The Westergaardian vertical dimension, Part 1

Having done enough harmony-bashing for the moment, I’ll try to advance the discussion in a more positive manner, by presenting some ideas of Westergaardian theory. In fact, I’ll even offer an olive branch to harmonicists, by suggesting a possible legitimate use for Roman numerals! Whether or not you continue to use Roman numerals, if you find something here that’s useful, or that you hadn’t considered before, then I will have accomplished my goal. (The treatment given here is far from exhaustive, or even adequate — so I encourage readers to refer to ITT for more information. Material of my own that is not in ITT will be written in colored text, with the color depending on whether it describes ideas that are fairly clearly implicit in Westergaard, or more substantial extrapolations.)

We’ll begin by recalling Scott Spiegelberg’s query:

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… 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?

That’s certainly a good question to ask. Clearly, no theory of tonal music (or, I would argue, any music) can be of much use unless it takes into account how simultaneous lines interact with each other to create an overall structure. So let’s see what Westergaard’s tonal theory has to say about the vertical dimension. (As it turns out, it has quite a lot to say.)

Let’s first review some basic concepts about the horizontal dimension, i.e. the structure of individual lines. We’ll stick to simple cases, to keep the focus on the main ideas. Basically, the purpose of a line is to move from one tonic triad pitch to another over a particular timespan:

The notes of the line are contructed using the basic linear operations: Rearticulate (R), Embellish By Neighbor (N), Arpeggiate (A) Connect By Step Motion (P).

Each operation segments a timespan into two or more subordinate timespans, over which the operations may again be applied, and so on indefinitely. (The fact that the operations may be iterated at will to build structures of increasing complexity is one of the most fundamental concepts of musical thought, but one which receives far less emphasis than it deserves in introductory treatments of music. I attribute this to an inappropriate focus on objects (such as chords) and their uses, rather than processes (like these linear operations) and their applications.)

Aligned and Parallel Structures

Now suppose we want to combine two (or more) lines together simultaneously over a span of time. The simplest way to do this would be to apply the same operations, segmenting the span in an identical manner, in both lines:

In this situation, we say that (the structures of) the two lines are functionally parallel. Note that this does not imply parallel motion; two lines might have functionally parallel structures and yet move in contrary motion:

A more general concept is that of alignment. We say that (the structures of) two (or more) lines are aligned when they segment the span identically, and in such a way that simultaneously sounding notes also correspond hierarchically:

Obviously, all functionally parallel structures are also aligned; but the converse is not true.

Now, to prove that I’m merely an open-minded seeker of increased theoretical power, and not a single-minded zealot who thinks he’s found the Ultimate Final Theory contained in the One True Book, I’ll make a definition that isn’t in Westergaard. Behold my olive branch, O devotees of harmonic theory:

Definition: A chord progression is a structure consisting of three or more aligned lines.

(I also like to refer to this as a type of “conspiracy”.)

Example:

Here is an example of something that is not a chord progression:

Only the lower voices are aligned, while in the upper voices the span isn’t even segmented. Finally, here is something that isn’t a chord progression itself, but contains a chord progression within:

It goes without saying that status as a chord progression is dependent upon the structural level under discussion (indeed, this is already true for the Schenkerian theory of chord progressions).

If you want to have a (tonal-syntactic, as opposed to motivic) notion of “chord progression”, this is how you do it. (The astute reader will of course realize that the important point here is that the phenomenon of chord progressions, or “harmonic motion”, is reducible to that of alignment of linear operations, and hence need not — should not — be taken as a primitive concept.)

This concept suggests a possible use for Roman numerals: to denote triads resulting from chord progressions. For example,

– a construction, incidentally, that would not be tolerated at the surface in the “common practice period”, presumably because composers wanted to keep the focus on elaborating (single) triads, not progressing among them — might be labeled “I-II-I”.

Cadences and Hierarchical Levels
The alignment of multiple lines, especially over longer timespans, is an extremely powerful structural device; when more than one long-range structural line arrives at its destination simultaneously, the effect is unmistakeable. This is the phenomenon of the cadence.

Of course, what counts as “simultaneously” depends on the structural level under discussion. Cadences often lie beneath the surface:

while not technically a cadence, obviously stands for:

which is a cadence, in the literal sense of simultaneous arrival.

Ultimately, in fact, the principal structural lines of a work can always be understood as aligned at a sufficiently remote level of structure. Even a twentieth-century “atonal” composition with a background looking like:

may be understood, at an even more remote level, in terms of:

and thus, ultimately, as an elaboration of:

the lines of the latter structure being trivially aligned!

The point is that what determines the “strength” of cadences is precisely the extent to which deep structural alignment is realized explicitly as alignment at surface levels. In particular, it is not a question of mere pitch-class content. The difference between twentieth-century music and earlier music is not that the pitch-class content of the latter is “too strange” to be understood as generated through the tonal operations; it is that you have to go very far back into the background to find the degree of linear alignment that one is used to hearing at or near the surface in works of the “common practice period”. It is simply a question of increased complexity; there is no need to invoke new systems of musical grammar. (Or rather, there would not be, if people were accustomed to conceiving musical grammar in terms of operations rather than objects.)

Let us pause to reflect on the irony of this state of affairs. For all that Schenker railed against new music, the reason it seemed so foreign to him was because he could not hear deeply enough into the background! (Although he did come closer than most: cf. his analysis of a few bars from Stravinsky’s Piano Concerto, about which — speaking of irony — I once heard a lecture by Joseph N. Straus, the very same author of a well-known article arguing that Schenkerian theory doesn’t apply to “post-tonal” works ["The Problem of Prolongation in Post-Tonal Music", JMT Vol. 31 No.1 (Spring 1987), pp. 1-21]. Why, I have always wondered, haven’t theorists like Straus ever asked themselves what specifically Schenker would have said had he chosen to tear apart a piece by Webern, for example, rather than the Stravinsky concerto?)

Next time: Vertical Sonorities in ITT