
appreciation of the value of my lattice formula in its ability
to explain the entire scope of JI theory.

And I have indeed made some (IMO, quite successful) attempts
to follow my lattice formula in ASCII in a few Tuning List
posts.

I know you (Paul) disagree with my use of 144-EDO notation,
especially to describe Partch's scale, but this ASCII lattice:
http://www.onelist.com/messages/tuning?archive=147
gives a clear picture of the placement in ratio-space of the
pitches in the 4-dimensional 11-limit system Partch used.

> [Dan Stearns, TD 533.19]
>
> [a template for odentities of harmonics 16-32]
>
> 5/
> / \
> 19/ \
> / \
> / \11
> / 7, \
> 17/ 29, ` \23
> / , `13\
> 1/31, ` \3
>

I think that's brilliant, Dan, as an alternative solution
to the one I've found.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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YOU'RE PAYING TOO MUCH FOR THE INTERNET!
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Try it today - there's no risk! For your FREE software, visit:
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🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

Joe Monzo wrote,

>So by omission it seems that you're saying my lattices don't
>do the job at least as well as Erv's?

Sorry for the omission, Joe, but you're right, that is what I'm saying.

>I know the copy of my book that you have is very sloppy-looking,
>Paul, and I also know that there are some deep disagreements
>in the way the two of us look at the subject, but I do wish
>you'd really read it from start to finish and try to get what
>usefulness you can out of it. If you get nothing else from
>it, you (and any other reader) should at least gain an
>appreciation of the value of my lattice formula in its ability
>to explain the entire scope of JI theory.

I have read it from start to finish. We've discussed the merits of various
lattice schemes before.

>I know you (Paul) disagree with my use of 144-EDO notation,
>especially to describe Partch's scale, but this ASCII lattice:
>http://www.onelist.com/messages/tuning?archive=147
>gives a clear picture of the placement in ratio-space of the
>pitches in the 4-dimensional 11-limit system Partch used.

Putting aside the notation (which implies, in 144-tET, some pretty
un-Partchian "3:2"s), I think your lattice is perfectly OK, but fails to
convey the essential information that Wilson's lattice would convey. For
example, in Wilson's lattice, you would be able to immediately pick out each
otonal hexad as a right-side-up pentagon with the 1 odentity in the center,
and each utonal hexad as an upside-down pentagon with the 1 udentity in the
center. These compact, symmetrical, easily discernible structures are a
perfect visual analogue for how Partch regarded the hexads from a musical
point of view.

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

Joe Monzo wrote,

>>The great advantage of this approach (as I see it) is that,
>>in theory, no matter how complex the JI description becomes,
>>each ratio can be given an absolutely unique plot on the
>>2-dimensional face on which the lattice must be drawn.

I wrote,

>That's also the case with most of the types of lattices Erv Wilson
designed.

I was wrong. Sometimes the same pitch will appear in more than one place in
the lattice. But that's the price you pay for trying to show all the
relationships symmetrically. Paul Hahn and I discussed other options a while
back . . .

Even preserving your basic concept, Joe, it would be better to restrict the
mapping of prime cent-values 0-1200 to a half-circle, instead of a full
circle, since (a) the negative powers wouldn't be confused with new "primes"
with a cent-value 600 cents away from that of the positive power, and (b)
the secondary consonant relationships will tend to be shortened, while
relationships based on dissonant products of primes will tend to be
lengthened.

🔗Joe Monzo <monz@juno.com>

> [Paul Erlich, TD 536.2]
> Even preserving your basic [lattice formula] concept,
> Joe, it would be better to restrict the mapping of prime
> cent-values 0-1200 to a half-circle, instead of a full circle,
> since (a) the negative powers wouldn't be confused with new
> "primes" with a cent-value 600 cents away from that of the
> positive power, and (b) the secondary consonant relationships
> will tend to be shortened, while relationships based on
> dissonant products of primes will tend to be lengthened.

Wow! - how interesting that you should propose that at just
this time. Robin Perry, who hasn't been on the List for a
while, came to visit me a couple of months ago, and then very
recently sent me some diagrams based on exactly this adaptation
of my ideas.

I haven't really had a chance to explore it more fully,
but I did think that Robin was on to something good here,
and now you've given me even more food for thought. I may
change my formula to make the lattices like this.

-monz

🔗D.Stearns <stearns@capecod.net>

[Paul H. Erlich:]
>I don't think there's a good way to portray 11- and higher limit
lattices in 2 dimensions, let alone in ASCII, but the best attempts
have probably been Erv Wilson's: for the chord 1 3 5 7 9 11,

I was digging around to see if I had used any scales that contained
this hexad, and I found a 3 1 3 3 3 2 2 scale with three step size
cardinality that I used in a section of a piece in 17e that was based
on a 1/1, 9/8, 7/6, 4/3, 3/2, 5/3, 11/6, 2/1, which of course also has
a 1 3 5 7 9 11 (which is so pronounced that it really seems to want to
be a 1 3 5 7 9 11 13) in it's 1/1, 9/8, 5/4, 11/8, 3/2, 27/16, 7/4,
2/1 rotation...

5/3
/ \
/ \
/ 11/6
/ ,7/6. \
/,' '.\
4/3---------1/1---------3/2---------9/8

40/27
/ \
/ \
/ 44/27
/ 28/27 \
/,' '.\
32/27-------16/9---------4/3---------1/1

10/7
/ \
/ \
/ 11/7
/ ,1/1. \
/,' '.\
8/7--------12/7---------9/7--------27/14

5/4
/ \
/ \
/ 11/8
/ ,7/4. \
/,' '.\
1/1---------3/2---------9/8---------27/16

10/9
/ \
/ \
/ 11/9
/ 14/9. \
/,' '.\
16/9--------4/3---------1/1---------3/2

11/10
/=
/ =
1/1 =
/ \ =
/ \=
/ (12/11)
/ ,7/5. \
/,' '.\
8/5---------6/5---------9/5---------27/20

(11/6)
/=\
/ = \
/ = 1/1
/ 14/11 \
/,' = '.\
16/11-20/11-12/11--------18/11-------27/22

As these are seven note scales that aren't using 11-limit intervals in
a way that really demands a different orientation, I think this sort
of a lattice representation works out... Anyway, an interesting and
obstinate scale.

Dan

🔗D.Stearns <STEARNS@CAPECOD.NET>

"As I was walkin'
I saw a sign there
And that sign said no tress passin'
But on the other side
It didn't say nothin!"

This is a 22-tone set built on two series, 10-20 and 14-28 that I used
on a small ensemble piece (which if they're instruments I don't play,
really just means a less than satisfying -- and really "small(!)" --
MIDI sound module instrument imposter ensemble piece...) loosely based
on a pair of American patriotic songs (_America the Beautiful_, and
_Battle Hymn of the Republic_):

25/14
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
10/7--------------15/14
/|\ |
/ | \ |
19/14|11/7 |
/| | |\ |
/ | | | \ |
17/14| | |23/14 |
/| |,1/1.+---\--------,3/2.
/ |,'| /|\ 13/7 \ ,' /|\ `.
8/7'+--+/-+-\+-+`12/7'----/-+-\----`9/7--------------27/14
| |19/10|11/10 | / \ |
| | / | \| | / \ |
| |/ | \ | / \ |
|17/10 | |\ | / \ |
| / ,7/5. | \ | / \ |
|/ ,' 13/10\|/ \|
8/5'-------------`6/5---------------9/5

While the above placement is a compromise of:

5/
/ \
19/ \
/ \
/ \11
/ 7, \
17/ 29, ` \23
/ , `13\
2/31, ` \3

which in itself is not ideal, I still think that this (two dimensional
ASCII compression) method of plotting higher primes can (on occasion)
work out OK.

Yankee Doodle Dandy