“Oh dear, where to start?

Well, you certainly have put your finger on it when you write:

‘Forgive my generalizations, but it seems to me that the compositional approach stems from a time when composition and theory were basically the same thing, hence, this approach is favored by an earlier generation of pedagogues.’

Yes, indeed! The whole distinction on which your post is premised, namely that which is alleged to exist between “compositional” and “analytical” approaches to music, exists only because, once upon a time, “theory” (or “analysis”) stopped yielding insight into composition! And instead of saying “Oops, we must have gotten our theory wrong” and fixing the problem, which would have been the proper thing to do, people instead decided that they were involved in a new distinct field of study called “analytical theory”. That way, they didn’t have to discard the erroneous ideas to which they had become attached; they could simply relabel their occupation and move down the hall.

Sadly, people do this kind of thing all the time, and not just in music. The modern concept of religion is another example. Once upon a time, people believed that supernatural agents such as gods were needed to explain the natural world; then along comes science and what do people do? Instead of simply biting the bullet and admitting that the whole God theory was just plain wrong, they invent the concept of non-overlapping magesteria and assign new purposes to religion (“it gives us morality” or “provides meaning and purpose”, etc.).

My comment has to do with that last paragraph. You sound like someone who doesn’t subscribe to religion; is that a safe assumption? I get the impression that you believe that science can sufficiently take the place of God – that “God” was simply an ignorant human’s substitute for something he didn’t understand. It also seems that you believe that now science can explain those unknowns, and therefore the concept of God is not rational or necessary.

How do you account for the IRrationalities in science? For example, science can THEORIZE that everything we currently see originated in a miniscule particle that exploded and gave us all the building blocks of our universe that have, by chance over billions of years, come together in an infinitely precise way to give us the amazing functions of complexity in a single cell of the human body (let alone in the trillions of cells that all work together in the body as a whole).

BUT YET, where did that miniscule particle that exploded in the Big Bang come from?

Science can’t explain that. I believe it would take just as much faith to look to science – which is based on the discoveries of human beings – for the answers to our existence and the wonders of our universe as it does to believe that there is an all-powerful God who created everything. Actually, it’s more comforting to believe in God because there is a terminus in that: anything that we can’t reason out (such as where did God come from, or why did he create everything) can be attributed to the almighty power of God that is beyond us to understand and there it stops, whereas with science there are certain things that it will NEVER be able to answer.

Just my two cents worth!

No, I haven’t abandoned the idea — just like I haven’t abandoned the idea (believe it or not) of answering funkhauser’s latest question.

Alas, in thinking that I was going to be able to do these things imminently, I appear to have been a victim of the planning fallacy.

I actually have a rather large number of unfinished drafts for various posts and pages. Some subset of these will be finished over the weeks, months, and years to come.

(Funkhauser won’t have to wait for more than another month or so. The mathematics lectures will take longer, however.)

I thought #25 was humorous; I mean when did ‘they’ start calling it ‘classical’ music any way?

Anyway I thought there were going to be some lectures on Mathematics, but maybe you abandoned that idea? Just wondering.

Your analysis also reveals something I have believed for a long time–namely, in bass lines with a basic structure of 1 – 6 – 4 – 5, the 6 and 4 arise as neighbor tones to the 5. The 6 is an upper neighbor while the 4 is a lower neighbor.

It is very interesting to me to see the contrast between this analysis and the one given in the Part I, Ch. 2, Ex. 1 of “Structural Hearing” by Felix Salzer. Of course, Salzer uses a bizarre mixture of harmonic progression and linear progression in his analyses, but they are still of a largely Schenkerian/linear nature.

In his analysis of this piece, Salzer asserts that the Bb – G – Eb – C in the bass line is the result of an inversion of an ascending major second Bb – C to obtain a descending minor seventh, followed by a “filling in” by thirds to get Bb – G – Eb – C.

I personally like your analysis more, because I it seems more “structural” (i.e. new bass notes arise as neighbors to more fundamental notes). But I’d like to hear your thoughts on it. What is your reaction to Salzer’s analysis of the bass line? Why would you disagree, and why would you argue that your analysis is better?

What term should be used for “European art music from the time of Hadyn, Mozart, and early Beethoven”? If you ruled the world and no-one used “classical music” to mean “art music” or something similar, “Classical music” (like “Baroque music” to refer to the art music of an earlier time) would do fine. But as long as “Classical music” gets used with other meanings, there seems to be no good term.

A well-known characterization of the real number
system is as the unique complete ordered field (up
to isomorphism). Hence, if one wants to extend
this field, one will have to give up either
ordering or completeness (or perhaps both).

If one gives up ordering, one can construct an
algebraic field extension of the real numbers,
which has the pleasant property of being
algebraically complete as well as complete in the
sense of convergence. One can then proceed
to extend the operations of analysis to functions
over this larger field and finds that, even when
one is only interested in some question involving
real numbers (say deriving trigonometric
identities or computing a definite integral),
one can often save labor by first working with
complex analysis and then taking the real part
at the end of the day. (As Watson once quipped,
the shortest path between two real facts often
passes through the complex plane.)

Suppose instead that one wants to extend the
real numbers to a larger ordered field. One
then needs to give up completeness and arrives
at non-standard analysis. The epsilons arise
naturally — one statement of completeness is
Archimides principle, the negation of which is
tantamount to the assertion that the field
contains infinitessimal elements.

The basic methods of non-standard analysis are
reminiscent to me of complex analysis. Just as
in the latter, one analytically extends real
analytic functions to complex functions in a
unique manner so too, in the latter, one extends
real functions defined in first order logic
to non-standard functions in a unique fashion.
Just as the principle of persistence of relations
allows one to carry over functional equations and
differential equations to the complex plane, so
too does the transfer principle work.
Just as one might take the real part at the end
of some complex operations, so too one often
takes the standard part after some non-standard
operations to obtain a result in good old
real analysis.

As for the ultrafilters, they arise from looking
at limits of sequences algebraically. (I learned
this from Ebbinghaus’ wonderful book on numbers.)
When talking about taking limits of sequences,
what we are considering is assigning numbers to
sequences is some reasonable way. As for what the
term “reasonable” might denote in the last
sentence, we presumably don’t want to assign
numbers to sequences haphazardly, but in a way
which is consistent with the structure; at the
least, we might want the limit of a sum of sequences
to be the sum of limits and likewise the limit of
products to be the product of limits, for instance.

Algebraically, this desideratum about sums and
products says that the limit operation should be
a morphism from the ring of sequences to that of
real numbers. Now, in real analysis, we only
obtain such a morphism for the proper subalgebra
of convergent sequences; let’s see if we can do
better by picking a different field as target of
the morphism. A basic result of commutative algebra
states that a morphism from a ring to a field is
isomorphic to quotienting by a maximal ideal.
Hence, we should llok for the maximal ideals of
the ring of real sequences. A little work shows
that these are in one-to-one correspondence with
the maximal ideals of the Boolean algebra of
subsets of the integers, which are trivially
related to ultrafilters. This too, is a parallel
to complex analysis — just as one can obtain
complex numbers by extending the real number
system so as to make every polynomial have a
root, so too one can obtain the non-standard
nunmber system as an extension in which every
sequence has a limit.

By the way, I am no more put off by the term
“non-standard” than by the term “imaginary”
(which could suggest that complex analysts are
dayreamers) or the term “degenerate” or the
term “irrational” — mathematicians have a
habit of using negative, even pejorative terms
to describe new concepts which challenge
naive intuition.

Agreed, and thanks for your reply. I suppose I was reacting (or over reacting!) to your statement

“Rameau did recognize that inverted chords are essentially equivalent to those in root position.” I do not find this to be the case, the inner tension of a 6/4 chord is completely different from root position triad.

I will admit to finding Rameau’s works quite hard going, often a real maze, so it is likely I am getting some of it wrong. I have ordered the Thomas Christensen book and will see if it makes things more clear. I do feel I understand the principles of functional harmony and I also understand how it can be re-considered from a more linear perspective. It is obvious that voice leading is part of Rameau’s ideas, my main point is that it was a bad idea to replace thoroughbass practice with the theory of fundamental bass even if it gets quicker results, and modern figured teaching generally bass does not make this distinction.

(1) Schenker certainly did not subscribe to Rameau’s principles. The point I was trying to make is that Rameau’s thoughts are driven by voice leading more than history has acknowledged. We can make some indirect connections between the two, however. Simon Sechter was a prominent music theorist and teacher in the middle of the 19th century. He taught Schubert and Bruckner, and Bruckner was Schenker’s teacher. Of all his predecessors, Schenker’s theory is most like Secther, who had a Stufentheorie that clearly predates Schenker’s. Furthermore, Sechter was a very conservative thinker who still thought of music in terms of the Rameau’s fundamental bass.

(2) Reameau analyzes “cadential 6/4 chords” as suspensions or as chords of supposition. His analysis depends on the preceding chord. When the tones of the 6/4 are prepared—when they follow a tonic chord for instance—Rameau analyzes the chord as a suspension. Otherwise, he thinks of the chords as having a supposed bass. Again, voice leading is very important. So, at least in relation to the 6/4 chord, Rameau does not seem to be propagating an “‘equivalence’ dogma” as you suggest.

I don’t mean to be a Rameau apologist and actually feel sort of strange in the role, but I think it is important get the facts straight before criticizing anyone.