Issues (was Set theory)

After a weekend to think about it, and catching up on the digests, there
are several things more I have to say. I do have to make this as short as
I can this time (not likely) since I really have to write an overdue
program note for an Irving Fine program. So I'll just try to clear up one
thing here -- what I mean by "theological arguments."

I can't remember the logician's technical term for such arguments (maybe
someone can fill in the term?), but it's roughly a form of circular
reasoning that assumes what you are trying to prove as a key in the proof
itself, e.g., you start out with a premise and then, after an often
lengthy (and even "faultless") series of arguments, reach a conclusion
that is virtually identical to the original premise. I call such circular
reasoning "theological," not because it's related to the more famous
medieval proofs of God's existence (which it is not), but because all such
arguments begin with a (subjectively) unassailable article of faith such
as "the Bible is the literal truth" -- the only way to "prove" this (which
one who believes it isn't compelled to do), is to line up all observations
and evidence in such a way that "the Bible is the literal truth" is true.

One of the most famous cases is presented by Galileo's defense of his work
before the Inquisition. The Church held, as an article of faith, that the
Earth was unmoving at the center of creation. Galileo reasoned otherwise
from his invention and use of the telescope, and published a treatise with
an essentially Copernican bent. The Church would have been happy if
Galileo could take all his observations and fit them into the immovable
earth theory, but that's not how he saw it. So his defense was this: God
wrote TWO books -- the Bible and the Book of Nature; he was simply trying
to read the latter and would gladly bow to the theologians in reading the
former. The Inquisition didn't buy it. As Galileo was packed off to
"house arrest" for most of the rest of his life, he made the still
quotable statement: "Still, it turns!"

There's no arguing that the perfect fifth is a simple ratio (fact). And I
certainly wouldn't argue that this is a salient feature of many scale
systems (fact). But as soon as one starts to say something uncomfortably
close to: THEREFORE any scale system (natural or synthetic) that does NOT
exhibit the attribute of, say, being generated by the fifth, is of lesser
importance or even inadmissable is logical poppycock. Not only has it not
been established that fifth-as-generator is a NECESSARY or even desireable
attribute of all compositionally viable scales (I don't think I'm
misconstruing intent here), but it's a back-door (or possibly
unintentional) attempt to logically derive a *value* from a *fact* contra
David Hume and 2 1/2 centuries of attempts of some pretty bright people to
discredit his argument. You just can't LOGICALLY get an "ought" from an
"is" without positing another "ought" statement as a minor premise.

When you attempt to discredit or shrug off something like ic vectors, this
attempt is just absurd to me, since ic vectors, like much of what we're
talking about here are simply "tools" for seeing or expressing
relationships. It's probably a better strategy to continue to look for
what others see so much in. For example, as it has been correctly noted
here, the ic vector for the usual diatonic is V(D) = [254361]. The only
other (12tet) heptachord with such a unique distribution of intervals is
the 7-note chromatic whose vector is V(C) = [654321]. If you know a
little bit about vectors you'll soon realize that, not only are the
chromatic's entries in reverse consecutive order, which is not surprising,
but that you can "get" V(C) "from" V(D) and vice versa by transposing
V(D)'s entries for ic1 and ic5:

254361
>|||<|
654321

Now, some ask "So what?" Others ask "Why?" The reason is that this
reflects the arguably intuitive relationship between a circle of fifths
and the chromatic circle, an operation commonly refered to as M5 or M7 --
when you multiply any sonority by 5 or 7, you get the same ic vector with
the reversal noted above. For the circle sonorities such as the chromatic
and 5th/4th stacks, this is hardly earth shattering news, but by
generalizing it, you now have a leg up (though some might think this is
still unimportant).

Again, others ask "So what?" while others ask "What happens if we push
onward." By "collapsing"/"enlarging" all the fifths/m2nds in the diatonic
(without even necessarily referring to an underlying chromatic there are
several other conceptual ways to do this), you get a parallel system in
which all the typical diatonic triads change with the exception of the
diminished triad (which one might use to surf between the two systems),
and in which all the usual *relationships* stay the same. Here's an M5
(multiply by a perfect fourth) version:

C D E F G A B (C)

C Bb Ab Db B A G (C)

Unisons, m3 & tritones stay the same; M2 & M3 are inverted (e.g., M2 up
becomes M2 down); and P5s become m2s and vice versa. The "primary
triad" C-E-G (43) becomes C-Ab-B (83) and V-I is some version of this:

B -> B
Bb-> Ab
G -> C

I.e., Agmon's "rules" which he demonstrated only work for the diatonic,
work in this parallel universe as well. All chord functions work the
same. All neo-Riemannian relations work as well under re-defined
(re-tuned?) intervals and voice leading. E.g., one of Richard Cohn's
parsimonious cycles works the same and produces the same "hexatonic"
scale:

C Ab B
C Eb B
C Eb E
G Eb E
G Ab E
G Ab B
C Ab B

(but don't look for this version in Brahms!)

In abstract geometry, we say that these parallel systems "cover" one
another under an M5/7 transformation (yes, we're now into geometry). This
means that the objects themselves may "distort" (predictably), but the
relations between those objects remain the same (are invariant) -- like
the earth projected onto a flat map.

The ONLY thing you can say judgementally about the process or result is
that you don't like the sound of the distorted objects (the new basic
chords) or the weird voice leading the new geometry implies (but doesn't
have to be adhered to compositionally). That's entirely subjective -- and
how often have "casual listeners" said they don't like the sounds of other
cultures? But what you can't say is that no composer could ever come
along and convince a suitably open ear to accept them via a suitable
compositional treatment -- and the door opens to the unimaginable.

[Afterthought: This basic result can be generalized still further, if you
wish, into other n-tet tunings. I don't know if it has been proven
anywhere, but I think it could be demonstrated that ANY chromatic segment
of any (only even?) chromatic universe could be transformed by a suitable
multiplier into a non-repeating ME scale (in a unique way?) and vice
versa. Since coordinate systems have been changed, it's even open to
argument that the short non-ME chromatic segment itself is "warped ME" in
these "curved" coordinates -- an idea worth at least thinking about since
Block & Douthett have identified these chromatic collections as "minimally
even," a category Douthett has extended into his particle physics
applications of ME.]

More on other "issues" at a later date.

Still your colleague, Paul, but still rattling your cage ;-) and still
hoping to convince you and others of the absolute value of the many-paths
approach.

Steve

Hi Stephen.

>a category Douthett has extended into his particle physics
>applications of ME.

May I ask you what you mean by "particle physics" here, and what
these applications are?