J.S. Bach: Air from Orchestral Suite No. 3, mm.1-2

May 31, 2008

Recall that in a previous post I challenged readers to analyze the first two measures of the Air from Bach’s Orchestral Suite No. 3 in D major (a piece, incidentally, that might be better referred to as “Air Off The G-String” than by its usual nickname). The time has come to reveal the answer.

In the Pachelbel analysis, we started from the underlying basic structure and showed how the passage was constructed via the Westergaardian operations. This time, for the sake of variety, we’ll proceed in the reverse direction, starting from the passage itself and “undoing” the operations until the basic structure is revealed.

Our passage is the following:

Stage 1

Call this Stage 12. The first thing we’ll undo are the explicit arpeggiations in the first violin and continuo lines:


Actually, I did a bit more than that, as you can see. I skipped a stage in which the first violin part looks like:

How did I know that D was the span pitch of the second half of beat 2 rather than C#? That is, why did the first violin part not reduce to:

Is it because G#-E-B (or even G#-E-B-D) is a Certified Chord, whereas G#-E-B-C# isn’t? Fat chance! As an exercise in eliminating harmony, see if you can explain the real reason. (I’ll likely explain it in a future post, but probably only after we’ve formally developed more Westergaardian theory. Hint: It has nothing to do with Certified Chords.)

Eliminating the borrowed G and B from the first violin, we obtain stage 10:


What an odd interpretation of beat 2! Instead of hearing a passing motion from E to C, I am interpreting the E as a borrowing from the viola line:


(Note also the elimination of the A borrowed from the second violin line.) Why on Earth is this interpretation to be preferred to the seemingly simpler one? The answer is that the seemingly simpler one isn’t in fact so simple. Notice that the D in the second violin line is left hanging (ITT, p. 30), and therefore not displaced, after beat 1. If the D in the first violin line were to be interpreted as a passing tone, that would leave us without a D among the sounding span pitches of beat 2. However, we know from the C# of beat 3, as well as from the fact that D was left hanging in the second violin, that D must be a span pitch for some span that includes beat 2 (deeper levels will make this clearer; see below). We would therefore be compelled to regard the second violins’ D as being temporarily displaced during beat 2; that is, it must move by step to some note borrowed from another line. (The only alternative would be to regard it as (entirely) undisplaced during beat 2, but this is made difficult because of the simultaneous E: since in this scenario we’re not considering D as a local span pitch of beat 2, we’re left with understanding an implicit dissonance, which is quite problematic indeed.) Since E is a span pitch of beat 2 and C# is not, we must therefore hear the D-line as moving up to a borrowed E during beat 2. But why should we go through the trouble of understanding such a conceptually difficult situation as the D-line effectively “merging” temporarily into the F#-E line? Given the stated step motion D-C# in the first violin, isn’t it easier to regard that D as a span pitch over the span of beat 2?

Stage 8 shows transferred pitches (ITT, sec. 7.7) reassigned to their rightful homes:

Stage 1

This stage represents the transition from instrumental lines to structural lines; I have symbolized this by switching from the alto clef to the treble clef in the third line.

Next the transferred pitches are reassigned to their rightful registers:


The suspension in the top line is removed:


Rearticulations in the bottom three lines:


Rearticulation of a suspension in the second and third lines; chromatic step motion in bass:


Suspensions eliminated:


Neighbor note removed:


Finally, then, we have the basic structure of the phrase:


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?

Coming (back) soon: this blog!

February 29, 2008

That’s right: I am pleased to announce that, after a hiatus of more than four months, regular posting (or quasi-regular, at any rate) will resume shortly – probably some time in early-to-mid March

Many thanks to all of you who sent greetings during the holidays.

(As for what this post is doing here, how could I resist creating an entry dated February 29 while I had the chance? 🙂 )

Schoenberg op. 19 no. 2

August 10, 2007

Score here (p.2).

Analysis here.

Read it and weep. 🙂

Update (8/11): A couple of minor errata in the analysis: in m.6, the G in the bottom staff should be parenthesized; likewise for the half-note G-B dyad in m.7. About the parenthesized notes in general, I should add that their purpose is to clarify the function (or “meaning”) of the surrounding notes in the same line; they are not meant to be understood literally as “hallucinated” pitches sounding at the same time as the other notes. (Of course, since all the notes in the analysis are conceptual anyway, the distinction may not be all that important — especially in view of the fact that every parenthesized note is doubled,  up to pitch-class,  by a “real” conceptual note, which is one of the things that makes this piece so easy to understand.)

For those who are wondering how this analysis fits in with other recent posts (not that it necessarily has to, of course!), the point is the following: the traditional classification of op.19 no 2 (and many other 20th century works) as “atonal” depends upon a bad theory of “tonality”; that, indeed, was the main point of my Chomsky post. After all, how was it originally decided that this piece wasn’t “tonal”? Presumably, someone looked at the score, saw the final sonority, or the one in m.6, and said “What chord is that?” After looking around further, they proclaimed, “I don’t see a coherent harmonic progression anywhere in here.” Perhaps they even asked, in desperation, “Where’s the V-I cadence?”  

If, however, we take the Westergaardian view  as our point of departure, such questions never get asked. Our theoretical vocabulary does not refer to chords and progressions, but rather to lines and elaborations. Consequently, the fact that a particular coincidence of notes is “unusual” is never an issue, so long as the notes are individually comprehensible as elements of lines, any more than the fact that 3574.37562 is an “unusual” number poses problems for arithmetic. (I suspect most musicians intuitively realize that this should be the case, but attribute their analytical difficulties to the inadequacy of music theory in general, rather than to the true culprit, which is the particular music theory they have been taught.) 

The above Schoenberg analysis is, I hope, a dramatic illustration of the power of this type of theoretical framework — dramatic because it shows how easy it is to hear a so-called “atonal” composition as “tonal” once we start thinking about “tonal” music in the right way. Contributing to the drama is the fact that Westergaard himself never intended his “tonal theory” to apply to the middle-period music of the Second Viennese School, but yet it does apply, simply because it was the right way to approach music in the first place.  

How to make a Chopin Prelude

August 5, 2007

Or the first twelve measures of one, at any rate. (The score to the piece [op. 28 no. 4] can be found here.)

In four parts (be sure to enlarge all the way):

Stage 0:

Stages 1-2

Stages 3-5
Stages 5-8

(The rest is left as an exercise for the reader! :-))

 Now, what prompted this? Answer: this 2004 post by Scott Spiegelberg (who, incidentally, was kind enough to link to me — much appreciated!). Alas, here I go again, discovering another interesting music blog, only to immediately start using it as a foil.

Of this piece, Spiegelberg has some interesting and worthwhile things to say; unfortunately, his comments are also contaminated with harmonic theory, which never illuminates and usually obscures. So let’s see if we can do anything to help. Spiegelberg says:

[The Prelude] starts innocently enough with a simple tonic chord, though the E is not in the bass so the chord is slightly unstable.

The beginning of this analysis seems as innocuous as the beginning of the piece itself; but, as in the case of Chopin’s opening, the veil of innocence merely serves to conceal difficulties that will rear their head in due course. In this case, there is already something subtly misleading about saying that a work “starts with a tonic chord”. It’s as if the composer came up with the opening of the piece by consulting a list of possible “chords”, and choosing to go with the tonic rather than the dominant or perhaps the doubly lowered raised submediant (if the composer happens to be, say, Max Reger). One presumes that after making the choice, the composer then goes back to the list and decides what the second chord of the piece will be. (Oh, and I almost forgot — he has to pick an “inversion” too!) And so on.

Although this model of musical construction sounds ridiculous (or so I hope), it is nevertheless precisely the model that is being invoked whenever anybody speaks of “harmony” or “chord progression”. It is the model that some people think music students need “a thorough grounding” in before they can study Schenker. It is a model so hallowed by tradition that even Schenker needed several decades to break free of it — and he almost succeeded.

What would be a better model? Returning to the Chopin work under discussion, I think that, instead of saying it begins with a “tonic chord”, we ought rather to say that it begins with a B in the top voice, which is counterpointed by a G in the bass, along with a couple of inner voices starting on B and E. Each of these notes then sets off on a journey of its own through some region of diatonic space — in the process of which it elaborates (or “composes-out”) some particular gesture that the composer wished to convey.

Spiegelberg continues:

The next chord is the dominant chord, though with a suspension: the E refuses to let go.

Except for the “next chord” business, this is very well put.

When this suspension does resolve, Chopin “misspells” the chord with an Eb instead of a D#. The melody turns this dominant chord into a diminished seventh chord, which resolves as a common-tone chord to a secondary French augmented-sixth chord!

This is where my head starts spinning, so let’s see if we can translate the harmonic jargon into English. To say that “the melody turns this dominant chord into a diminished seventh chord” is an extremely awkward way of saying “the B moves up to C”; but it also carries the suggestion that there is a sort of “conspiracy” among the voices — as if they said, “let’s now form a diminished seventh chord!”. Now, conspiracies of that sort can certainly happen in music, but this is not one of those occasions. Here, it seems, we simply have a note moving to its upper neighbor, without any concern whatsoever for what its fellow notes are doing at the same moment. (Just as in real life, it takes quite a lot of work to establish a musical conspiracy.)

“Secondary French augmented sixth chord” — oh, boy. Well, a “French” augmented sixth chord is the kind that has scale degree 2 in it; thus in E minor we would be talking about pitch-classes C, A# (the augmented sixth), E, and F#. However, our chord is a secondary chord, meaning evidently that it is a French sixth when viewed from the perspective of some other key. Now, the pitch-class content of the sonority I presume Spiegelberg is talking about (namely the one on the first half-note of m.3, immediately following the “diminished seventh chord”) is F, A, Eb, B; if we thought of the Eb as D#, this would spell a French sixth in A minor.

What Spiegelberg is claiming, then, is that, at least for the first half-note of m.3, we are locally in A minor — and in particular the Eb is a raised scale degree 4! Needless to say, I have absolutely no idea how one could arrive at such an analysis: as far as I am concerned (see the graphs above), there is nothing in the entire Prelude (least of all in the first three measures) that requires one to think in terms of any key other than E minor — not so much as a single secondary dominant, let alone a secondary French sixth!

This stands in stark contrast to what Spiegelberg says:

By this point, only the third measure, the listener is quite confused as to where tonic is, even though the chords progress by very small steps with many common tones.

Not only am I not confused about where the tonic is, I don’t even see how one could be confused about that in this context. What note besides E is even a candidate for tonic status?

Spiegelberg is apparently quite serious about A minor being a contender:

The augmented-sixth chord does not resolve correctly, instead shifting to a chord progression that fits best in the key of A minor: iiø43 – viio42 – V7. By half-steps the dominant chord gets transformed, leading us back to the key of E minor. A minor is hinted at several times, and the final cadence of each phrase (there are only two phrases in the 25-measure prelude) includes an oscillation between the dominant B7 chord and the A minor triad.

First we have to decode the Roman numerals: ii43 (the “ø”, meaning half-diminished, is of course redundant in a minor key) is the “second inversion” of ii7, which means the fifth is in the bass; ii7 in A minor is B-D-F-A, so we’re looking for F-A-B-D (or some other arrangement with F on the bottom). Likewise, vii42 is a version of vii7 (G#-B-D-F) with F in the bass. V7 (the only one I can spell without having to think!) is of course E-G#-B-D.

Well, we do indeed find an instance of that particular (partially-ordered-by-register-pitch-class-set)-sequence (for that is what a “chord progression” is) in mm. 3-4 — provided, of course, that we don’t take into account the C on the last quarter of m.3! (Remember that the presence of an exactly corresponding C in m.2 compelled Spiegelberg to posit a “diminished seventh chord” for that timespan — what’s the difference here?) The question, however, is whether this “progression” has the analytical significance that Spiegelberg is attributing to it. I don’t see any good argument for this at all. A “V7 chord”, for example, is by definition composed of scale degrees 5, 7, 2 and 4 — but is the E in the bass in m.4 a scale degree 5? Is the G# a scale degree 7? If so, when exactly did E cease to be scale degree 1, and why?

(In case you’re worried that my question reflects too much of an “in time” analysis rather than a “final state” analysis, I can put it this way:  why is the A-minor analysis of these notes given by Spiegelberg preferable to the E-minor analysis that I have given above?)

Spiegelberg and I agree that there are exactly two “phrases” in the prelude (although this is something of a contradiction on his part, since he has analyzed mm. 3-4  as a cadence in A minor!). However, to speak of “an oscillation between the dominant B7 chord and the A minor triad” is once again misleading, even if literally accurate. Yes, we do get the pitch-class set C-E-A occurring in measures 10 and 11; but this is the purely accidental result of simultaneous neighbor-note motions in the two lowest voices — a very mild conspiracy, in which the A is not involved at all (it just happens to be there, like an innocent bystander). In fact, far from being the “root” of an A-minor triad, this A is actually a dissonant 7th, as you will see by referring to Stage 2(b)  of the above analysis (second page).  Needless to say, I do not understand how this phenomenon could possibly be said to reinforce any sense of A minor, which is what Spiegelberg implies.

This prelude is all about the tensions between the melody and the harmony, with the harmony clearly winning. But what is so striking is that the exotic harmonies are created by simple means, small little movements of the left hand, and this slow harmonic rhythm creates such emotional intensity

Well, since I don’t believe there is such a thing as “harmony” (in the traditional sense), obviously I can’t agree that the harmony “wins”. The fact is that these “small little movements of the left hand” are perfectly comprehensible — if ingeniously and subtlely timed — melodic motions through various parts of the E-minor scale. It is, indeed, Chopin’s highly refined sense of timing (and not any exotic modulations to other keys) that is responsible for the mysterious magic of this Prelude.

Spiegelberg concludes with a challenge:

For a real brain twister, try to analyze the second prelude!

Sure! I’d be happy to do this, if there’s any interest. The A-minor Prelude is, if you will, opposed to everything the E-minor Prelude stands for: it, unlike the E-minor, actually does keep you guessing about where the tonic is, and doesn’t in fact reveal the truth until the very end. Quite a contrast!

Contempt for “Contempt”

August 5, 2007

One of the nice things about WordPress is that it allows you to create “pages” that are separate from blog posts, thus allowing the user to turn his or her blog into something like a complete personal website. Terence Tao in particular has made excellent use of this feature: his blogsite contains a number of pages on career advice, for example, that are independent of the regular blog. Unfortunately (and as if to compensate), WordPress also places stifling constraints on the visual layout of the site: you have to choose from among a list of predesigned “themes”, which you essentially cannot modify. Now, it turns out one of the properties of the default theme (the visual style of which I happen to like) is that it lists, on the sidebar, all of the pages you create — and there’s apparently nothing you can do about it! Since (as I will explain presently) I am envisioning creating independent pages numbering perhaps into the hundreds(!), this obviously is not an acceptable state of affairs.

So, I had to change themes. But (of course) none of the available themes are really satisfactory: they either have the same profligate-linking property of the default theme, or else they are aesthetically inappropriate or displeasing. The best compromise seems to be the one I have currently selected (called “Contempt”), which is described as a “more professional” version of the default theme (“Kubrick”). There are, however, at least three problems with “Contempt” : (1) the color scheme, which would be perfect if this blog were only about, say, mathematics and linguistics, but which contains far too much “technical” gray for a blog one of whose principal topics is something as “artsy” as music (yes, of course I know that’s a spurious contrast to make, but we’re talking about colors here, folks); (2) the fact that the text size used for comments is tiny-issimo; and (3) it lacks the feature of “Kubrick” where the “archive” view of a post links to the posts chronologically before and after it — a feature I like because it encourages browsing.

So, people at WordPress: please fix this and give me what I want, which is “Kubrick” with the link format of “Contempt”. Better yet, allow me to choose exactly the links I want to display, regardless of the “theme”.

Update (8/07): Just as I was beginning to really like the look of the “Contempt” theme, I have discovered a highly inconvenient bug that is specific to this theme: it is impossible to highlight text for copying/pasting. This really pisses me off. I mean, how hard can it be to have f***ing text highlighting work properly, like on every other theme?

So, for now, if you want to copy text from my posts, I suggest doing so from the HTML source file (which your browser hopefully allows you to access).