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.
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.)