Pachelbel’s Canon

April 23, 2008

As you might expect, the demise of the IMSLP has put something of a damper on my grandiose plans of analyzing musical works on this blog. Today, however, we’re in luck, as Wikipedia provides all the source material I’ll need for this post.

The context for this is a question, of sorts, posed by a commenter named “funkhauser” (the reason I say “of sorts” will hopefully become clear):

It seems to me that a surprisingly large number of progressions of 8 chords found in say, your favorite piece, have a IV chord as the “middle” chord (i.e., if each chord lasts a quarter note and we are in 4/4 time then the IV chord is the first chord of the second measure). Two familiar examples are Pachel[bel]’s Canon and the beginning of Bach’s Air on the G string. The chord progressions are, roughly:
I – V – vi – iii – IV – I – IV – V
I – iii – vi – vi7 – IV – V7/V – V – V7

(…)However, I can’t think of an 8-chord progression I’ve heard in which the V chord is the middle chord…Imagine putting the V in the place of the IV in Pachelbel’s Canon or Bach’s Air. It would completely change the feel of the piece (and arguably, it would ruin it).

(…)What I’m wondering is: In Westergaard’s theory can we derive the important difference in function between IV and V? And can we derive the fact that the IV should occupy stronger beats and larger time spans than V?

Tisk tisk. It’s obvious that the questioner has not yet managed to throw off the Rameauvian shackles, and is still laboring under the impression that musical passages are constructed by juxtaposing “chords” in time. Well, funkhauser, you’ve come to the right place — disabusing innocent souls of this mistaken notion has become one of my missions in life.

The best way to start, I think, is to take a look at these passages and see what’s actually going on. Here’s how to construct the opening of Pachelbel’s canon:

1. The underlying basic structure is the usual \hat{3} - \hat{2} descent (with the \hat{2} on its way to \hat{1}, of course):

2. These structural lines will be realized as three textural lines, with span pitches assigned as follows:

(When I say “assigned” I technically mean borrowed, of course.)

3. The top two lines will both descend from the upper note to the lower:

4. The A is delayed by a lower neighbor, in familiar fashion:

5. We connect the F# to the A and the D to the F# by step motion. In fact, we’d like to have continuous quarter notes in these two voices, so on beat 3 of m.2 we’ll also elaborate the C# by a lower neighbor passing tone in the top voice (producing functional parallelism alignment with the bass) and borrow a G from the bass for the middle voice:

6. Actually, we’d like to have quarter notes in all three voices, so we elaborate the bass by means of borrowing :

(The A and the F# are of course borrowed from the span pitches of stage 2 above.)

7. Now, since this is supposed to be a canon, we’ll present the voices one by one.

8. Finally, this is how the texture is actually realized, in terms of which instruments play what.

Now, having analyzed the passage, let’s see if we can address funkhauser’s question. The first thing to note is that nowhere in the above derivation sequence is there any mention of “chords” at all. As a matter of fact, I didn’t even bother to check whether the progression claimed by funkhauser

I – V – vi – iii – IV – I – IV – V

is “accurate” or not — so that as I’m typing this, I literally don’t know what the “chords” of this passage are! I It’s important to emphasize this, because I just got through analyzing the passage in precise detail, attributing a specific function to every single note, and I have the passage itself, as well as my analysis of it, firmly entrenched in memory. Indeed, I can’t mentally replay the passage without instantly and simultaneously reconstructing my analysis. And yet — and yet — when it comes to selecting the appropriate Roman numeral for each of these quarter-note simultaneities, I am — at least at this immediate moment — about as clueless as a typical freshman theory student. (Though I do already know the first one will be I and the last one V.)

Having made that point, let me now pause to reflect on what the chords are…Okay, yes, funkhauser has got it “right”; though I suppose there is an ambiguity about beat 3 of m.1, since there are only two distinct pitch-classes in that simultaneity. Come to think of it, the same is true of both “IV” chords in m.2. Oh, and it’s also true of the very first chord!

(Notice how very different this type of thought is from the instinctive, intuitive reasoning that I used to construct the above analysis. Actually, “instinctive, intuitive” is not the correct description; what I meant to say was specifically musical. Whereas what I am doing here, in verifying funkhauser’s chord progression, is the totally abstract (if trivial) mathematical problem of verifying that two finite sets are equal to each other.)

Funkhauser asks about the difference in function of the IV and V chords. What I would like to point out is that there is no “IV chord” at all! The simultaneity on beat 1 of m.2 is just the coincidence of two passing tones, and that on beat 3 is just the coincidence of two neighbors a passing tone and a neighbor. To pick out these chords as fundamental objects in their own right (and as the same fundamental object, no less!), is to carve up musical reality in the wrong way, like putting dolphins in the same category as fish.

Strictly speaking, then, the answer to funkhauser’s question is “mu” — i.e., “your question depends on incorrect assumptions”. The “chords” of harmonic theory are simply not legitimate music-theoretical entities, any more than Earth, Air, Water, and Fire are chemical elements. Yes, these four things do exist, but they don’t play anything like the theoretical role that people once attributed to them. In fact, today we understand that not only are they not fundamental, but they’re not even the same kind of thing: “Earth” is a planet, “air” is a state of matter (gas), “water” is a chemical compound (H2O), and “fire” is a process (combustion).

So it is with “IV”, “V”, and all the rest. Yes, there are collections of notes in musical compositions to which you could give these labels, but to do so is to presuppose the wrong theory of music.

Like Aristotelian chemistry, harmonic theory may not seem obviously wrong until you’ve had considerable experience with the alternative. This explains why I invariably get reactions like “But…but…of course harmonic theory is correct (or useful) — look how ubiquitous progressions like I-IV-V-I are!”

Yes, and the “Four Elements” are also ubiquitous in the natural world.

For the moment, I will leave it as an exercise to come up with the correct analysis (or at least an analysis of the correct type) of the first two measures of the Air from Bach’s Third Orchestral Suite. Here’s a big hint:


The joy of “pathology”

April 12, 2008

Recently I had a conversation with a mathematician who had worked on the theory of Banach spaces early in his career, but had since left that particular subject. He explained that he had become disillusioned by the fact that “all the natural conjectures turned out to be false”; indeed, Banach spaces can have some “strange” properties, such as having uncomplemented subspaces, lacking a basis, admitting operators with no invariant subspaces, or admitting almost no operators at all. (Actually, in fact, they can even be unexpectedly well-behaved!) The last straw for this fellow, apparently, had been the Gowers-Maurey space (for which Gowers won the Fields Medal) , which has a whole bunch of “weird” properties.

Disparaging language is used with disturbing frequency by mathematicians to describe mathematical concepts. Examples are labeled “pathological”; objects are described as “badly behaved”; functions are called “nasty”; problems are said to be “ill-posed”. In a library once I encountered a book whose title actually was Differentiable Functions On Bad Domains — where “Bad” here is not the name of a mathematician, but the ordinary English word meaning the opposite of “good”.

You might think this is nothing but picturesque language — like calling a certain group “the Monster” — except that there are plenty of mathematicians who actually seem to think the way the labels suggest they do. The ex-Banach-spacer I mentioned above is only one example; spend some time among mathematicians and you will find many more. Indeed, such aversion to the unexpected has a distinguished historical pedigree, I am sorry to say. Who can forget the dismay with which Weierstrass‘s construction of a nondifferentiable continuous function was greeted? And even now there are still some people who are pissed off about Cantor’s discoveries, and who would sooner overthrow the standard axioms of mathematics than confront the “paradoxes” of the infinite. Even Hilbert, who had the good sense to regard infinitary mathematics as paradisal rather than paradoxical, nevertheless reacted with anger (!) to Gödel’s results on the limits of formalization, according to Constance Reid. One is reminded of the discoverer of the irrationality of \sqrt{2}, who, you will recall, was allegedly thrown overboard by his Pythagorean comrades.

I have never sympathized with this way of thinking. As far as I am concerned, unexpected “pathological” phenomena are a large part of what makes mathematics interesting in the first place. Indeed, this accords with the attitudes of other kinds of scientists with regard to their own fields. You generally don’t find physicists crying in agony about the discovery of black holes, or biologists resenting the existence of extremophiles. Why, then, do so many mathematicians insist on doing the equivalent?