lattices vs./pro st :) :) :) :) &

>Answers should probably go to the tuning-math list, and
>so perhaps so should this question... oh well... I'd like
>to see more on the mathematics of manipulating sets of
>exponents in a JI tuning system, similar to the way
>academic theorists use set-theory in 12-EDO. Any ideas
>on this?
>

Lattice diagrams basically *are* set theory, for JI, no?
I mean, it's trivial to map out a lattice of triads for an ET, since any
set can happen on any note. But in a JI system, where triads are (or some
other sonority), is not trivial, cause they ain't everywhere.

I guess this is obvious.

Question: What about latticing non-triads in JI?
1) is there a point? (i.e. the whole point of JI seems to be to be
able to have vertical sonorities that are otonal/utonal, preferably lower
on the series).

2) has anyone fooled with the idea?

***From: Christopher Bailey******************

http://music.columbia.edu/~chris

**********************************************

Hi Chris,

--- In tuning@y..., Christopher Bailey <cb202@c...> wrote:

/tuning/topicId_24271.html#24271

> Lattice diagrams basically *are* set theory, for JI, no?

Sounds right to me. But I like to see the formal mathematical
language which I'm not capable of producing ...that's *why*
I use the diagrams. :)

> I mean, it's trivial to map out a lattice of triads for an
> ET, since any set can happen on any note. But in a JI
> system, where triads are (or some other sonority), is not
> trivial, cause they ain't everywhere.
>

That's a good point about reasons for using lattices.
The interval gaps between pitches in strict-JI tuning
systems are frequently symmetrical but always unequal.

> I guess this is obvious.
>
> Question: What about latticing non-triads in JI?
> 1) is there a point? (i.e. the whole point of JI seems
> to be to be able to have vertical sonorities that are
> otonal/utonal, preferably lower on the series).

Triads should only be assumed to be the basis of vertical
sonorous structure for the "common-pracice" Eurocentric
harmonic model (c. 1600-1900), and the modern remnants
derived from it (as in much pop music).

You seem to me to be answering your own question and
recognizing my above clarification in your parenthetical
comment. Certainly you know that latticing Partch's
harmony makes it easy to visualize the mathematics of
its 4-dimensional hexadic structure.

>
> 2) has anyone fooled with the idea?

Adriaan Fokker, Erv Wilson, Paul Erlich, Graham Breed and I have
all latticed many non-triadic structures in our work.
(apologies to Dave Canright and Kees van Prooijen if
I'm neglecting something of yours)

In the cases of Fokker and Erlich, most of their lattices
have to do with illustrating periodicity-blocks. Wilson
has used them to illustrate his Combination Product Sets.
I have used them primarily to illustrate complete tuning
systems proposed by others.

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_24271.html#24302

>
> That's a good point about reasons for using lattices.
> The interval gaps between pitches in strict-JI tuning
> systems are frequently symmetrical but always unequal.
>

Personally, I found the visual comparisons using lattices to be VERY
valuable. Especially with Paul's Periodicity Blocks, it was
interesting to see the degree of "interconnectedness" between various
just intervals of the scales.

And, of course, with the "blackjack" scale, mapping in 72-tET, the
lattice goes on, apparently forever in each direction.

>
> > I guess this is obvious.
> >
> > Question: What about latticing non-triads in JI?
> > 1) is there a point? (i.e. the whole point of JI seems
> > to be to be able to have vertical sonorities that are
> > otonal/utonal, preferably lower on the series).
>

Hasn't a lot of the latticing been TETRADIC in nature, using the 7th
partial as the THIRD dimension, going off the page??

Or, am I misunderstanding something....

>
> Triads should only be assumed to be the basis of vertical
> sonorous structure for the "common-pracice" Eurocentric
> harmonic model (c. 1600-1900), and the modern remnants
> derived from it (as in much pop music).
>

I would say that there could be a NEW usage of triadic and tetradic
harmony using just intervals that could be possible... maybe
reverting back to certain forms of progression and voice leading from
the past, but not entirely.

Certainly, though, that would be a different "animal" from writing
NEW works which use PRIMARILY "standard-practice" harmony... (Not
that YOU mentioned this, Monz, but it has been a recent topic on this
list...)

__________ _______ _____
Joseph Pehrson