Imminent resurrection possible

February 29, 2012

Continuing a tradition begun last February 29, I am posting to announce that…I may start posting again.

(The hiatus was a bit longer this time, of course. But perhaps the coming non-hiatus may be also.)

I would be picking up where I left off, more or less.


Yes, exactly!

March 22, 2009

Two blog posts — one recent, the other less so — that have me jumping up and down in excited agreement:

  • You’re Calling *Who* a Cult Leader? — in which Eliezer Yudkowsky (one of my favorite bloggers) points out that that it’s okay to be really enthusiastic about something:

    Behold the following, which is my true opinion:

    “Gödel, Escher, Bach” by Douglas R. Hofstadter is the most awesome book that I have ever read. If there is one book that emphasizes the tragedy of Death, it is this book, because it’s terrible that so many people have died without reading it.

    I know people who would never say anything like that, or even think it: admiring anything that much would mean they’d joined a cult (note: Hofstadter does not have a cult)[…]

    But I’m having trouble understanding this phenomenon, because I myself feel no barrier against admiring Gödel, Escher, Bach that highly.

    He continues:

    You know, there might be some other things that I admire highly besides Gödel, Escher, Bach, and I might or might not disagree with some things Douglas Hofstadter once said, but I’m not even going to list them, because GEB doesn’t need that kind of moderation. It is okay for GEB to be awesome. In this world there are people who have created awesome things and it is okay to admire them highly! Let this Earth have at least a little of its pride!

    Yes! As I have noted before, there is an inhibition in our culture against expressing strong feelings. Away with this!

    I’m embarrassed to admit that I still haven’t read GEB, even though everybody raves about it and it’s got both a mathematician and a composer in the title. Well, it’s now (higher) on my to-do list. But anyone who has ever visited this blog will know that I harbor a similar level of enthusiasm for Westergaard’s ITT. And that does not make me some kind of crazed fanatic.

  • What’s in a number? — in which a member of the Texas Tech music theory faculty correctly explains the meaning of figured-bass symbols (link added by me):

    I often tell my students that figured bass evolved as a shorthand notation for species counterpoint.That is, figured bass actually suggests lines, not chords. Consider the example below:


    If you look at those examples without worrying about vertical sonorities, the figured bass makes quite a lot of sense. Once you begin trying to assign Roman numerals, the task becomes a bit muddier. In the first example, we can easily understand the E-F motion in the soprano as some kind of neighbor motion or perhaps as the beginning of a passing motion. I prefer that interpretation to one which says the first chord is a root-position tonic and the second chord is a first-inversion submediant.

    And rightfully so. In fact even to speak of this measure as being composed of two “chords” is a misleading distortion. If there are two entities into which this measure is divisible, they are a second-species line on the one hand, and a complex of three first-species lines on the other. (As a minor quibble, I will point out that “worrying about vertical sonorities”, which species counterpoint does just fine, is not to be confused with “assigning Roman numerals”, the discredited province of harmonic theory.)

    In short: figured bass tells us diatonic intervals above the bass and nothing else. If notes are to be altered, the accidentals will appear in the figured bass. Figured bass is simply a shorthand for linear motion.

    So true! Whether or not it is technically accurate that figured bass evolved in connection with species counterpoint per se, this is much closer to the truth than to suppose, as many still do, that it indicates some sort of preexisting awareness of Rameauvian “harmonic” concepts on the part of Baroque-era musicians. (Of course anyone who thinks that hasn’t read Schenker, but that’s for another time…)

(Not too long ago it finally occurred to me why this confusion exists. The reason for it is that people mistake figured basses, which are a form of musical notation, for some sort of analysis of the music. When you look in a treatise and see a figured bass at the top of a page, say, followed by a realization below, perhaps it’s natural to suppose that the figured bass on top is in some sense a “more primitive” structure, from which the realization is derived. But this is a misunderstanding. Figured bass was a performance practice; it was not the purposes of such treatises to engage in music theory as we know it, of the sort practiced by Westergaard — the subject had not yet come into existence as an explicit discipline. So one is by no means obliged to think of a passage in terms of some underlying figured bass. Quite the contrary, in fact: the figured bass is but a shorthand for the realization, and thus if anything the latter “explains” the former, rather than the other way around.)

Nice Boulez site

March 15, 2009

The London Sinfonietta has a website about Pierre Boulez. It, or at least some parts of it, must be rather old: Boulez’s 75th birthday was in 2000.

(Boulez, for non-musical readers, is pretty much the leading figure — or at any rate the leading European figure — in the art music of our era [i.e. post-WWII]. For mathematical non-musical readers, an approximately equivalent person would be Jean-Pierre Serre.)

The interesting part of the site is, of course, the series of pages devoted to Boulez’s 1984 work Dérive I, which is described as:

an elegant, shimmering and vibrating eight minute work which explores harmony and texture from a chordal starting point using material which “derives” from three earlier pieces, Répons (1980), Messagesquisse (1976) and Éclat (1965).

(Obviously, “harmony” here is to be understood in the sense of vertical pitch collections — nothing to do with harmonic theory! I would probably have used the word “sonority” here instead.)

It seems the excerpts included comprise virtually the entire piece, so go have a listen. The commentary hardly constitutes a detailed analysis, but overall the quality is pretty good for something on the Internet.

(Via Complement.Inversion.Etc.)

Adagio agitato

March 11, 2009

After reading the recent anecdote at Texas Tech Music Theory about the music student who didn’t know the meaning of “Adagio”, I was amused to find the rather strange marking “Adagio agitato” in the score of Beethoven’s Christ on the Mount of Olives (p.11).

You can hear the passage in question (which, of course, features Jesus in agony) beginning at 8:54 or so in this clip:

I admit, this could easily be a misprint for “Allegro agitato” (though the tempo in the above performance doesn’t strike me as quite fast enough for that; unfortunately I don’t remember the other recordings I’ve heard well enough to compare). Still, I can’t resist indulging, at least for a moment, in the thought that Beethoven is seeking some mysterious nuance here. He did after all quite deliberately create a surreal atmosphere by opening the oratorio in the highly unusual (at least in 1802) key of E-flat minor — a stroke that has permanently endeared this piece to me, whatever its flaws.

Every continuous function bounded implies compact

February 18, 2009

It occurs to me that it might be nice to post solutions to miscellaneous mathematical exercises at least once in a while.

I saw this one on a chalkboard earlier today; evidently the room was serving as the venue for an analysis class. It’s exactly the sort of elementary exercise that usually takes me a day to solve, if I’m lucky. But this time, I’m happy to report, I managed to figure it out in just a few minutes (while ostensibly listening to a lecture on something else).

Problem: Let U \subset \mathbf{R}^n be such that every real-valued continuous function on {}U is bounded. Prove that U is compact.

Read the rest of this entry »

Bach by popular demand…

December 31, 2008

Well, as requested, at any rate…

Here is an analysis of the first two measures of the B-flat major Prelude from WTC I.

Warning: this analysis breaks some of the rules (well, one in particular) of strict Westergaardian theory as expounded in ITT. In fact, it does so twice (at two distinct stages). Exercise: see if you can identify the rule that is broken, and give a convincing rationale for relaxing it.

1. The basic structure:

2. Segment the final two beats of the first span with: an incomplete neighbor in the bass, a complete neighbor in the soprano, a borrowing from the bass in the alto, and a rearticulated suspension in the tenor:

3. Anticipate the G in the soprano:

4. Borrow from these structural lines to create the texture of the passage:

5. Delay the fourth half-note:

6. Elaborate further (the operations being, I hope, clear):

7. Elaborate still further to obtain the passage as Bach gave it to us:

Happy New Year!

Principles of Westergaardian theory: Lines

June 2, 2008

Last time we discussed notes, the atomic units of musical structure. The topic of today’s installment is probably the single most important idea in Westergaardian tonal theory: the concept of a line. This material comes from Chapter 3 (probably the most important chapter) of ITT.

Lines, in Westergaardian theory, are the things that notes live in. This, you will observe, is the most salient and probably the most important way in which Westergaardian theory contrasts with harmonic theory. In harmonic theory (or, if you prefer, the “harmony-and-voice-leading” model), the things that notes live in are chords. More on this below.

A line is a chain of consecutive notes that we think of as being connected in a special way. From ITT, p. 29:

Take the notes

If we consider the lines to be

we are in effect saying that there is a special sense in which the first E and F are connected, or the C and the D, that is not true of the first E and the D, nor of the C and the F.

If, on the other hand, we consider the lines to be

we are saying that the E and the D or the C and the F are connected in this special way and that the C and the D or the E and the F are not.

Usually, we have reason to consider only some of the possible ways of analyzing a given set of notes into lines. We may, however, wish to consider different parsings of the same notes for different purposes. Lines, in fact, can be of a number of different types. As Westergaard says (ITT, p. 289):

There are different kinds of reasons for understanding one note as connected another and, hence, there are different kinds of lines. Where a series of notes is played by a single instrument or sung by one voice, we speak of an instrumental or vocal line, for example, the clarinet line or the tenor line. When a series of notes maintains the same registral [i.e. pitch-space order] relations to the other notes present, we speak of a registral line, for example the top line or the middle line. [Footnote concerning the ambiguity of terms like “alto line” omitted.] Finally, when a series of notes forms a [time-]span and pitch structure that gives us a way of understanding other notes in terms of that structure, we speak of a structural line.

These categories, incidentally, are not disjoint. In fact, since a line itself is a way of understanding notes, we could even regard the category of structural lines as encompassing all other types, including the first two mentioned above.

Not only are these categories not disjoint, but lines of the various types are frequently involved in complex dependence relationships. For example, we understand instrumental lines such as

(from Beethoven, Symphony No. 8 ) in terms of the “pseudo-structural lines”

which, in turn, we understand in terms of the “real” structural lines

(The question of how we understand the lines in this way will of course have to await future posts.)

Contrast with harmonic theory

Westergaardian theory makes virtually no a priori assumptions about musical texture (i.e. how many lines, of what types, are unfolding at once during a composition). All that Westergaard says is:

[W]e can conceive of a piece of music as being made up of two or more such lines unfolding simultaneously.

(ITT, p. 29.) (One point that, unfortunately, is not emphasized in ITT, but which I think needs stressing, is this: lines, like the notes of which they are made, are associated with time-spans. Thus, some lines may have longer durations than others; there is no a priori assumption that the texture should somehow remain constant. Some lines may extend through an entire composition; but a line could also theoretically consist of a single note.)

Furthermore, lines do not have to be of a particular type (e.g. structural, as opposed to instrumental) in order for Westergaardian theory to apply to them; you may (and, ultimately, must) begin a Westergaardian analysis of an orchestral passage, for instance, by looking directly at the individual instrumental lines — lines to which Westergaardian theory already applies in its full official formality. This should be contrasted with traditional “harmony-and-voice-leading” theory, which operates in the setting of a four-part texture, into which all other textures must (by some voodoo magic that is never quite specified) be transformed.

As noted above, however, the most important difference between Westergaardian theory and harmonic theory is the mere fact that Westergaardian theory views music as being composed of lines in the first place. Harmonic theory, on the other hand, views music as being composed of chords — simultaneities consisting of three or more notes. Although harmonic theorists acknowledge the existence of linear structures in music, for them the chord, not the line, is the fundamental note-generating entity; lines are then the epiphenomenal byproducts of chord progressions. The Westergaardian theorist views the situation is exactly the opposite way: lines are where notes are generated, and chords are the result of more than one line unfolding at the same time. This may be illustrated visually as follows:

In neither model is it a question of “slighting” one dimension or the other; both vertical and horizontal are present in both theories. The question is, rather, which dimension comes first; that is, to which dimension do notes themselves belong?

The distinction is readily apparent in the way that compositional exercises are conceived. In the harmony/voice-leading model, the task is to construct a progression of chords, taking care that the horizontal connections between notes obey certain rules (e.g. retention of common tones, no parallel 5ths, etc.). In the Westergaardian model, the task is to construct a complex of simultaneous lines, taking care that the vertical coincidences between notes obey certain rules (e.g. in first species intervals must be consonant; no parallel 5ths, etc.). In both models, one dimension is where “creation” occurs, and the other imposes constraints; the two models differ as to which is which.

In harmonic theory, the function of a note is defined by whether it is the “root”, “third”, or “fifth” of “the chord”. In Westergaardian theory, the function of a note is defined by the linear operation used to produce it (passing tone, neighbor, etc.– as will be discussed in a future post).

(Thus we see, for example, that a question that one often confronts in a harmony exercise, namely which component of the chord to “double”, makes no sense from the standpoint of Westergaardian theory. “Doubling”, as we shall see, is an operation that applies to lines, and not to notes. The latter do not exist independently of lines! Each and every one of them must be generated from within some line by some linear operation. Hence the collection of pitches (and thus also pitch-classes) that are sounding at a given moment is not determined except by the combination of linear operations that are being applied at that moment. The question is always “What operation?”; never “What note?”!)

(Warning: polemical passage follows.)

I don’t work as a professional music theorist, so I don’t have to be diplomatic about the fact that only one of these models is correct. The fact is that harmonic theory just has things totally backwards, and it’s high time this was acknowledged.

It’s no use trying to weasel out of reality in postmodern fashion with some nonsense about how all models have something to offer. For this would be nothing less than to deny the possibility of ever making mistakes in music theory — which in turn would be to deny the possibility that such a thing as musical knowledge can ever be attained. But as the acquisition of musical knowledge is after all the fundamental aim of music theory, we must expect that sometimes we will just need to say “Oops” and move on. (Just as the student composer or writer must learn that not everything he or she comes up with in the course of composition needs to be preserved in the final product.)

Rameau’s theory of the fundamental bass was simply a mistake — arguably an understandable one, given the historical circumstances, but a mistake nonetheless. Had Rameau never lived, no one need ever have thought up the idea of “root progressions”, and musical history would have been none the worse for it. Rameau’s theory was in fact controversial in its own time — two noted opponents having been J.S and C.P.E. Bach — so why should it not be in ours, when its flaws are, if anything, even more manifest than they were in the eighteenth century?

(Harmonic theory is unfortunately so deeply ingrained that it is frustratingly difficult even to get people to understand that we are talking about a comparison between two alternative models of musical structure, as opposed to simply disregarding one aspect of the traditional model. It’s as if it never occurred to them that harmony-and-voice-leading theory might have any competitors. Witness for example this comment of Scott Spiegelberg from last year’s discussion:

What you are doing is focusing solely on voice-leading, ignoring harmony completely, so you are like Rameau in ignoring one important aspect of music.

Now, I don’t want to claim that this would still represent Spiegelberg’s view after all the subsequent discussion; but it is at any rate a common type of reaction.)

As I have previously indicated, the Rameauvian directive to parse music into chords rather than into lines is what is responsible for the Myth of Atonality — the idea that diatonic scale degrees are not relevant to certain twentieth-century music such as that of the Second Viennese School. The Myth arose because theorists could not locate any of the familiar “chords” in the music of Schoenberg, Berg, and Webern, and thus concluded — by a complete and total non sequitur — that this music must be based on principles of organization radically different from those of earlier music. Had earlier theorists been clever enough to invent Westergaardian theory, we could have been spared the whole “atonality” business, along with all the accompanying theoretical, compositional, and even philosophical hand-wringing.

(End polemic.)

To summarize: Westergaardian theory is not “harmony-and-voice-leading without the harmony part”. It is Westergaardian theory — an alternative model of music that stands in opposition to the “harmony-and-voice-leading” model. The two models make conflicting claims about the structure of music. One of them tells us to conceive of a passage as a horizontal juxtaposition of vertical pitch-class sets called chords; the other tells us to conceive of the same passage as a vertical juxtaposition of horizontal chains of notes called lines.

(Schenkerian theory, by the way, is the result of Heinrich Schenker’s gradual realization — over the span of three decades, and never quite carried to completion — that the first model is incorrect, and that a model of the second type is needed. Westergaardian theory, however, is already a model of the correct type, right from the outset.)


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