RE: [tuning] How "infinite" the tonal lattice?

Polychroni wrote,

>After having read the _JI Primer_, much is made out of the "infinite" tonal

>lattice. But, clearly, their are limits, both physical and perceptional,
>on how fine an increment can be discerned. And, since some intervals are
>indiscernible, it would seem that the _practically_ the lattice is finite.

>Is it reasonable to superimpose a "discernability grid" or a "precision
>grid", or some other such grid, on top of the tonal lattice, thereby
>joining like-tones (hmm..., 'homotones'?), into a finite, practical
>lattice?

This is exactly the concept of _finity_ that forms one of Joe Monzo's main
interests. He's been kind enough to put lots of material on the subject on
the Web, and will no doubt be adding more once he gets settled in
California. You could start with
http://www.ixpres.com/interval/dict/finity.htm.

Also go to http://www.ixpres.com/interval/dict/pblock.htm and follow the
links from there. When Joe visited me in Boston, he made it a point to ask
me about finity and I mentioned that Fokker's periodicity block formalism
seemed to get at the same idea. Since then, I've looked into the concept
more deeply and posted many articles on it (search the archives for
"periodicity").

I'd be happy to help you if any of the material you find is unclear.

Kraig Grady wrote,

>I believe that the research is very limited and much more work really needs
to be done before we start drawing lines in >the sand.

I feel that any composer should be free to draw the line anywhere she sees
fit, based on her own perception of her compositional materials, her likely
uses of duration, timbre, etc. The Fokker periodicity block formalism I
pointed Polychroni to allows one to take any equivalences, whether as large
as 31¢ or as small as a fraction of a cent, and construct a finite pitch
system which mirrors the entire infinite lattice using only those
equivalences. No one is saying that 31¢ or even much smaller intervals can't
be perceived as structurally distant on the lattice if the composition uses
them that way. At the same time, finity can open up compositional
possibilities absent on the infinite lattice through the use of "punning" --
at which point temperament and/or adaptive tuning become useful options. The
commonest Western progressions rely on "punning" through the syntonic comma,
an interval which in pure just intonation would come out to 21.5¢. However,
if one wrote a piece in just intonation in which the melodic interval of
syntonic comma was a motivically emphasized feature, one could
compositionally project it as a significant move in the harmonic lattice.
These are not contradictions, they are merely alternate possibilities in the
vast field of musical composition, and a demonstration that musical context
is everything when dealing with these abstract "lines in the sand".

Kraig Grady wrote,

>Partch work with limits was an examination of the Lambdoma at its different
size levels. For a great part the structural >Possibilities is what
attracted him to build instrument in this tuning. As Dan could also point
out , he did not limit >himself to this singular structural perspective.
Many of his composition were based on ideas outside the diamond but >could
be done with a diamond master set.

If I'm understanding you correctly, Kraig, this is precisely the concept of
finity at work! Partch created a tuning system for his instruments that, as
Wilson helped show, was a 41-tone periodicity block with two auxillaries.
Ideas using pitches in the JI lattice outside this set were simply
"translated" by Partch to within the set, and even without any use of
temperament or adaptive tuning, he essentially had access to the whole
infinite lattice. The set of "unison vectors" or intervals of translation
that results in a periodicity block which corresponds most closely to
Partch's set seems to be:

896/891 = 9.7¢
441/440 = 3.9¢
245/243 = 14.2¢
100/99 = 17.4¢

The last interval is so large that Partch kept it in his set between two
sets of auxillaries, 10/9 and 11/10, and 20/11 and 9/5, making it a 43-tone
set in total (another reason for these auxillaries was their presence in the
29-tone diamond that formed the starting point for Partch's construction).
Due to the properties of periodicity blocks, we can say for certain that any
"translation" Partch may have done to get his ideas into his pitch set was
through one of the intervals above, or less likely a combination of more
than one of them. I would venture a guess that due to Partch's sensitive
ear, the 3.9¢ "comma" is the one Partch would have been most likely to make
use of.