'octave'-specific lattices

[Paul Erlich:]
> Your lattices do not make it possible to see the
> order of pitch of the notes in a scale.

They do now.

Not quite the same way that you're talking about,
but it's there, especially when taking bridges
into account.

I've incorporated the 2-axis in my lattice-plotting
formula, and so now i can plot ETs and also plot
the exact ratios of a tuning system in an 'octave'-
specific fashion.

Including the exponents of 2 warps the shape of the
whole lattice, so it's quite a bit different from the
old ('octave'-reduced) ones, but the basic outlines
are still recognizable.

Also, ETs were being plotted with ascending pitches
going *down* the page, so I revised my formula to
invert the entire lattice. So the new ones are
180 degrees from the old ones. In many cases the
shapes don't change, because theorists and composers
have often used symmetrical systems (i.e., Partch, Wilson).

There's an example of my new 11-limit tonality diamond
representation at:

http://www.ixpres.com/interval/monzo/lattices/xl/partch11.htm

-monz

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

___________________________________________________________________
You don't need to buy Internet access to use free Internet e-mail.
Get completely free e-mail from Juno at http://www.juno.com/getjuno.html
or call Juno at (800) 654-JUNO [654-5866]

I wrote,
>> Your lattices do not make it possible to see the
>> order of pitch of the notes in a scale.

Joe Monzo wrote,
>They do now.

Looking at the page below, I don't see it.

>There's an example of my new 11-limit tonality diamond
>representation at:

>http://www.ixpres.com/interval/monzo/lattices/xl/partch11.htm

Here's some comments on this page:

>Meantones can also be viewed as a fractional division of the 1:5 ratio.

Though the fifths in meantone can, the fourths (for example) can't. So this
is not correct.

>and it's not as attractive as my paper copy, where I used solid vectors for
the otonal connections and used >dashed lines for the utonal, as did Partch
on his diamond.

What do you mean? A connection is just a ratio, and a ratio can't be utonal
or otonal -- it takes 3 or more notes to decide whether one has an otonality
or a utonality.

I'll save more for later.

[Paul Erlich/aka Brett Barbaro:]
>>> Your lattices do not make it possible to see the
>>> order of pitch of the notes in a scale.
>
> Joe Monzo wrote,
>> They do now.
>
> Looking at the page below, I don't see it.

I added a comment on the webpage below the 12-eq lattice
that the scale for that was very different from the
scale on the Partch lattice. On the Partch lattice,
there are two 'octave' ratios plotted: a pair of
4/3s and a pair of 3/2s. They appear on the central
'spine' of the lattice. I should have connected them,
but I haven't yet.

They are about 1/4-inch apart on my screen. So that
entire 12-eq 'octave' that's plotted on that lattice
would fit into that little 1/4-inch space on the Partch
lattice.

I say at the end of that page that I'm not entirely
happy with how this turned out. But it's the first
time I've been able to plot ETs and a 2-axis according
to a formula that works for the other primes.

>
>> There's an example of my new 11-limit tonality diamond
>> representation at:
>
> http://www.ixpres.com/interval/monzo/lattices/xl/partch11.htm
>
> Here's some comments on this page:
>
>> Meantones can also be viewed as a fractional division of
>> the 1:5 ratio.
>
> Though the fifths in meantone can, the fourths (for example)
> can't. So this is not correct.

I realized re-reading the webpage last night that this
was not really clear the way I wrote it, so before I saw
this I had revised it, adding a remark that 5^(1/4) is a
1/4-comma meantone '5th'. Perhaps I'll emend it again.

>> and it's not as attractive as my paper copy, where I used
>> solid vectors for the otonal connections and used dashed
>> lines for the utonal, as did Partch on his diamond.
>
> What do you mean? A connection is just a ratio, and a ratio
> can't be utonal or otonal -- it takes 3 or more notes to
> decide whether one has an otonality or a utonality.

This is all true, but I simply connected the ratios in the
same way Partch did on his diamond, as I explained.
So even tho the connections *can* go both directions,
Partch's use of them (on his diamond diagram) was
direction-specific, except for a few cases which were
direct from 1/1.

In other words, he portrayed the pitches as elements
in certain sets (otonalities and utonalites), but not
in others - for instance, your A.S.S. chords.

(for those who need to learn about A.S.S. chords, see:)
http://www.cix.co.uk/~gbreed/ass.htm

I think the area we're entering with this discussion
is the kind of thing Erv Wilson was exploring by
creating sets that fill ratio-space. This is exactly
what you and Carl were just talking about in regard
to Ming's lattice applet and the Partch diamond (not doing).

Wilson's designs incorporate maximum combinatoriality
of pitch-classes within a given lattice-block.
I don't know if it's a periodicity block - I haven't
studied his work enough yet.

So anyway, with the exception of those few connections
like 3/2 and 4/3, which have to 'go both ways',
the vectors are either solid or dashed. This makes
the lattice look quite different, even tho the linearity
itself is the same. I'll try to incorporate the dashed
lines on the webpage when I can.

> I'll save more for later.

Always interested in *your* feedback, Paul/Brett.

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

___________________________________________________________________
You don't need to buy Internet access to use free Internet e-mail.
Get completely free e-mail from Juno at http://www.juno.com/getjuno.html
or call Juno at (800) 654-JUNO [654-5866]

Monz:

>Wilson's designs incorporate maximum combinatoriality
>of pitch-classes within a given lattice-block.

Roughly speaking, and so do Partch diamonds, roughly speaking another way.

>I don't know if it's a periodicity block - I haven't
>studied his work enough yet.

Since a Wilson CPS can be arbitrarily rotated about any of the axes in the
triangular lattice, unison vectors can become non-unison vectors and vice
versa without changing the set, which obviously couldn't be true of a
periodicity block.

>Always interested in *your* feedback, Paul/Brett.

Stop that! My name is Paul, and you met Brett yourself.