Schoenberg op. 19 no. 2

August 10, 2007

Score here (p.2).

Analysis here.

Read it and weep. ūüôā

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: ii√ł43 – 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 “√ł”, 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).