Harmony: still undefended

I would have thought that my vigorous theoretical attacks on the venerated notions of “root progressions” and the like would have elicited vigorous theoretical defenses of those notions from those who do all that venerating. Alas, I have thus far been disappointed. Here’s a brief recap of some points that have been made by various interlocutors:

• The Texas Tech Theory Department noted that Roman numerals are easy to teach. (So are lots of bad theories; the headaches come only later, when one actually tries to apply them.)
• Scott Spiegelberg pointed out that Heinrich Schenker wrote a book called Harmony and even used Roman numerals in his magnum opus Free Composition. (The implication being, I guess, that Schenker must not have had any problem with the ideas of Rameau!)
• Matthew Guerrieri expressed a lack of interest in reexamining harmonic theory in view of its alleged “usefulness” (lucky for him that it didn’t stymie his own musical education for a decade), and the fact that, after all, no theory will ever be perfect (hence no need to bother replacing a bad theory with a better one!).
• Michael Monroe suggested that it might be unreasonable of me to be so emphatic and impractical in my approach (a fair criticism perhaps, but hardly a defense of harmonic theory).
• And, in the latest installment, Scott Spiegelberg, upon returning to the blogosphere, declines to address the specifics of my argument, being content merely to assert that harmony does, in fact, exist.

Spiegelberg’s “defense” of harmonic theory is particularly disappointing, because I actually went to the trouble of explaining in detail how it led Spiegelberg himself astray in his analysis of Chopin’s E-minor Prelude — and giving my own harmony-free analysis of the piece for comparison. Spiegelberg not only declines to challenge my analysis in favor of his own, he explicitly says that he doesn’t have a problem with my analysis! Why then does he bother trying to “defend the honor” of harmony, when he apparently agrees with me?

The answer, perhaps, is that he misunderstands what my “voice-leading analysis” (as he calls it) represents. It’s not merely a description of one aspect of the music (“voice-leading”); it’s a derivation sequence of the actual notes of the piece. Its purpose is to provide enough information to allow one to identify the specific “grammatical” function(s) of every single note in the score (i.e. whether it’s a passing tone, neighbor, borrowed tone, or what have you) — with the implication that reference to “harmony” is nowhere necessary for this purpose. If Spiegelberg disagrees with this conclusion, then he is obliged to tell us which notes in the score require the invocation of harmonic theory in order to be understood. Or, at the very least, he needs to explain how the virtues of a “harmonic” analysis (whatever those may be) are not provided by my analysis.

According to Spiegelberg, I am “attempting to move us back to pre-Rameau (1723) days where chords don’t exist, everything is counterpoint alone”. The implicit praise of Rameau aside, this is not accurate. It would be fairer to say that Spiegelberg and others are attempting to keep us back in pre-Westergaard (1975) days where notes weren’t properly accounted for.

Spiegelberg also claims that

Heinrich Schenker’s greatest realization was that the rules of counterpoint – set by 16th century compositional practice – had been altered by the evolution of tonality.

If that were true, then Schenker’s historical significance would be that of just another theorist with his own set of “exceptions” to the “rules of counterpoint”. Practically every theorist in Western history came to the “realization” that the “rules” that were set by his predecessors “had been altered” by the evolution of musical practice (whether or not they actually favored such “alterations” of “the rules”). Why else would they bother to keep writing new books?

No, indeed: Schenker’s most important contribution was the idea that complex musical structures can be explicitly and systematically understood as elaborations of simpler ones — all the way down to the simplest possible musical structures (the Ursatz, and finally the tonic triad itself).

If you study Schenker’s works chronologically, you’ll notice that the concept of harmonic progression becomes less and less fundamental as time goes on and his analyses become more refined. By the time you get to Free Composition, the only irreducible “progression” left is I-V-I; all the others have been reduced to “contrapuntal-melodic” or “voice-leading” events — in other words, operations on notes (see Figures 14-19 as well as §278; I-V-I itself was finally disposed of by Westergaard in section 8.2 of ITT). Of course, he had already said, in Masterwork II (“Elucidations”):

There are no other tonal spaces than those of 1-3, 3-5, 5-8. There is no other origin for passing-note progressions, or for melody. The first passing-note progression comprised by the Urlinie generates dissonance (second, fourth, seventh). Dissonance is transformed into consonance because only consonance, with its tonal spaces (as shown above), unlike dissonance, can promote new passing-note progressions and freshly burgeoning melodies. This comes about through prolongations in ever-renewing layers of voice-leading, through diminution, through motive, through melody in the narrower sense; but all of these hark back to the initial tonal space, and to the passing-note progressions comprised by the Urlinie. As the outcome [my emphasis -- J.C.] of all these transformations and unfoldings, there emerge the harmonic scale steps…

Now it’s true that if you don’t read Schenker very carefully, and in context, you might (depending on the passage) come away with the impression that he’s just as much a proponent of harmonic theory as everyone else. For instance, the very next sentence after the passage I just quoted is:

Despite the notes being sounded successively, the arpeggiation of a chord remains a harmonic phenomenon…

If, however, you’re reading this passage in the larger context of his analytical work, you’ll realize that all he really means is that the notes of the arpeggiation are to be thought of as sounding simultaneously at a deeper level of structure. The point here is that he’s contrasting the conceptual status of melodic skips (which create compound lines — i.e. those that are understood as generated by more than one line at a deeper level) with that of steps (which serve to define a single line). The defining notion of harmonic theory, namely that of “root progression”, is not involved here at all.

But we needn’t actually go into these kinds of subtleties in order to settle the question of which side of this debate Schenker would be on. For the purpose of dismissing once and for all the idea that Schenker believed in harmonic theory in the usual sense of the term, I submit for your consideration his essay “Rameau or Beethoven? Creeping Paralysis or Spiritual Potency in Music?” from Masterwork III (the one with the Eroica analysis). Here’s how it opens:

The histories of music all draw attention to Rameau’s Treatise on Harmony of 1722, extolling it as a major contribution to the field of music theory. They proclaim the new doctrine of fundamental bass and the inversions of a chord, of chordal construction in thirds and harmonic relations among those chords, as contained, virtually full-grown, in its pages.
What no music historian — or theorist, for that matter — has yet realized however is that even while J.S. Bach and Handel were still living, and before Mozart and Beethoven were even born, with this doctrine [i.e. "fundamental bass and the inversions of a chord, of chordal construction in thirds and harmonic relations among those chords" -- J.C.] the seeds of death had already been sown in [music] theory, and indirectly also in music composition!

Later, he continues:

In that he reduced all musical phenomena to fundamental basses and the progressions proper to them, Rameau detached what not even the layman can avoid seeing in front of his nose, namely the superimposition of notes, from the flux of horizontal voice-leading in which every superimposition has its origin [my emphasis -- J.C.]…

Still later,

What is more, even granting the significance of fundamental basses on his terms, Rameau really ought to have asked himself the question: Why is it that if generation of content (i.e. diminutions and their cohesion) is solely a matter of the vertical axis and its cadences — why, if this is so, do not these selfsame cadences left to their own devices give rise to a perpetuum mobile, so to speak, thereby turning the content into a perpetuum mobile too? What is it that offers resistance to such a perpetuum mobile? Where does the impetus come from ever to bring a composition to a close? If not from the vertical and its cadences, then does it not perhaps come from form? But where does the latter come from? The vertical and its cadences? Or is it not much more likely that it comes from the horizontal, and from the impetus at work there, the impetus of the law of the passing note? Instead of giving primacy to the horizontal, as the composing-out of the fundamental chord that yields content, and subordinating to it as a mere counterpoint the vertical, with its first arpeggiation of the fundamental chord and the derivatives of that, Rameau right at the outset shunned the horizontal in favor of the vertical, which offered his more lackluster French musical taste the enticing possibility of a cosier schematization. So it became Rameau’s sorry task in life to lure the musical ear away from voice-leading, instead of being the first to identify the latter and its laws.

And so on.

Let not my opponents in this debate dare cite Schenker as one of their own, whatever Roman numerals he may have written.

I’ll observe that, except for the francophobic outburst, Schenker got it exactly right in that latter paragraph. It is the “impetus of the law of the passing note” (which in other contexts he might have called “the necessity of composing out the triad via the Urlinie” or something similar), and not the alleged phenomenon of “harmonic pulls”, that differentiates “free composition” (i.e. real music) from counterpoint exercises (as Schenker conceived them).

To conclude, let me restate my challenge to the would-be defenders of Rameau and his successors. Here are my three main attacks on traditional harmonic theory (in order of increasing controversiality):

1. It doesn’t explain anything that Westergaardian theory can’t explain (whereas Westergaardian theory explains quite a lot that harmony can’t explain).
2. When both kinds of explanations are available, the Westergaardian explanation is always to be preferred to the harmonic explanation.
3. The pedagogy of 20th century music would be greatly facilitated by adopting the Westergaardian viewpoint with respect to earlier music, because Westergaardian theory (or a slight modification thereof) is capable of unraveling the complex tonal structures that govern so-called “atonal” music. In contrast, harmonic theory has hidden these structures from us, because it does not permit a “tonal” analysis of music in which any collection of notes may be sounded simultaneously.

If 1. and 2. are true, that’s already enough to give harmonic theory the boot; 3. should just make us indignant that we had to put up with it for so long.

If you think that any of the above three claims are false, I challenge you to explain why. I’ll even help you out, by telling you what you would need to do:

• To refute 1., the weakest claim, your task will be the hardest: you will need to exhibit a passage of tonal music and show that Westergaardian theory is incapable of addressing some phenomenon in the passage that is successfully addressed by harmonic theory. Be prepared for two kinds of objections from me: (a) the phenomenon isn’t real (so you can’t just simply invoke harmonic theory in its own defense, and say something like: “the harmonic progression of this passage”); (b) you’re incorrectly characterizing the phenomenon (“what you think is Phenomenon X is in reality (Westergaardian) Phenomenon Y”).
• To refute 2. should in principle be a bit easier: all you need to do is compare and contrast a harmonic and a Westergaardian explanation of a passage or phenomenon and explain why the harmonic explanation is better.
• As for 3., I’d recommend holding off until I start posting some more analyses of this type (I’ve already done one on Schoenberg op. 19, no. 2). However, if you really feel strongly that this approach is destined from the start to be wrong, by all means, hit me with your best shot.

Well, there it is; now have at it!

Stay tuned…

The inevitable has happened: Scott Spiegelberg has returned fire on the harmony question. Unfortunately, I’ll be busy in meatspace over the next couple of days, so I won’t be able to get to it immediately — but fear not!

In the meantime (and because every blog needs a gimmick once in a while!), I’ll open a contest for the comments section: whoever can, after reading what Scott has to say, most accurately predict my reply, will win some prize that I’ll think up. (Perhaps I’ll follow in the great tradition of another Scott and let the winner ask me any question they want.)

Further clarification

Michael Monroe has weighed in on a recent post, raising some important issues. In this post I attempt to address them.

First of all — and I should perhaps have been clearer about this all along — while I may come across as a flag-waving radical, crusading for fundamental change in the way we approach music, such an attitude should not be attributed to Westergaard, and it is not really what his work represents. Indeed, I don’t think he would necessarily approve of all of the things I have said on behalf of his theory (in fact I’m sure he wouldn’t endorse — not immediately or automatically, anyway — my application of it to “post-tonal” music). He doesn’t even bother to mention anywhere in ITT that the book happens to overthrow harmonic theory! That’s because his primary purpose in developing his theory, and in writing the book, was not to revolutionize music theory, but was simply to explain the workings of tonal music to college freshmen (and other interested readers). It would probably help if I let Westergaard speak for himself. For the benefit of readers who haven’t yet ordered the book (everyone should get their hands on a copy, if they can find one!), let me quote a passage from the Preface (this passage is addressed to “the theorist who wonders whether this book has anything new to offer him”):

This book owes its greatest debts to Schenker, Fux, and Bernhardt. Its central pedagogical means — species counterpoint — is that developed by Fux in his Gradus ad Parnassum. But the end to which that means is applied is that developed by Schenker in Kontrapunkt. Fux’s goal, the Parnassus he wanted to lead the student to, was the ability to write imitation Palestrina. Schenker’s goal — and mine — was the ability to understand the complex and varied voice-leading patterns of actual eighteenth- and nineteenth- century music in terms of the simpler patterns available under the artificial contraints of species counterpoint. To that end I have

a. adapted the linear operations later developed by Schenker in Der Freie Satz to the rhythmic limitations of species counterpoint (Sections 4.1, 5.4, 5.6) and

b. tried to show how dissonant skips and skips to and from dissonant notes can arise from species situations in much the same way that Bernhardt shows how such departures in seventeenth-century voice leading had their basis in sixteenth-century practice (Sections 6.7 and 6.8).

Thus, most of the book is at least as old as Schenker, and much is as old as Fux or even Bernhardt. The exceptions are:

a. the modification of traditional species just referred to;

b. an attempt to shift from the acoustic to the physiological domain the assignment of certain interval sizes to certain structural functions (Chapters 1,2, and the Appendix); and

c. a theory of tonal rhythm (Section 2.2 and Chapters 7-9).

I offer this quote (with my own added emphasis) by way of suggesting that adopting a Westergaardian approach to musical analysis would hardly require forgetting everything one has learned about music and starting over ex nihilo. It is, in fact, quite a striking feature of Westergaard’s theory that virtually all of the concepts it uses — diatonic collections, passing notes, neighbor notes, tonic triads, anticipation, delay,… — are already in the vocabulary of working musicians trained in the “traditional” manner! (I put “traditional” in quotes here in light of what Westergaard says about the historical origins of his theory.) Michael rightly points out that you don’t have to have studied Westergaardian theory to understand the horn’s premature entrance in the Eroica as an anticipation — but then again you don’t necessarily have to have studied Westergaardian theory to understand a Westergaardian analysis in general!

What musicians may not realize until they read ITT is that this vocabulary (with just a couple of crucial additions, principally the highly intuitive “borrowing” operation, of which “doubling”, “transfer”, and “arpeggiation” can be viewed as special cases) is not only powerful enough to eliminate any need for a concept like “harmony”, but allows one to explicitly discuss certain critically important long-range phenomena that they may previously have thought inaccessible to words.

What I’m really pleading for is not so much theoretical reorientation for its own sake, but better analyses of music. Let’s remember why it is that people invoke harmonic ideas to begin with: it’s a way of explaining notes that they don’t know how to explain otherwise. What is that A doing there? “Well, we’re ‘on’ an F-major chord now, and A is part of the chord.” That’s a bad analysis. Why are we “on” an F-major chord now? “Because the previous chord was a C-major chord, and I-IV is a standard progression.” That’s an even worse analysis. Instead of resorting to these non-explanations, we should just think a little harder about what that note is doing, and what it is we really want to say about it.

An analogy I am fond of using (because it’s both intrinsically apt and familiar from contemporary intellectual life) is with theistic explanations in the sciences. To say that a note is there because it’s “part of the chord” is like saying that dogs have tails because God made them that way: it sounds like an explanation, and it may be temporarily satisfying (especially to the unsophisticated), but it’s really no explanation at all.

That’s the difference between Westergaardian theory and harmonic theory: the former actually makes a serious attempt to address what’s going on in the music. In that light, I was rather surpised that Michael would raise the following point, apparently against my position:

the ultimate point for most of us is the music – theories exist to help us think about how music works, but the theory isn’t usually the point

Perhaps it would be appropriate for me to indulge in some autobiography here: the whole issue with harmonic theory for me was that it got in the way of my own musical understanding. Because I read Piston, I thought that the way one understood music, and thus the goal of ear training, was to be able to identify by ear, at every moment of a piece, “what chord that is”. It never occurred to me that one could “legitimately” attempt to compose music without possessing this ability, because it never occurred to me that such an apparently well-respected book as Piston’s would be less than truthful about what the building-blocks of music were. To be sure, I had my own personal way of understanding notes, but I never dared think of it as a competing “theory”; until I could understand “proper theory” and integrate my own understanding into that system, I considered myself “untrained”. For years, I despaired at my lack of progress at raising my competence to the level of expert musicians, who presumably could listen to a recording of Brahms or Bruckner and rattle off Roman numerals in real time. (Obviously, you had to be able to do that before you could think up your own Roman numeral progressions and know what they would sound like!) Can you imagine the combination of liberatory euphoria and retrospective resentment that I felt when I later discovered that my secret, private, unacknowledged, “untrained” way of thinking about notes had all along existed as a legitimate “proper” music theory written down in a book by a leading composer and theorist? Why hadn’t the music theory community allowed me to discover that book in my middle-school library?

Still, I have an idea of what Michael may be getting at, and this may have to with a fundamental psychological difference between me and other people. Observation of other musicians, past and present, has led me to the inescapable conclusion that many of them never take music theory seriously to begin with. That is, they would never “trust” a book like Piston’s Harmony in the way that I did. It’s as if their actual musical training occurs in some totally separate cognitive sphere, while the conventional verbalizations are learned as a sort of rite of passage that “we all have to go through”. That’s why they don’t seem to care that the theories they have “learned” are bad — it’s not as if they actually use them! (Except maybe for talking, once in a while, or for writing papers in school — but certainly not for the important stuff!)

(There’s no way, for example, that Walter Piston could actually have “meant” what he wrote in his harmony book. I can’t believe for a moment that when he sat down to write one of his pieces (which are invariably of high quality), the musical part of his brain was actually thinking about chord progressions. I daresay this would still hold even if he were writing a fugue in the Baroque style. Obviously, I can’t actually prove a proposition like this — but come on!)

However, I am different. Perhaps I don’t possess this “musical unconscious” that everybody else has. Perhaps I’m just obsessively introspective. Whatever the reason, I consider it an unacceptable situation when discourse does not reflect thought. I don’t demand that it reflect thought perfectly (so it won’t do for Matthew and others to retort that no theory is perfectly accurate), just that it make an honest attempt.

So let’s just try to speak (as well as think) more clearly and precisely about music, shall we? It won’t hurt, and it can only help.

Now you tell me

Well, it turns out there’s a website that lets you use LaTeX in webpages without having to host some kind of program at your own site: it’s called Texify. Here’s an example:

So, texnically speaking, I probably didn’t need to move my blog after all. Oh well; I like it better here anyway.

(Via Ars Mathematica.)

Once again…

In response to my last post, commenter Matthew writes:

In that sense, “harmony” is useful for predicting the behavior of so much music from such a wide swath of traditions and periods that to toss it away seems a baby-bathwater proposition. It’s not a coincidence that the majority of Western music written between 1750 and 1900 adopts what we call a V-I cadence at the close.

If I had a nickel for every time I heard this…

Seriously, folks: if that is your reaction, then you need to reread the second paragraph of the post. I wasn’t kidding when I said

There is nothing that can be explained with harmonic theory that can’t be explained without it.

What’s being proposed is a more precise description of musical phenomena, not a less precise one. We’re not talking about merely jettisoning harmonic theory; we’re talking about superseding it. Whatever phenomenon you think you’re referring to when you talk about “I-IV-V-I” has a much better description in terms of Westergaardian theory (or something similar). Hence it’s not a “baby-bathwater proposition”; it’s a proposition about cleaning the tub.

What’s so bad about harmony?

Regular readers know by now that I am not a fan of certain music-theoretical ideas, namely those having to do with “harmony” and “chord progressions”. They also, I hope, understand my reasons, which I have tried to convey (and will continue to try to convey) within the context of musical analysis. However, it occurs to me that I should probably devote at least one post specifically to presenting my case against these notions — to help avoid misunderstandings of the sort that tend to arise in these discussions. It will also provide an opportunity for those who may disagree to explain their reasoning. (I am entirely open to having my mind changed if someone can give a good enough argument.) So, here goes.

The first thing to point out is that the concepts in question are nothing more than excess theoretical baggage: that is, they are not needed in order to explain or describe any musical phenomenon. One should already begin to suspect this upon reading Schenker’s later witings, such as The Masterwork in Music and Free Composition; but in any case it was later demonstrated conclusively by Peter Westergaard, who built on Schenker’s work and devised a theory of tonal music that is completely harmony-free. Westergaard’s efficient and elegant theory allows one to explain every note in a tonal compostion in terms of operations on pitches and rhythms alone — no abstract superstructures such as “chords” or “progressions”. What I like best about this theory is what I would call its “honesty”: if you understand it, you will also understand how to actually make (compose and perform) tonal music (rather than just how to talk about it).

That is actually the crux of the matter, and leads directly to the second point. If harmony were merely redundant, that would be one thing. After all, sometimes theoretical concepts that are superfluous from a strictly logical standpoint can nevertheless be illuminating. For example all of mathematics can in principle be reduced to set theory, but no one actually wants to do away with numbers, groups, topological spaces, or even triangles (except for Jean Dieudonné in the last case). Harmony, however, is not in this category. It belongs instead with gods, witches, phlogiston, and élan vital in the hallowed hall of Bad Theories — those that are such that to retain them after they have been “reduced away” would actually obscure the true explanation for the phenomenon they were invented to explain. The simple fact is that “harmonic” explanations of musical events do not accurately reflect musical intuition. (Again, just as I did before, I invite readers to compare and contrast a harmonic explanation and a Westergaardian explanation of a simple passage.)

All this, of course, has been (or certainly should have been) known since at least 1975, when Westergaard’s book An Introduction to Tonal Theory was published. However, harmonic theory has yet another bad consequence, which I haven’t yet seen anybody point out explicitly (though it was hinted at, for example, by Roger Sessions): it encourages people to think about music in ways that cause them to totally miss the deep connections between 20th-century music and earlier music. Discourse about the art music of the past 100 years makes it seem as if the activity that Bach, Mozart, and Beethoven were involved in has ceased to exist as a profession. It was for this reason that, as a child (when the only writings on music I knew were those that can be found in bookstores), I resented composers like Schoenberg and Boulez — not for their music, which I had never heard, but for what I thought their music represented, based on what I had read in music dictionaries and the like.

But it didn’t have to be that way. If Westergaardian theory had been the standard theoretical framework for discussing music in 1900, the concept of “atonality” would probably never have been invented in the first place. Why not? Because no one would have thought that unfamiliar coincidences of notes required a new theory of music. Oh sure, people would still have bitched about Schoenberg’s music, but they would have done so within the confines of the existing theory. Instead of saying “This music fundamentally violates the principles that have governed music for centuries”, they would have said “You know, Herr Schoenberg, when you pile on more than 50 layers of elaboration over a 20-measure timespan, I really have a hard time following you.” And they might have added: “Even Reger, curse him, only goes up to about 30 layers over such a span.”

Can you imagine how much more reasonable the history of 20th-century music would have looked? Babbitt’s notion of The Composer as Specialist would have been understood from the outset; people would have realized that the new music was difficult for the simple reason that it was quantitatively more complex than, and not qualitatively different from, the music of the past. There would be none of these silly arguments about whether it is appropriate for today’s music students to study the “common practice style”; it would be axiomatic that one begins with older music, because older music is simpler — and later music builds on earlier music by adding additional layers of structure. Most importantly, young aspiring composers would not have experienced existential crises upon discovering that “one doesn’t write that kind of music any more”, since it would have been understood that, from a composer’s standpoint anyway, there is only one “kind” of music. (Recall Schoenberg’s comment about twelve-tone composition, to the effect that one devises the row, and continues to compose as before.)

To sum up:

1. There is nothing that can be explained with harmonic theory that can’t be explained without it;
2. Harmonic theory provides only bad explanations in the first place; and
3. If harmonic theory were replaced with a better theory, then 20th-century (and later) music wouldn’t seem so alien.

So that is why I feel so strongly that we need to get rid of “harmony”. The sooner, the better.

Regerian “harmony” and the tonal system

More than one websurfer has happened upon this blog recently by putting the phrase “reger chord progressions” into a search engine. (Evidently they were led to my Chopin post, where Max is given a brief mention.) You have to be kidding me, folks.

There is perhaps no corpus of music that more effectively demonstrates the futility of the Rameau-Riemann-Piston conception of music theory (“chord progressions”, “functional harmony” and the like) than the oeuvre of Max Reger. That is despite the fact — and how’s this for irony — that Riemann had been Reger’s teacher, while Schenker (who actually did have something close to the right theoretical idea) couldn’t stand Reger’s music.

I know whereof I speak: as an undergraduate, I spent an entire semester (and the summer afterward) racking my brains in an attempt to “analyze” the first movement of Reger’s clarinet sonata, op. 107, using a version of harmonic theory. Suffice it to say that the results of that effort were nowhere near as beautiful as the music itself.

Even to attempt to describe — let alone analyze — Reger’s music in terms of “chords” and “progressions” is sheer madness. I can’t locate my score for op. 107 (which, thanks to me, is a mess anyway), and there isn’t one online, so we’ll have to “make do” with another of my favorite Reger works, the Piano Concerto in F minor, op. 114 (which is online, in both full score and two-piano reduction). Let’s start right at the beginning (click for larger version):

Well, I’ll grant that the first measure isn’t terribly problematic, even for harmonic theorists. But take a look at the second measure! What the heck is that chord on beat 3?! F-G-C: does this have a name besides [027]? The first chord, a [036], we can at least call “viio7 of V without the D” (though that raises the question of why Max would gratuitously choose to omit the D that “belongs” there — a question to which harmonic theory provides no answer); but what Roman numeral goes under F-G-C?

A solution can be cobbled together, but it requires a bit of clever cheating. A viio7 of V, you see, is a chord of “dominant function” with respect to V, and is required to resolve to its tonic, namely V = {C, E, G}. Well, if we ignore the F in our sonority and declare that an E has been omitted, we can interpret it as the “resolution” of the preceding chord, and put a “V” under it. To appease our guilty consciences, and make ourselves feel better about this theoretical patch-up job, we can remind everyone that, after all, F is the tonic of the piece, and it’s stuck way down there in the bass as a sustained tone or “pedal”.

Unfortunately, to do this is already to give the game away to the other side. For as soon as we allow ourselves to “ignore” notes, we have left Rameau-Riemann territory and entered Schenker-Westergaard territory. Once we admit that a line in a texture does not necessarily have to “conspire with” or “ask permission from” other lines in order to move where it wants, we have ceased to conceive of music in terms of a sequential “progression” of predetermined “chords”, and have begun instead of think in terms of a simultaneous superposition of melodic lines — a superposition that may or may not be tightly coordinated at any given time. And, once we have admitted this way of thinking into our theoretical repertory, there is no excuse (other than intellectual cowardice) for failing to follow it to its logical conclusion — which happens to be the complete and total abolishment of harmonic theory.

In this post-harmonic conception of music theory, each line has an agenda over a given timespan; these local agendas may reinforce each other to a large extent, a small extent, or hardly at all. Let’s look again at the second measure of the above excerpt:

Over the timespan of this measure, there are basically three distinct lines:

1. A descent from the stable A-flat to the (even more) stable F;
2. An ascent from the unstable B-natural to the (slightly less) unstable D-flat, which wants to continue upward;
3. A static and stable F.

(Notice that, once a diatonic collection and tonic note have been established, the various degrees of “stable” and “unstable” are fundamentally properties of scale degrees, and thus only incidentally apply to simultaneities.) Line 1, which reinforces Line 3 to a great extent, completes its task (to get from A-flat down to F) over the timespan under discussion. Line 2, however, though it goes along for the rhythmic ride with Line 1, is on a fundamentally different tonal mission, one which is not yet accomplished at the end of m.2. Its job is to introduce us to the unstable B-natural that will form part of a larger-scale structural line (it is picked up again in m.2, where it then resolves very temporarily to C) and then to ascend to…well, we have to keep listening to find out! (Exercise for the reader: see if you can follow this line to its “destination”, and describe the function of its “journey” within the structure of the orchestral introduction (mm.1-24). What, indeed, is the structure of the orchestral introduction, and what role does it play within the larger context of the movement? Answers will be given in a future post.)

By the way, if there are any Roman numeral enthusiasts left out there, I have a question for you: Consider the third chord of m.2 (on the second half of beat 4); I assume you would label this “VI without the A-flat”. But what about the F? Is it part of the “chord”, or is it the same “pedal tone” it was during the previous three eighth-notes? Or is the low F a pedal tone, while the other F’s are part of the chord? Think carefully about your answer, because I am entirely serious about the question.

Now, at this point someone is bound to object that no sophisticated theorist (and certainly, no one of the caliber of, say, David Lewin, whom I respect enormously, but who happens to have instigated the current revival of Riemann’s ideas) would ever approach this passage in such an idiotic way as reflected in my “Roman numeral analysis”. Agreed, but I must point out, to use an analogy, that this is like responding to Richard Dawkins’ The God Delusion by pointing out that no sophisticated theologian (and certainly, no Oxford don such as Alister McGrath) would ever seek to undermine evolutionary biology. Perhaps not, but by clinging to a superfluous concept such as God, they are helping to prevent the spread of a rational understanding of the world, and thus indirectly lending support not only to the battles of some people against modern science, but to other negative effects of religion as well. Likewise, by continuing to invoke unnecessary and, frankly, deleterious theoretical concepts such as “harmonic progression”, even in conjunction with better ideas such as those of Schenker, music theorists and pedagogues are preventing their students from gaining an honest, explicit understanding of what music is actually made of — and in the process they are lending undeserved legitimacy to all sorts of bad analyses (such as my Roman numeral analysis above).

Those readers who teach freshman theory know perfectly well that the thought process I mocked above is exactly the sort of thought process that many of their students use on their analysis assignments. You can urge them all you like to look at the horizontal dimension, at the phrase structure; you can implore them to (God forbid) play or listen to the passage and use their musical common sense to arrive at an analysis. But, at least for many students, it is all for nought — simply because, while your mind, your soul, and your lips all say one thing, your theoretical vocabulary says something different.

Another possible objection (closely related to the previous one) is that this discussion is old hat, and that passages like the above are common, nothing particular to Reger, and readily tractable within a theoretical framework that includes both “harmony” and “voice-leading” considerations. For example, a Schenkerian theorist might (especially if he or she is familiar with Westergaard) agree entirely with my analysis of m. 2, but simply maintain that mm.1-4 as a whole represent a “I” chord (or Stufe).

My first response to this is to ask, “Why bother?” There is no explanatory work being done by such a concept; to invoke it anyway is to patently violate Occam’s razor. (Again, imagine a theologian agreeing that Darwinian evolution explains life but also holding that life was designed by God.) However, I suspect that a reply might be offered along the following lines: Stufen do in fact perform explanatory work, in that the presence of “coherent progressions” of Stufen (particularly V-I) is a necessary condition for a piece to be “tonal”, and thus for Schenkerian or Westergaardian theory to be applicable in the first place. (Cf. Michael Monroe’s comment regarding the lack of “familiar harmonic progressions” in Schoenberg’s op. 19 no. 2.)

At first glance, this seems like a vicious circle: how does the listener know to listen for Stufen if he or she is not already using Schenkerian theory? But perhaps what is really going on is that my imaginary interlocutor is imagining a sort of “blank slate” model of musical perception, in which the listener starts off having no idea what system to use to interpret the notes, and is subsequently “cued in” by observing certain musical behaviors. This is where things start to get interesting, and where the example of Reger becomes especially pertinent. For I must now ask: where are the “familiar harmonic progressions” in the op. 114 Concerto? (Or the op. 107 Sonata? Or…?) Reger clearly thought of the piece as tonal (specifically in the key of F minor) — but if you do a Roman numeral analysis of, say, the first movement, you won’t find “f: V I” appearing very often. Indeed, you will probably end up with more colons than “I”’s! Here, for instance, is an entirely typical passage:

Yet, despite the proliferation of such “unfamiliar” progressions, the movement does, in fact, sound “tonal” — it sounds very F-minory to my ears — so what accounts for this?

Well, let’s ask ourselves when it is that we first “discover” that the piece is tonal and in F minor. Speaking for myself, I am already thinking in terms of F minor when the above-cited chromatic passage is reached — well before the first explicit F-minor “V-I cadence”, which occurs in the first measure of p. 6 in the two-piano score. In fact, I can’t imagine anyone getting past the first four measures with any doubts whatsoever about the tonality. But how can this be the case, when there hasn’t yet been any “harmonic motion” at all (if we’re to believe our imaginary Schenkerian) — when only a single chord has been stated?

Ah, but that’s just it! I don’t know about you, but if I hear an F minor triad in isolation, I’m usually going to think of it as the tonic triad in F minor. Why? Because that’s the simplest way to think of it given the evidence I have available. Thinking of it in any other way would require me to refer to absent entities. For example, I could think of it as the mediant triad in D-flat major if I wished, but that would require me to imagine a hypothetical D-flat. (You might want to object and say that to think of it as the tonic triad — i.e. as scale degrees 1, 3, and 5 — in F minor requires one to imagine the rest of the F-minor scale; but remember that the notes of a diatonic collection are not all equivalent in their conceptual status: there is a natural hierarchy based on the complexity of the various acoustical phenomena. Thus, what we call “scale degree 5″, for instance, is actually conceptually prior to all other scale degrees except 1 and 8. So, when I say “tonic triad in F minor”, I’m actually referring to a slightly more primitive concept than “elements 1, 3, and 5 of the F minor scale”.) In fact, if I hear a single F in isolation, I am going to think of it as the “tonic” — in the precise sense that I will be prejudiced toward hearing any subsequent events (or any simultaneous events that I choose to imagine) as conceptually subordinate to the F, rather than the other way around. So, given these constraints, and given the way the Reger concerto opens, I am going to be hearing F as the tonic from the very beginning.

Furthermore, I will continue hearing it this way until a different analysis is forced upon me by the events of the piece. In order for that to happen, the F-minor analysis has to become excessively complex — not according to some absolute standard of complexity, but only relative to alternative analyses. Thus, for example, if immediately after the initial F a B-flat were to follow, I may be strongly tempted to change my analysis to one with B-flat as tonic, since, all other things being equal, it’s easier to think of F as scale degree 5 and B-flat as scale degree 1 than it is to think of F as scale degree 1 and B-flat as scale degree 4 (“scale degree 5″ is a simpler notion than “scale degree 4″). (Of course, since the F came first, all other things are not equal, so the specific way in which the notes were articulated would play a vital role in my choice of interpretation.)

Thus, at least as far as my own hearing is concerned, the “blank slate” model of musical analysis is totally inaccurate. (I suspect that, if readers are honest with themselves, they will find that the same is true in their own cases.) I do not listen for “cues” to decide which of several possible systems to hear the piece in terms of (even ignoring the problem of how the “cues” themselves might be interpreted before such a system has been chosen!). Rather, the analytical system (“tonality”) is given at the outset, and the events of the composition merely constrain the analyses produced by the system. Tonality is therefore not a property that certain musical compositions have and others don’t; it is a cognitive procedure by means of which I (and, I suspect, you too) attribute structure to all music.

I hope you see where I am going with this. If tonality is what I have said it is, then it necessarily applies just as much to the music of Boulez and Babbitt as it does to that of Bach and Beethoven.

But, regardless of whether or not you accept my definition of “tonality”, you cannot have it both ways. “Tonal” is either a historical/descriptive category that applies to works, or a theoretical/explanatory category that applies to listeners’ analyses. Either you test for “tonality” by looking for paradigmatic musical events (in which case Reger works such as this fail the test), or else you assume it a priori and interpret the events of the piece accordingly (in which case there is no such thing as “atonal music” — it becomes an incoherent concept, constructed out of a category error). What you absolutely may not do, however, is apply different definitions to different composers or works. If Schoenberg’s music is atonal because it lacks “familiar progressions”, then so is Reger’s. On the other hand, if Reger’s music is tonal because its notes can be organized hierarchically according to their level of stability (as determined ultimately by acoustical simplicity), then so is Schoenberg’s (and everyone else’s).

Score one for rebellion

From a recent radio interview:

Interviewer: How did you get on in high school?

Scott Aaronson: Well, in high school I kind of felt like if I was going to be in what essentially amounted to a prison camp, that I should at least have been sentenced by a judge for having done something wrong. I didn’t feel like I was learning anything, I didn’t feel like anyone was learning much of anything; I didn’t feel like that was the purpose of it, or that it had ever been. So I actually dropped out of high school.

Interviewer: And now, as you mentioned, you’ve just become a professor at MIT.

Scott: You know, I don’t think they’re anything to sneeze at; I think they’re all right.

Need I say any more? 

(Via Shtetl-Optimized.) 

(Further reading: Return to the Beehive.)

Death to triangles! (and life to ultrafilters…)

The (in)famous rallying cry of Jean Dieudonné is the title of a new blog by “Nick Bornak” (hmmm, I wonder where the name came from…). The author, a student of mathematics and philosophy, started the blog in order to have a place to write about mathematics (for that, WordPress is pretty much the only way to go at the moment).

One of his major interests is nonstandard analysis, the very concept of which I used to despise until I read this brilliant post by Terence Tao. If a Fields Medalist says it’s okay, then it must be okay! Actually, what Tao’s post helped me to realize was that so-called “nonstandard analysis” can be regarded as just another item in the “soft” vocabulary of (“standard”) analysis, like asymptotic notation, or the definition of continuity in terms of inverse images of open sets. Since I absolutely adore “soft” mathematics (to the point where one of my missions in life is to “mollify” as much “hard” mathematics as I possibly can), you can see why I would find this point of view so appealing!

Before this, I used to see nonstandard analysis as, at best, an obfuscatory way of formulating ordinary analysis; or at worst, a sort of contrary position in the ontology/epistemology of mathematics, like constructivism (except perhaps on the “other extreme”). The way nonstandard analysis was described — as a way of making Leibniz’s infinitesimal reasoning rigorous — irritated me to no end. Didn’t these people realize that the work of the likes of Cauchy, Weierstrass, and Cantor had already made that reasoning rigorous? Not to understand this — simply because of Leibniz’s “different” vocabulary — is to exhibit the sort of over-concrete literal-mindedness that is all too typical of certain historians — particularly historians of mathematics — and, for that matter, music theorists. It thus seemed as if (not for the first time, alas) a whole field had been founded on a simple lack of intellectual agility!

Luckily, thanks largely to Terry Tao, I’ve since moved beyond this. Perhaps Nick Bornak will over time be able to still further illuminate the virtues of the “nonstandard” way of thinking. (That word “nonstandard” really is a turn-off, I have to say — to me it carries a suggestion of crackpottery.)

(Incidentally, though I’m a great admirer of Dieudonné, I don’t have anything against triangles. In fact, I think I could discuss them in a way that he would approve of. But that’s for another occasion…)

Colorless green music theory makes me a furious insomniac

Commenting on my Schoenberg analysis, Michael Monroe writes:

It doesn’t take an analytical graph to see that the G-B sonority functions as a pitch center for this piece – I’m no student of the Second Viennese School, but I’m pretty sure this ostinato focus on two pitches (a consonant major 3rd, at that!) is quite unusual for most music that would be called atonal. However, it still fits somewhere along that spectrum because it lacks the sort of familiar harmonic progressions that show up in what most people call tonal music. Again, I don’t think it’s news to anyone that Schoenberg maintained a strong interest in counterpoint and voice-leading principles, but what makes this music different than what might be called “common practice,” even when we hear the G-B reinforced by frequent repetition, is the lack of such progressions. We also don’t generally refer to medieval polyphony as tonal, even though the modes tend to reinforce certain home pitches as well. Still, in my experience, Op.19/2 is often used as an example of a transitional piece, so it’s odd that you should choose this to make your point. I’d be interested to see you take on something more obviously atonal. (I understand that you don’t see “atonal” as a useful word, but you know what I mean.)

All right, I had a feeling someone would object to my choice of Schoenberg op. 19 no. 2. In my defense, let me say first of all that “making a [theoretical] point” is not necessarily the primary purpose of analyzing a work. Why not simply regard the previous post as a presentation of my take on this charming little piece, which after all exists (or can be thought of as existing) for its own sake alone? The fact that it happens to help illustrate certain theoretical ideas of mine (I doubt they’re totally original with me, of course) is an incidental consequence of the fact that I’m the one doing the analysis — and you’ll notice that my explanation of how the analysis supports what I had already been saying was an afterthought.

The fact is, I chose op. 19 no. 2 for a reason no more profound than that a score happened to be lying around nearby. (Although I might also note that this piece, and its op.19 siblings, are among a select few piano works that I can actually play, so that may contribute to my affection for it.)

Now with regard to the analysis itself: I think Prof. Monroe may have missed the point — which is probably my fault, since I wasn’t as explicit as I could have been. I’m not claiming that the “G-B sonority functions as a pitch center for this piece”, or that the piece has “tonal features”, or makes “tonal allusions”. Lord knows, these things have been said many times before, to the point where one’s eyes want to start rolling. No, what I’m claiming is something stronger: the G is a scale degree 1, the B is a scale degree 3, the D of the top line is a scale degree 5, and so on. That is: the G-B sonority isn’t just a “pitch center”; it’s literally “do-mi”.

In attempting to explain what sets op. 19 no. 2 apart from “tonal” works, Monroe makes reference to “familiar harmonic progressions”; I would like to know specifically what he (and everybody else) means by this. Are we talking about particular sequences of actual notes in a score? If so, then what does the lack of such “familiar” sequences in op. 19 no. 2 have to do with anything (theoretical) at all? Look: the sentence

Colorless green ideas sleep furiously.

was, at the time it was first composed (1957), about as unfamiliar as can be imagined (though its familiarity, like that of Schoenberg’s music, has grown with time). And yet, it is — manifestly — a grammatical sentence. Thus, the notion of “grammaticality” is quite distinct from “familiarity”, which was basically the point. Similarly, 684 may not be a particularly “familiar” number; but all of us nonetheless “understand” it, since we would know how to determine (in principle) whether a certain box contained exactly that many balls. So clearly, the fact that a musical composition does not contain “familiar” events is not grounds for putting it in a separate theoretical (as opposed to, say, historical) category. Surely, therefore, this cannot be what Monroe means by “familiar harmonic progressions”.

On the other hand, the only alternative meaning that I can think of is a certain theoretical concept (way of understanding notes) that was rendered obsolete three decades ago!

Hopefully, I’ll be able to take up Monroe’s challenge and analyze a “more atonal” work very soon. For the moment, however, let me try to get to the heart of the theoretical matter, in a very clear and simple way. Consider the following bit of music:

Here are two derivation sequences, representing two different analyses of those two bars:

1. Westergaardian derivation sequence

2. Pistonian derivation sequence

Which is better? (Hint: one of them should seem like a caricature, but actually isn’t.)