Principles of Westergaardian theory: Lines

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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5 Responses to Principles of Westergaardian theory: Lines

  1. funkhauser says:

    Okay, so I’m back, and I have another question. I’m going to phrase it like a challenge to Westergaard’s theory, but actually it’s not. I actually more or less agree with the Schenkerian/Westergaardian approach to music. But after all, a good way to better appreciate a theory is to try and break it.

    So in Westergaardian theory, arpeggiations, that is borrowings, can be anticipatory. Say we start with a basic Schenkerian 3-2-1 Urlinie over 2 measures (by which I mean scale degree 1 occurs on beat 1 of measure 3). If in the third measure we have this scale degree 1 harmonized by a scale degree 3 (so as to start another Urlinie say), then we could pull back this 3 to the previous measure, say to beat 4 of measure 2. In the end, what we would have is a 3-2-1 where the 2 – 1 motion is interrupted by an anticipatory 3.

    As far as I am aware, this doesn’t commonly happen. One would say, in harmonic terms, that you usually don’t see a “IV -V – iii – I cadence”. In fact, the effect sounds “weak”. It’s as if you’ve drained some of the momentum of the music when you place scale degree 3 between the dominant and tonic chords. But as far as I can tell, Westergaardian theory admits this very type of construct.

    There is one passage in ITT (Ch. 4, p. 58) in which Westergaard says “Warning: inserting the mediant between the final statement of the dominant and the final tonic of the bass arpeggiation may obscure the fundamental relationship of dominant to tonic–if in doubt, don’t do it”. But he does not explain this further, leaving the statement sounding rather ad hoc. I mean, he just pulls this “fundamental relationship of dominant to tonic” business out of thin air. I understand instinctually what’s he’s talking about, because I’ve studied music. But Westergaard never makes such a notion a formal part of his theory, as far as I can tell.

    So I’m wondering–IS there a good Westergaardian explanation for why one shouldn’t insert the mediant between the final dominant and tonic? Thank you.

  2. James Cook says:

    funkhauser,

    First off, sorry for taking so long to respond; the explanation is that I’m currently in the process of moving (so posts and comments may continue to be sporadic for a while).

    I think it’s important to correct a certain misunderstanding that shows up frequently. The task of Westergaardian theory is neither to mandate what one “should” or “should not” do in constructing music, nor to account for statistical patterns that occur in historical works. The task of Westergaardian theory (on my interpretation at least) is simply to take the data of musical compositions — whatever they happen to be — and break them up into their simplest conceptual components. Therefore, you certainly shouldn’t be surprised that Westergaardian theory admits any particular construction, however uncommon it may be. (In fact, as you may know, I personally take the radical view that Westergaardian theory admits everything found in Schoenberg, which obviously includes things much more complicated, problematic, and work-specific than what we are talking about here.)

    For what it’s worth, I don’t share your impression that the anticipatory 3 before final 1 is a rare construct. Just off the top of my head, see, for example, the end of the slow movement of Bruckner’s 4th symphony (where there’s no danger of missing it, this being Bruckner). Also, numerous passages by Elgar, who used this effect so frequently that it amounted to something of a personal mannerism.

    Now, to your main point about the “fundamental relationship of dominant to tonic”. You’re right that Westergaard does not adequately explain this. The reason may be that the material of ITT started out as a set of ad-hoc rules for counterpoint classes that Westergaard taught at Columbia in the 1960s; only later did this evolve into a full-blown theory of tonal music. In fact, I suspect that the evolution continued to occur as the book was being written. The upshot would be that chapters 4-6 belong to an earlier stage of the theory’s history than the rest of the book, before Westergaard set himself the task of redoing tonal theory from first principles. (I’m just speculating here; I haven’t actually spoken with Westergaard about this.)

    However, we can try to piece together an account based on the entirety of the various works of Schenker and Westergaard. Here’s a preliminary attempt of my own:

    Contrary to widespread (Rameau-inspired) belief, the “fundamental relationship” is not actually the fifth between 5 and 1, but rather the descending step 2-1. This is made clear in late Schenker, but even if it weren’t, one could construct the argument on one’s own. The 1-5-1 (or V-I, if you’re Schenker) bass arpeggiation emerges as a later byproduct of the 3-2-1 fundamental line. 5 comes into the picture because it makes a fifth with 2, and potentially gives rise to a new triad. Schenker would say that V harmonizes 2; the Westergaardian way of understanding it would be that 5 (like 3) is present throughout the whole span, in its own line, and is borrowed by the the bass in order to achieve alignment (ITT, sec. 4.4) with the passing 2. This alignment (or, for Schenker, the new Stufe), with the soprano and bass a fifth apart (the weakest sonority for two different pitch-classes), has the effect of emphasizing the 2, and thus the 2-1 descent; it is this emphasis which is responsible for the power of the “V-I cadence” effect. But the essence of the effect itself is really the 2-1 in the upper voice, not the 5-1 in the bass.

    Understanding this, one can see that inserting something between the 2 and the 1 (such as a borrowed 3), has the potential to undermine the force of the progression. In order to achieve the cadence, one needs to keep the 2 prolonged until the 1 arrives. An interpolated 3, being step-related to 2, makes this sense of prolongation difficult to retain (regardless of which voice the 3 appears in). It can work provided it is made clear that the 3 is an anticipatory borrowing, as opposed to a displacement of the 2.

  3. funkhauser says:

    James,

    Don’t worry about taking your time to respond. I have plenty of crazy stuff going on this summer too, so I understand.

    “It can work provided it is made clear that the 3 is an anticipatory borrowing, as opposed to a displacement of the 2.”

    I totally agree with this. In fact, I started thinking about this soon after my post. After all, an anticipatory borrowing of the 3 is what I suppose is responsible for the “V add13″ chords one finds in jazz.

    Also, I’m really glad you cleared up the point about Westergaard’s theory admitting any type of construct. I am certainly guilty of the misunderstanding you mentioned. I haven’t listened to the Bruckner symphony yet, but thanks for the suggestion!

  4. drieway says:

    Занимаюсь дизайном и хочу попросить автора mathemusicality.wordpress.com отправить шаьлончик на мой мыил) Готов заплатить…

  5. [...] evolved as a shorthand notation for species counterpoint.That is, figured bass actually suggests lines, not chords. Consider the example [...]

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