No wonder!

Okay, I know everyone else took The Springer GTM Test a long time ago, but in this case my tardiness is timely:

If I were a Springer-Verlag Graduate Text in Mathematics, I would be Robin Hartshorne’s Algebraic Geometry.My creator studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. After receiving his Ph.D. from Princeton in 1963, he became a Junior Fellow at Harvard, then taught there for several years. In 1972 he moved to California where he is now Professor at the University of California at Berkeley. My siblings include “Residues and Duality” (1966), “Foundations of Projective Geometry (1968), “Ample Subvarieties of Algebraic Varieties” (1970), and numerous research titles. My creator’s current research interest is the geometry of projective varieties and vector bundles. He has been a visiting professor at the College de France and at Kyoto University, where he gave lectures in French and in Japanese, respectively.My creator is married to Edie Churchill, educator and psychotherapist, and has two human sons and one daughter. He has travelled widely, speaks several foreign languages, and is an experienced mountain climber. He is also an accomplished musician, playing flute, piano, and traditional Japanese music on the shakuhachi.Which Springer GTM would you be? The Springer GTM Test

As it turns out, I have just recently started reading this very book! And, well…yes. It’s written in pretty much exactly my preferred style. (I like it better than Shafarevich — but don’t tell anybody, or I’ll get objects thrown at me!) Is this a sign that algebraic geometry is my true calling? Or does it have something to do with the fact that the author is a polyglot musician, not unlike your humble blogger? (Meanwhile, Ponder Stibbons, who would probably not like Hartshorne’s expository style, may want to take note of his experience in mountain climbing — and draw whatever conclusion seems appropriate!)

Controversy in music

A comment of mine at Musical Perceptions has apparently thrown Scott Spiegelberg off balance. (That certainly wasn’t my intention, of course.) One could perhaps detect as much from the fact that, in the discussion in question, Spiegelberg very bizarrely accused me of violating civility. This led me to wonder whether there is some sort of etiquette in the field of music theory, or in the music blogosphere, that I am unaware of, according to which it is impolite to challenge one’s interlocutors and raise points of controversy. (I certainly never got this impression during all the time I spent reading the scholarly literature in the field, but that was a few years ago, so maybe times have changed.)

If so, I would respectfully ask readers to get over it. If you think I’m a raging fanatic with an axe to grind, if you think my attacks on harmonic theory are too harsh, if you think my tone is too polemical, then you need to put things into perspective. Start by having a look at the writings of Schenker — to take only one particularly relevant example. By comparison, I think I have been remarkably restrained — especially when you consider that, at times anyway, I am hardly less frustrated than Schenker was. And for many of the same reasons, too!

I don’t mean to rub anybody the wrong way; but the fact is, there is an intellectual debate here to be had. Theories of music are no less subject to criticism than theories of anything else; and the specific criticisms that I have raised about harmonic theory (I don’t claim they’re original with me) are serious and warrant discussion. To leave them undiscussed, indeed, is to concede the issue to my side.

Towing the line of infinity

After several months of long walks and taxi rides, yesterday I finally worked up the courage to pick up the phone and have my malfunctioning automobile towed to a mechanic. (It turned out that only one phone call was required; I had feared that two or three would be necessary, and that I would characteristically place them in the wrong order.) The long wait (by which I mean both the months of procrastination and the hours I spent yesterday waiting for the guy to show up) did not go unrewarded.

Upon climbing into the truck (it was, of course, news to me that tow trucks also transport the owners of towed vehcles), I instinctively reached for the seat belt — beginning a search which was destined to be futile. In short order, the driver offered me a lesson on the folly of seat belts; apparently, wearing one increases the risk of death or injury in the event of a crash! (Evidently, yet another case of the government conspiring to kill as many people as possible.) Well, I thought, at least I won’t have gotten through this day without learning something.

But that was only the beginning. It was not long before the inevitable conversation began:

– “So, I guess you’re at the university?”

– “That’s right.”

– “What do you study?”

– “Mathematics,” I replied, bracing myself.  

– “Ma-the-ma-tics!” the driver exclaimed, drawing out each syllable as if to emphasize the esoteric nature of the subject. “Tell me, what do you do with a degree in ma-the-ma-tics, other than teach other misguided souls that there’s actually some use for it?”

(Oh boy, here we go…)

– “Well,” I said, “one can also study mathematics itself as a discipline — do research, you know.”

– “And to what purpose?” he immediately asked, with the clear implication that only certain purposes were acceptable.

– “Curiosity, I suppose.”

– “Curiosity, eh! Curiosity killed the cat!” (It was around this time that I noticed the Confederate flag sticker above the windshield — rather unusual in a region of the U.S. where, for example, a university cancels its classes on the occasion of Rosh Hashana.)

Before I could say anything, he quickly conceded:

– “Actually, I think curiosity’s an okay thing.” That was good news. I mean, maybe I’m willing to have my mind changed about mere life-and-death matters like seat belts, but if this fellow was going to question the value of curiosity, that would be going too far!

He continued:

– “I have a degree in English, you know…”

– “Well, it doesn’t seem to have stopped you!” I pointed out. (Would you have guessed that a degree in a subject as “impractical” as English could qualify one for an occupation as “practical” as that of tow-truck operator? Me neither.)

–”Yeah, well….I could never get interested in mathematics. Geometry and logic and such I could handle, but when it came to real mathematics… My father was always trying to get me interested in mathematics. I’ll tell you exactly when I realized that mathematics was useless. One day my father sat me down and said ‘I’m gonna show you something really interesting: that 0.999 to the n is equal to 1′. And he wrote out a proof that 0.9 to the n is equal to 1, and he said to me, ‘Isn’t that interesting?’ And I said, ‘I have a question’, and my father said ‘Okay’. So I said, ‘A decimal is a way of writing a fraction, right?’ And he said ‘yes’. And I said, ‘A fraction is by definition a part of the whole, right?’ And he said ‘yes’. So I asked ‘How can a fraction like 0.9 to the n be equal to 1, which represents the whole?’ And my father scowled and went away. And I realized that if mathematics allowed you to prove things that contradict logic, like a part being equal to the whole, then I had no use for it.”

(Obviously, by “0.9 to the n” the driver and/or his father must have meant , i.e. an infinite string of 9’s after the decimal point.)

– “Surely you don’t still hold to this view, do you?” I asked incredulously, referring to the idea that fractions are necessarily less than “the whole”.

– “Why yes, of course!”

– “Oh, well, it’s quite wrong,” I said, in that softly authoritative ivory-tower tone of voice. The driver smirked and snorted, as if to say, “here comes another one of these pointy-headed academics trying to tell me that things don’t mean what they mean!”

I asked:

– “What about 1.5? That’s a decimal — hence a fraction. But it’s greater than 1, which shows that a fraction can be not only equal to the whole, but even greater than the whole!”

– “No, no, that’s one, point five. That point five is zero point five, a fraction, less than one. Any time you have zero point some number, it’s always less than one. It doesn’t matter if you have a million nines, or a billion, or any number, it’s still less than one.”

– “No mathematician would argue with that!”

– “Then how come my father proved that 0.9 to the n is equal to 1?”

– “If you have infinitely many 9’s, that’s a different story — that means the 9’s never stop. If they stop somewhere, then it’s less than 1, of course. But if they never stop, then it’s equal to 1.”

– “That would defy logic! It doesn’t matter how many 9’s you put on; you’ll never get to 1.”

I was of course totally unprepared for this discussion, in addition to working under the stress of time pressure (by this time we had already arrived at our destination), so I wasn’t able to employ the obvious arithmetical demonstration of who was really “defying logic”. Nor did I remember to ask the driver the most important question: What, exactly, was his father’s argument that 0.99999….. should be equal to 1, and where was the flaw in his reasoning? The best I could do was:

– “So what you’re saying, basically, is that you don’t believe infinity exists — that it simply doesn’t mean anything to you to write infinitely many 9’s after the decimal point.”

– “What I’m saying is that you can put as many 9’s as you like — infinitely many or whatever — and it will always be less than 1.”

– “Okay, let me ask you this. Suppose you take this number 0.999… with infinitely many 9’s, which you say is less than 1, and subtract it from 1. What do you have left?”

– “Who knows? Some infinitely small number.”

I certainly didn’t have time to go into a discussion of infinity, and of Cantor and Kronecker, let alone Abraham Robinson and John Conway. It would have been complicated anyway, because the driver contradicted himself once we were outside the vehicle:

– “The problem is that mathematics and science and physics and geology and all that — these are finite subjects. Once we start introducing ideas of infinity into the picture, then we’re talking about philosophy. And philosophy should not be mixed into mathematics and science.”

I really had no idea where to even begin with this, so I just kept listening, while he unfastened my vehicle from the tow truck.

– “I guess what I’m saying is that I don’t want it to be true.” (Always a knockdown argument!). “If all the sudden 5.9 were equal to 6 and so on, my whole world would be out of whack — buildings that were supposed to be straight would be slanted…”

And then…

– “It’s like religion, organized religion. One of the biggest is Catholicism, you know, Roman Catholicism. They used to say that it was a sin to eat meat on Fridays. So when I was young the schools couldn’t serve meat on Fridays. And then all of a sudden they changed their minds and said it was okay after all. Now if something could go from being a mortal sin to being perfectly okay on a dime, just because somebody said so, then that’s a way of thinking I don’t have any use for.”

I don’t happen to know the details (historical or current) of the Catholic Church’s policy on meat-eating on Fridays or any other day, but at any rate, here at last was a principle that this Confederate-flag-displaying, 0.999…= 1 denying seatbelt skeptic and I could agree on!

When I wrote out a check to this guy, I was careful to write “60.00″ and not “59.9999…”.

Gowers on mathematical pedagogy

Fields Medalist Timothy Gowers has entered the blogosphere, and right off the bat has started a couple of very important discussions on the pedagogy of elementary mathematical topics (specifically logarithms and linear algebra). You can read my contributions here and here.

Those following the harmony debate and my discussion of music pedagogy should pay attention, as some of the same issues (in particular the damage done by a lack of an appropriate level of theoretical sophistication) are involved.

(Fortunately the situation is a somewhat less severe in mathematics, if only because students concentrating in that subject do eventually get to see it done properly in “advanced” courses. The situation would be analogous in music if every music student, perhaps after sitting through a year or two of harmonic theory, had to take a course in Westergaardian theory at the 300- or 400-level. )

It’s a small namespace

It turns out that computer scientist Stephen Cook (of NP-completeness fame) has a son named James. This may not seem like a terribly remarkable fact until you read the latter’s “Creation Story”:

In the beginning, there was Everything. And James cried out in terror, for it was too much: and He cast everything away, so that there was Nothing.

And James was sad, for He had destroyed all of existence. A tear flowed out of his eye, and He spoke to the tear:

“You are all that exists, for I have cast all else away. I give you all that I am, that you may some day forgive me for destroying Everything.”

And James poured his essence into the tear, giving it the powers of Music and of Mathematics and of Order, till there was nothing left of James and He had given all his essence to the tear. And the tear was silent for immeasurable time as it reflected on what it had been given.

And the tear resolved to create Matter and Time and Space: so it became a great point of light, and suddenly it exploded. And thus there was Space, and it was filled with Matter which moved through Time. All this the tear brought forth from Music and Mathematics and Order, which James had given it.

As the tear exploded, eventually the Matter grew wise and reflected upon itself. And thus was born Consciousness. And the wisdom of the Consciousness grew until it was wiser than James Himself had been, and great civilizations of sentient beings rose and fell on many worlds.

And the Consciousness basked in the glory of Music and of Mathematics and of Order, and saw that it had been created by James Himself before Time existed. And it took this essence of James and brought James back into being that He might see the glory of what his tear had created.

And so James the Human was born on the planet Earth. And He was filled with joy, and told the tear that had become a Universe:

“You have done well, for you have created something as great in potential as Everything was, yet not so overwhelming that I must cry out in terror at the sight of it.”

And James saw that the Universe would be alright, and that he could rest and live as an ordinary citizen of Consciousness.

James is now in the computer science and math joint specialist program at the University of Toronto.

Music, mathematics, computer science, consciousness? Is this a coincidence?

Yes, it is (the author of this blog being neither an undergraduate, nor a resident of Canada, nor the son of a complexity theorist, nor a violist, nor a singer, nor a dancer — nor, for that matter, even necessarily named James Cook, since for all you know that could easily be a pseudonym, being the name of a historical figure). But it’s a pretty neat one, I’d say.

Somebody gets it…and disagrees with me!

I have discovered a very interesting blog called The truth makes me fret. What’s so interesting about it? Well, its author, in addition to sharing some of my general intellectual interests (philosophy, physics, evolutionary biology, the follies of mathematics textbooks), and using the same WordPress theme, actually manages to simultaneously understand what the harmony debate is about and have something to say against my position! Naturally, I couldn’t resist leaving a comment, which I’ll reproduce here in the unlikely event that you’re too lazy to click on this link:

Wow! It’s so nice to see that somebody else understands what this debate is about — even if they (apparently) don’t agree with me!

Let me try to explain how I view the situation, in terms of your analogy from physics. It isn’t just that QED and the like are consistent with the Standard Model; rather, the former supervenes on the latter, in the sense that one is supposed to be able, in principle, to start with the Standard Model and deduce theories like QED as consequences. The “higher-level” theories, in short, are explained by the “lower-level” theories. If you ask why QED holds in this universe, the Standard Model gives you an explanation. (Something similar is true of Newtonian mechanics, incidentally, which is easily understood as a limiting case or approximation of relativity.)

If harmony were merely a higher-level theory (or even just an empirical characterization) of phenomena whose fundamental explanations were Westergaardian, I wouldn’t really have a problem with it (other than that it’s just plain superfluous–see below). The biggest problem I have is that this is not how harmony is treated; it’s treated as a fundamental explanation in itself — thereby intruding on the territory of Westergaardian theory. The situation would be comparable in physics if everybody had just ignored Einstein and continued to treat Newtonian mechanics as a fundamental cosmological theory. Of course, they couldn’t really do so, if they wanted to study the “extreme” phenomena of the universe; so what they would have to do would be to invent ad-hoc theories for such phenomena, and declare that the extreme situations (or even just 19th-century electromagnetism!) represented “exceptions” to the fundamental Newtonian laws. Note that this is precisely the theoretical situation that musicians are in with respect to twentieth-century music!

It’s important to remember that, just like the fundamental laws that cover even extreme non-Newtonian phenomena are the subject matter in theoretical physics, so too are the “fundamental” musical phenomena addressed by Westergaard the subject matter in music theory class — or music pedagogy generally. So the analogy with physical theories breaks down here, simply because, while chemistry, biology, and even astronomy at the scale of the solar system represent genuine higher-level phenomena that would be truly cumbersome to describe in terms of fundamental physics (even though it would be logically possible to do so), Westergaardian theory already addresses music at exactly the appropriate level for practicing musicians.

As an afterthought to this, and in response to another point in the post, I would add that I’m not sure that I believe all of those who claim that harmonic theory represents their musical intution — simply because so few of them turn out to be familiar with (or even aware of) Westergaardian theory.

Chopin: Round 2

Scott Spiegelberg has taken up my challenge! So now, it’s time to roll up our sleeves for the second round (the first round is here):

JC – [I'll ignore the straw man version of harmonic theory James describes first. I made none of the claims he states about composer intent, thus I do not have to defend them.]

Well of course he didn’t make such claims; they would be utterly ridiculous! What Spiegelberg has to defend is not any claims about the intents of specific composers, but rather the use of a vocabulary which is by its very nature suggestive of that sort of absurd compositional process.

JC – “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.”

SS – James’ statement leaves out the idea that we have begun with the tonic note that will be the eventual goal of this piece.

That idea is implicit in the fact that we’re thinking in terms of E minor — so that E is necessarily the tonic (and G the mediant, and so on).

James’ analysis also misses the idea that we begin with an uneasy sonority, since the triad is not in root position.

I specifically mentioned that the bass begins on G — which, if we’re thinking in terms of E minor, is not where it “wants” to be, simply by virtue of the fact that it’s not the tonic.

Plus James’ statement suggests there is no relationship between each of the four voices, since they each has a journey of their own. Nay, in tonal music the voices journey together, even when they take different paths.

No: what my statement suggests is that there is not necessarily any relationship among the four voices. Which is true: they could turn out to have any of an infinite number of degrees of relatedness or independence, according to the choice of the composer. The denial of this is for me one of the most offensive aspects of harmonic theory.

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

SS – How can one have a suspension if there is no chord? The E is in dissonance to the other voices only if one considers harmonic relationships. If you are thinking only in contrapuntal interval relationships, then there is also a dissonance between the A and the B, but James doesn’t state this.

A suspension is a rhythmic event, specifically a delayed stepwise resolution of a dissonant tone. The relevant dissonance here is that of the E with respect to the F# of the bass. If one understands the tied E as a delay of the chromatic passing tone Eb, then one can understand an underlying functional parallelism (ITT, sec. 4.4) between these two voices over the span of the first two measures. (If I had shown this in my graphs, it would have gone between Stages 7 and 8.)

JC – He actually says nothing about this verbally. His graphs also don’t say anything about the unusual nature of this Eb.

SS – Any musician would find the Eb odd, since it is not in the diatonic space of E minor. It sounds fine, since it is enharmonically the same as D#, and thus fits with the expected dominant chord. James’ analysis does not show this at all.

Perhaps any musician trained on harmonic theory would find the Eb odd; but for me it is a perfectly straightforward passing tone on the way from E down to D — scale degree 7.5, if you will. (And, by the way, D# isn’t in the diatonic space of E minor either — which doesn’t prevent it from having a perfectly comprehensible meaning in that key.)

As for the “expected dominant chord”, what is actually “expected” is that the alto voice will move downward in parallel motion with the tenor and bass:

Why might D#/Eb be expected instead of D? Are the mysterious laws of harmony at work here, dictating the content of a “V chord”? Not at all. The choice is between these two (linear) structures:

There’s nothing wrong with either of these, but if you’re expecting D# instead of D, it’s because you consider the first one to be simpler.

JC – “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.)”

SS – James doesn’t state his criteria for creating a conspiracy, much less show why my example is not one.

It’s a complex question, but I’d recommend taking section 4.4 of Westergaard as the point of departure.

My justifications are (a) the way Chopin chooses the notes spells a fully diminished seventh chord; (b) fully diminished seventh chords are very recognizable sonorities; (c) it is an expected sonority in the E minor pantheon, being an inversion of the viio7; (d) it is an expected chord to follow the V65 chord, since both are dominant functions and composers throughout the 17th, 18th and 19th centuries have had V7 chords move to vii07 chords; (e) it explains how the next chord makes sense, as it is quite unexpected in E minor diatonic spaces

All five of these justifications assume the harmonic framework, without comparing it to the Westergaardian one; hence all five of them can be immediately dismissed.

What is interesting here is that in the first three measures the melody is exactly the same, but the harmonic rhythm changes – measure 1 is one chord, measure 2 is two chords, and measure 3 is three chords. James’ analysis misses this, probably since he doesn’t have any truck for chords. But the speeding up of harmonic rhythm is something that Schenker appreciated, and an interesting way for Chopin to create variety while keeping the same melody and surface rhythm.

Again, Westergaardian theory provides an alternative (and much more refined and nuanced) account of what people refer to as “harmonic rhythm”; to construct this account, combine the considerations of chapters 4 – 8 of ITT. With respect to this passage of the Prelude, the point is basically that the linear operations of the various lines are applied over different timespans. To get specific: the soprano’s neighbor notes segment each measure; the alto moves chromatically from E to D over the three measures; the tenor and bass would be expected to arrive on G and E respectively in m.3, but the alto’s descent encourages them to spread out this motion over mm.1-4 rather than mm. 1-3 (with the bass edging chromatically downward in m.3). Thus, in m.1 only one linear operation is going on; in m.2 we have one at the measure level (soprano), one at a “3-measure level” that is delayed at the measure level (alto), and two others at what will turn out to be a 4-measure level (tenor and bass); in m.3 we have the same thing in the soprano, another measure-level delay in the alto, and a chromatic passing motion in the bass taking place over mm.2-4.

To emphasize: the thought process going on here is not a horizontal juxtaposition of vertical structures; it’s a vertical superimposition of horizontal structures.

JC – “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!”

SS – And this is exactly where James’ analysis misses something wonderful. First of all, the secondary French augmented sixth could resolve to an E minor triad, basically like a tritone substitution in jazz theory. This progression is somewhat rare, but found in Tchaikovsky and other High Romantic composers’ works.

In other words, “secondary French 6th to I” is on the “list of permitted progressions”. Again, since the very idea of such a “list” assumes the horizontal-juxtaposition-of-vertical-structures model that I have explicitly rejected, I don’t know what Spiegelberg’s purpose is in informing me of this piece of data. I suppose I should at least be glad that the progression in question

is “permitted”, since it would be most unfortunate (as well as futile) if harmonic theorists were to forbid the simultaneous resolution of a raised scale degree 7, a lowered scale degree 2, and a scale degree 4 to their respective destinations.

By the way, James doesn’t comment on this point, but a common-tone resolution of a diminished seventh chord holds one note steady (two, A and Eb, in this case) and moves the other voices by step, usually downward.

This is opposite of the normal resolution of a fully-diminished seventh chord, where one note resolves upward by half-step, the local leading-tone to tonic resolution.

I don’t know what comment I should make, other than the fact that these two examples represent midstream snaphots of different processes. Again, Spiegelberg is speaking as though he conceives of a passage in terms of chords juxtaposed in time (otherwise the question of “to which other chord should this chord resolve?” would not come up), rather than linear processes overlain on each other. The fact that the first simultaneity of both examples happens to be a type [0369] is, from an abstract standpoint, a complete coincidence. Now, there’s nothing stopping a composer from turning this kind of coincidence into a motivic process specific to the work, but if Spiegelberg is going to claim that’s what’s going on here, he needs to show how the [0369] idea pervades the work. Otherwise there’s no point in using the term “diminished seventh chord”.

JC – “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?”

SS – Play the first 2.5 measures to some listeners, stop right there and ask them to sing tonic. I guarantee that you will get at least two different answers, as long as your sample size is above 2.

That doesn’t answer my question. Maybe everybody in the world has their own opinion about what the tonic is. This doesn’t tell me anything about what the source of the alleged ambiguity is, or why my own hearing isn’t the best one.

I would, however, like to add some clarification to my own remarks about E minor versus A minor. I don’t actually want to deny that there are local aspects of A minor to be heard in various places in the Prelude. It is a consequence of the iterative, hierarchical nature of tonal operations that virtually all pieces temporarily focus the listener’s attention on other scale degrees besides the tonic. But only in some pieces does this process cause any ambiguity about what the underlying tonic actually is. The problem with Spiegelberg’s analysis in this regard is that it does not capture the fact that, for all that A may have the feel of a local “tonic” at certain points (especially if you focus on very short timespans), it never rises to the level of displacing E in the hierarchy. Contrast this situation with that of the second Prelude (op. 28 no. 2), which is an example of genuine tonal ambiguity. In that piece, not only are you forced to invoke subordinate diatonic collections in a way that isn’t true of the E-minor Prelude, but the clarification of the relative hierarchical status of these various tonalities takes place over the span of the entire composition. Whereas in op. 28 no. 4, E minor is established at the beginning and never displaced, despite a temporary focus on scale degree 4 at certain places.

You ask what harmonic analysis has that Westergaardian analysis doesn’t, and this is one of those things. Yes, you can explain abstractly how the voices progress by logical melodic motions, but it misses the tensions built up by the relationships of the voices to each other.

Bull-oney. One doesn’t just “explain abstractly how the voices progress by logical melodic motions”; one explains the hierarchical relationships within and among the voices, over various spans of time. These hierarchical relationships are the source of the tensions that Spiegelberg speaks of.

SS – First, sometimes a melody note is a chord tone, sometimes it is not. Even in Westergaardian analysis one would add notes at different stages (I presume, it is certainly the case with Schenkerian analysis). In this case the C in measure 3 is a dissonance just as the C in measure 1 is, whereas the C in measure 2 is not.

Aha! Thank you for so beautifully making my point for me. Harmonic theory completely obscures the fact that these C’s are all on the same hierarchical level — they’re all neighbors to the B, added at the same stage (Stage 8 in my illustration). Simply because of the fact that the accidentally-resultant simultaneity in m.2 happens to be a “recognized chord”, whereas the other resultant simultaneities are not, we are forced to completely distort the conceptual process going on in the top line. Who decided that [0369] was so special, anyway? Why can’t the other sonorities be “chords”, too? (And besides, I thought that even in harmonic theory, a diminished seventh chord was considered a dissonance.)

Second, a V7 is not just scale degrees 5, 7 , 2, and 4, it is also a sonority of the major minor seventh chord – a major triad with a minor seventh added – which is a highly distinctive sonority.

Would Spiegelberg agree that

is an example of a “sonority of the major minor seventh chord”? If so, then he must confront the fact that such a sonority is not an irreducible entity: it is made out of notes. In this example, we have G, D, F, and B. How does one understand such a thing? Answer:

In other words, if you play that chord on a piano, a listener is going to imagine a context for it, probably of the above form. But what does that context consist of? Answer: scale degree 7 progressing to 1, 4 to 3, 2 to 1, and 5 to 1. In other words, the effect of this “highly distinctive sonority” consists precisely in that it strongly suggests an interpretation of its constituent notes as scale degrees 5, 7, 2, and 4! So much for Spiegelberg’s claim that there is more to a V7 than that.

James is clearly a steadfast listener for this prelude (I have no idea what type he is for others, I find people often change with different pieces),

Could that be because some pieces lend themselves more easily to “steadfast” listening than others?

It’s not really that my hearing of this piece is “steadfast”; it’s that it’s hierarchical. Whatever happens with A, it happens on a lower level than what happens with E. Context matters a lot; you might think A is tonic in m.4 if you’ve forgotten the opening. But if you take the opening into account, it’s hard to come up with a plausible analysis on which A is superordinate to E. The sense of B as scale degree 5 (indeed, the primary tone of the fundamental line, in Schenkerian terms) in the first measure is just too strong.

How could A not be involved in the progressions of m. 10-11, when it is in the melodic line, accented by a grace note in m. 11?

The neighbor-note motions of the lower voices are not conceptually dependent on the A (as would be implied by saying that there is a “progression to an A-minor triad”). As for the A’s in the top voice, they are easily understood as borrowings.

I am curious how James distinguishes “perfectly comprehensible [...] melodic motions through various parts of the E-minor scale” that don’t form triads or seventh chords that are found in E minor tonality. As an example, shift the entire right hand part of the Prelude over by one beat, so it starts at exactly the same time as the left hand. The melodic motions of the top voice are the same, and the four voices have separate journeys, so the Westergaardian analysis should be the same, except possibly for the alignment of the upper voice with the lower voices.

As if the degree of alignment of the voices were a mere detail, rather than the very essence of the matter!

But then, each voice has a separate journey, so what difference does it make if they don’t line up? Is it the intervals? But if we care about vertical intervals, why not vertical sonorities created by all four voices? How would the analysis be changed without referring to harmonies? Would the modified piece still be in the E minor tonality, and if so, to the same extent as the original Prelude? The last question is one that can be answered by harmonic analysis, but I don’t see how it is answered by Westergaardian analysis.

I would simply refer Spiegelberg to section 4.4 of ITT, which he obviously has not read.

Perhaps I’ll think about giving a systematic exposition of Westergaardian principles on this blog at some point — it seems to be badly needed (especially since the book is out of print).