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.)

J.S. Bach: Air from Orchestral Suite No. 3, mm.1-2

Recall that in a previous post I challenged readers to analyze the first two measures of the Air from Bach’s Orchestral Suite No. 3 in D major (a piece, incidentally, that might be better referred to as “Air Off The G-String” than by its usual nickname). The time has come to reveal the answer.

In the Pachelbel analysis, we started from the underlying basic structure and showed how the passage was constructed via the Westergaardian operations. This time, for the sake of variety, we’ll proceed in the reverse direction, starting from the passage itself and “undoing” the operations until the basic structure is revealed.

Our passage is the following:

12.

Call this Stage 12. The first thing we’ll undo are the explicit arpeggiations in the first violin and continuo lines:

11.

Actually, I did a bit more than that, as you can see. I skipped a stage in which the first violin part looks like:

How did I know that D was the span pitch of the second half of beat 2 rather than C#? That is, why did the first violin part not reduce to:

Is it because G#-E-B (or even G#-E-B-D) is a Certified Chord, whereas G#-E-B-C# isn’t? Fat chance! As an exercise in eliminating harmony, see if you can explain the real reason. (I’ll likely explain it in a future post, but probably only after we’ve formally developed more Westergaardian theory. Hint: It has nothing to do with Certified Chords.)

Eliminating the borrowed G and B from the first violin, we obtain stage 10:

10.

What an odd interpretation of beat 2! Instead of hearing a passing motion from E to C, I am interpreting the E as a borrowing from the viola line:

9.

(Note also the elimination of the A borrowed from the second violin line.) Why on Earth is this interpretation to be preferred to the seemingly simpler one? The answer is that the seemingly simpler one isn’t in fact so simple. Notice that the D in the second violin line is left hanging (ITT, p. 30), and therefore not displaced, after beat 1. If the D in the first violin line were to be interpreted as a passing tone, that would leave us without a D among the sounding span pitches of beat 2. However, we know from the C# of beat 3, as well as from the fact that D was left hanging in the second violin, that D must be a span pitch for some span that includes beat 2 (deeper levels will make this clearer; see below). We would therefore be compelled to regard the second violins’ D as being temporarily displaced during beat 2; that is, it must move by step to some note borrowed from another line. (The only alternative would be to regard it as (entirely) undisplaced during beat 2, but this is made difficult because of the simultaneous E: since in this scenario we’re not considering D as a local span pitch of beat 2, we’re left with understanding an implicit dissonance, which is quite problematic indeed.) Since E is a span pitch of beat 2 and C# is not, we must therefore hear the D-line as moving up to a borrowed E during beat 2. But why should we go through the trouble of understanding such a conceptually difficult situation as the D-line effectively “merging” temporarily into the F#-E line? Given the stated step motion D-C# in the first violin, isn’t it easier to regard that D as a span pitch over the span of beat 2?

Stage 8 shows transferred pitches (ITT, sec. 7.7) reassigned to their rightful homes:

8.

This stage represents the transition from instrumental lines to structural lines; I have symbolized this by switching from the alto clef to the treble clef in the third line.

Next the transferred pitches are reassigned to their rightful registers:

7.

The suspension in the top line is removed:

6.

Rearticulations in the bottom three lines:

5.

Rearticulation of a suspension in the second and third lines; chromatic step motion in bass:

4.

Suspensions eliminated:

3.

Neighbor note removed:

2.

Finally, then, we have the basic structure of the phrase:

1.

Felix Salzer agreed with me

From Unfoldings: Essays in Schenkerian Theory and Analysis, p.4, emphasis mine:

[Joseph N. Straus]: What was the nature of your early work with Salzer?

[Carl Schachter]: I studied counterpoint with him. He didn’t like to talk about harmony as a discipline in itself, but we did all kinds of melody and bass settings and things of that sort, both written and at the keyboard. I had two years of analysis class with Salzer; I also studied music history with him. He was a very comprehensively educated musician, and so he taught everything other than subjects like orchestration or dictation or sight-singing. My basic musical training was with him.

Could that be because “harmony” is not in fact a legitimate “discipline in itself”?

(Note, by the way, how this undermines Scott Spiegelberg’s claim that his take on Schenker is the same as Salzer’s, since Spiegelberg very clearly does like to talk about harmony as a discipline in itself.)

Principles of Westergaardian Theory: Notes

Over the past week, I have been hard at work on a couple of rather involved music-analytical posts, as well as various of the Mathematics Lectures. It occurred to me, however, that I might take a bit of time out to begin the promised systematic exposition of Westergaardian theory. For one thing, it would be nice to have something online to refer to when writing up analyses; but, to be honest, the proximate reason I decided to start this now was that there are some things I would really like to get off my proverbial chest, and the appropriate place to do so will be in the second post of this series, which will be about the concept of lines.

First, we have to talk about notes. This material comes from chapter 2 of ITT, though my discussion of pitch differs in some minor respects from Westergaard’s. (For the moment, I’m skipping chapter 1, which deals with meta-issues, because 1) I’m in a hurry to get to chapter 3 and 2) there will be plenty of opportunity to talk about meta-issues as they come up.)

***

The most basic element of musical structure is the note. A note is defined to be a unit of sound that we think of as having

1. a particular pitch
2. a particular onset time
3. a particular duration

In addition, we may also think of a note as having

1. a particular loudness
2. a particular timbre

The first three attributes are mandatory: they are necessary to determine the syntactic value of a particular note. By contrast, the latter two attributes are in a sense optional: their function is to clarify or reinforce the syntactic value of a note.

I assume that readers are familiar with musical fundamentals, and so I won’t bother to go into too much detail here about how each of these dimensions is conceived; a quick run-through will have to suffice. If anything needs clarification, feel free to ask in the comments. Note that Westergaard gives a characteristically thorough exposition in chapter 2 of ITT. That exposition is far superior to the one given here, as will be obvious to anyone who reads both.

Pitch

The space of pitches is divided into semitones. A semitone is the interval from the pitch of one key on a piano to the pitch of an immediately adjacent key. The size of a semitone is such that the interval of twelve semitones corresponds to a doubling of the frequency (recall that pitch perception is logarithmic with respect to frequency, so that pitch intervals correspond to frequency ratios). Such an interval is called an octave. For some purposes we shall consider pitches an octave apart to be equivalent; the equivalence classes so obtained are called pitch-classes. We name pitch-classes by numbers 0,1,2,…10,11 (0 being the class of the pitch of the “middle C” key on a piano, 1 being the class of the next higher key, and so on), or by letters in a manner that will be discussed below (”middle C” being indeed an instance of this nomenclature).

We conceive of pitches not only as elements of the semitonally-divided pitch space, but also as elements of special subsets of this space called “diatonic collections”. Consider the pitches of the seven white keys on a piano starting from middle C and continuing upward (to the right); call this collection of pitches S. We define a diatonic collection to be a transposition of S by some number of semitones. (Thus S itself is an example of a diatonic collection.) By abuse of language, we also use the term “diatonic collection” to refer to the set of pitch-classes corresponding to the pitches of some diatonic collection.

This furnishes an alternative nomenclature for pitch-classes, defined as follows. For historical reasons, the pitch-class 9 is called A. The elements of the diatonic collection {9,11,0,2,4,5,7} are then called respectively A,B,C,D,E,F,G. Arbitrary pitch-classes, in turn, are named as if they were conceived of as transpositions of an element of this collection. Thus pitch-class 1 may be called (upward transposition of C by one semitone), (downward transposition of D by one semitone), (conventionally written ; upward transposition of B by two semitones), (downward transposition of E by three semitones; or indeed , downward transposition of by two semitones), etc. This system is convenient because we do indeed conceive of pitches in terms of some diatonic collection (though the particular collection is determined by context, and is not always {A,B,C,D,E,F,G}).

We also use diatonic collections (in the strict sense, as a collection of pitches, rather than pitch classes) to conceive of intervals between pitches. The interval from a pitch to itself (such as from middle C to middle C) is called a unison (or prime). An interval between adjacent members of a diatonic collection is called a second (or step). Other intervals are named according to the number of seconds from which they are built up:

Two seconds: third
Three seconds: fourth
etc.

A second may be either a semitone (half-step, or minor second) or 2 semitones (whole-step, or major second). Likewise, other intervals come in different varieties, depending on how many of the seconds used to construct them are major and how many are major. (Any pattern may be used provided that it fits into a diatonic collection; thus a third may be built out of two major seconds, or out of a major second and a minor second, but not two minor seconds.) The intervals of a unison, an octave (seven diatonic steps), a fourth of the type consisting of two major seconds and a minor second (as from a particular member of pitch-class C to the first member of F above), and a fifth of the type consisting of three major seconds and a minor second (as from C to G) are called perfect intervals. An interval obtained from a perfect interval by raising the higher pitch (or lowering the lower pitch) by a semitone is said to be augmented; thus the interval from a (particular member of the pitch-class) C to the first (member of) F# above is an augmented fourth. Likewise, an interval obtained from a perfect interval by lowering the higher pitch (or raising the lower pitch) is said to be diminished: thus the interval from C to Gb is a diminished fifth.

(Note that Gb and F# both refer to pitch-class 6, so that both an augmented fourth and a diminished fifth refer to an interval of six semitones; such pairs of pitches or intervals are said to be enharmonically equivalent.)

Other intervals (thirds, sixths, and sevenths) come in two types, as the reader can easily verify. The larger type of each is called major, and the smaller type minor. Expanding a major interval by a semitone yields (again) an augmented interval; contracting a minor interval by a semitone likewise yields a diminished interval. (Thus C to A# is an augmented sixth; C to Ebb is a diminished third.)

Time

From ITT, sec. 2.2:

We conceive of time in tonal music in terms of systems of equally spaced reference points…We call the reference points beats. If a note begins at a reference point we say it is “on the beat”; if note, we say it is before or after the beat or simply “off the beat”. We call primary reference points downbeats. Secondary reference points are sometimes called upbeats, but properly speaking upbeat is reserved for that secondary reference point immediately preceding the next downbeat. We call the span between consecutive primary reference points a measure. We say that a note that begins on the downbeat and lasts until the next downbeat “lasts a measure”. We call the segments formed by the secondary reference points beats*.

If a note begins on one beat and lasts to the next beat we say it “lasts a beat”. We call the way the secondary reference points divide the spans between primary reference points the meter. One secondary beat dividing each measure into two equal parts is called duple meter; two secondary beats dividing each meaure into three equal parts is called triple meter. We call the rate at which beats occur the tempo. A rate of around 85 beats per minute (time from one beat to the next is about seconds) is usually considered a moderate tempo; most tempos fall between twice and half that rate.

*An unfortunate double use of the same term to mean both a point in time and a period of time between two points.

Loudness

We conceive of loudness as measured by a scale whose only structure is that of a totally ordered set:

(For further discussion, see ITT, sec. 2.3.)

Timbre

Piano, violin, clarinet, etc.; see ITT, sec. 2.4.

Coffee and Math

The title of a new blog (discovered via Rigorous Trivialities). One of the first posts calls our attention to The Catsters, who have uploaded a collection of videos on category theory to YouTube — and also this nontechnical one on the Klein bottle:

Musical magisteria, or lack thereof

A comment I left (or attempted to leave; comment moderation is enabled, so I don’t yet know if it was successful) at this post on the Texas Tech music theory blog:

Oh dear, where to start?

Well, you certainly have put your finger on it when you write:

Forgive my generalizations, but it seems to me that the compositional approach stems from a time when composition and theory were basically the same thing, hence, this approach is favored by an earlier generation of pedagogues.

Yes, indeed! The whole distinction on which your post is premised, namely that which is alleged to exist between “compositional” and “analytical” approaches to music, exists only because, once upon a time, “theory” (or “analysis”) stopped yielding insight into composition! And instead of saying “Oops, we must have gotten our theory wrong” and fixing the problem, which would have been the proper thing to do, people instead decided that they were involved in a new distinct field of study called “analytical theory”. That way, they didn’t have to discard the erroneous ideas to which they had become attached; they could simply relabel their occupation and move down the hall.

Sadly, people do this kind of thing all the time, and not just in music. The modern concept of religion is another example. Once upon a time, people believed that supernatural agents such as gods were needed to explain the natural world; then along comes science, and what do people do? Instead of simply biting the bullet and admitting that the whole God theory was just plain wrong, they invent the concept of non-overlapping magisteria and assign new purposes to religion (”it gives us morality” or “provides meaning and purpose”, etc.).

Like the religious, music theorists are also adept in the art of post-hoc (re)justification. Thus, when harmonic theory (which holds that music is constructed out of “progressions” of “chords” built on “roots”) was finally and utterly disproved by 20th century music, theorists continued to teach it anyway, on the (newly invented) grounds that studying old music is a sort of “separate magisterium” from learning how to make new music.

Which brings us back to your post. If different musical repertories are all separate magisteria, then a student with a particular interest in only some of them may legitimately wonder why he/she should bother studying the others. The answer is that they aren’t separate magisteria; and in fact it isn’t a question of repertory at all. Different repertories do not have different theories of music, just like different planets do not have different theories of physics.

The reason musical study should begin with strict species counterpoint has nothing to do with any special virtue possessed by music of the sixteenth century; in fact it has nothing to do with the sixteenth century at all! (Despite generations of misunderstanding.) The actual justification is to be found in section 4.0 of Westergaard (”What Species Counterpoint Is And What It’s For): it is that species counterpoint is simpler than actual music. Even more to the point, it is a way to approach the study of music systematically. One concept at a time, in logical order. It has nothing, repeat nothing, to do with a particular “style” of music!

I read your post as advocating, or leaning towards advocating, a kind of eclecticism in music pedagogy: let’s bring in a lot of different complex things to throw at the students. In fact, let’s even throw different complex things at different students, in different years! But this is antithetical to what is needed. What is needed is, first of all, logical, systematic training in (and not eclectic exposure to) the actual practice of creating music; and secondly, a sufficiently unified conception and metalanguage so that people with immediate interests in different musics or aspects of music can speak to each other and have some understanding of why they are in the same department.

IMSLP coming back

Apparently, the Wikigods have decided to resurrect IMSLP. And all I had to do was say the word!

(Via Musical Perceptions.)

Pachelbel’s Canon

As you might expect, the demise of the IMSLP has put something of a damper on my grandiose plans of analyzing musical works on this blog. Today, however, we’re in luck, as Wikipedia provides all the source material I’ll need for this post.

The context for this is a question, of sorts, posed by a commenter named “funkhauser” (the reason I say “of sorts” will hopefully become clear):

It seems to me that a surprisingly large number of progressions of 8 chords found in say, your favorite piece, have a IV chord as the “middle” chord (i.e., if each chord lasts a quarter note and we are in 4/4 time then the IV chord is the first chord of the second measure). Two familiar examples are Pachel[bel]’s Canon and the beginning of Bach’s Air on the G string. The chord progressions are, roughly:
I - V - vi - iii - IV - I - IV - V
I - iii - vi - vi7 - IV - V7/V - V - V7

(…)However, I can’t think of an 8-chord progression I’ve heard in which the V chord is the middle chord…Imagine putting the V in the place of the IV in Pachelbel’s Canon or Bach’s Air. It would completely change the feel of the piece (and arguably, it would ruin it).

(…)What I’m wondering is: In Westergaard’s theory can we derive the important difference in function between IV and V? And can we derive the fact that the IV should occupy stronger beats and larger time spans than V?

Tisk tisk. It’s obvious that the questioner has not yet managed to throw off the Rameauvian shackles, and is still laboring under the impression that musical passages are constructed by juxtaposing “chords” in time. Well, funkhauser, you’ve come to the right place — disabusing innocent souls of this mistaken notion has become one of my missions in life.

The best way to start, I think, is to take a look at these passages and see what’s actually going on. Here’s how to construct the opening of Pachelbel’s canon:

1. The underlying basic structure is the usual descent (with the on its way to , of course):

2. These structural lines will be realized as three textural lines, with span pitches assigned as follows:

(When I say “assigned” I technically mean borrowed, of course.)

3. The top two lines will both descend from the upper note to the lower:

4. The A is delayed by a lower neighbor, in familiar fashion:

5. We connect the F# to the A and the D to the F# by step motion. In fact, we’d like to have continuous quarter notes in these two voices, so on beat 3 of m.2 we’ll also elaborate the C# by a lower neighbor passing tone in the top voice (producing functional parallelism alignment with the bass) and borrow a G from the bass for the middle voice:

6. Actually, we’d like to have quarter notes in all three voices, so we elaborate the bass by means of borrowing :

(The A and the F# are of course borrowed from the span pitches of stage 2 above.)

7. Now, since this is supposed to be a canon, we’ll present the voices one by one.

8. Finally, this is how the texture is actually realized, in terms of which instruments play what.

Now, having analyzed the passage, let’s see if we can address funkhauser’s question. The first thing to note is that nowhere in the above derivation sequence is there any mention of “chords” at all. As a matter of fact, I didn’t even bother to check whether the progression claimed by funkhauser

I - V - vi - iii - IV - I - IV - V

is “accurate” or not — so that as I’m typing this, I literally don’t know what the “chords” of this passage are! I It’s important to emphasize this, because I just got through analyzing the passage in precise detail, attributing a specific function to every single note, and I have the passage itself, as well as my analysis of it, firmly entrenched in memory. Indeed, I can’t mentally replay the passage without instantly and simultaneously reconstructing my analysis. And yet — and yet — when it comes to selecting the appropriate Roman numeral for each of these quarter-note simultaneities, I am — at least at this immediate moment — about as clueless as a typical freshman theory student. (Though I do already know the first one will be I and the last one V.)

Having made that point, let me now pause to reflect on what the chords are…Okay, yes, funkhauser has got it “right”; though I suppose there is an ambiguity about beat 3 of m.1, since there are only two distinct pitch-classes in that simultaneity. Come to think of it, the same is true of both “IV” chords in m.2. Oh, and it’s also true of the very first chord!

(Notice how very different this type of thought is from the instinctive, intuitive reasoning that I used to construct the above analysis. Actually, “instinctive, intuitive” is not the correct description; what I meant to say was specifically musical. Whereas what I am doing here, in verifying funkhauser’s chord progression, is the totally abstract (if trivial) mathematical problem of verifying that two finite sets are equal to each other.)

Funkhauser asks about the difference in function of the IV and V chords. What I would like to point out is that there is no “IV chord” at all! The simultaneity on beat 1 of m.2 is just the coincidence of two passing tones, and that on beat 3 is just the coincidence of two neighbors a passing tone and a neighbor. To pick out these chords as fundamental objects in their own right (and as the same fundamental object, no less!), is to carve up musical reality in the wrong way, like putting dolphins in the same category as fish.

Strictly speaking, then, the answer to funkhauser’s question is “mu” — i.e., “your question depends on incorrect assumptions”. The “chords” of harmonic theory are simply not legitimate music-theoretical entities, any more than Earth, Air, Water, and Fire are chemical elements. Yes, these four things do exist, but they don’t play anything like the theoretical role that people once attributed to them. In fact, today we understand that not only are they not fundamental, but they’re not even the same kind of thing: “Earth” is a planet, “air” is a state of matter (gas), “water” is a chemical compound (H2O), and “fire” is a process (combustion).

So it is with “IV”, “V”, and all the rest. Yes, there are collections of notes in musical compositions to which you could give these labels, but to do so is to presuppose the wrong theory of music.

Like Aristotelian chemistry, harmonic theory may not seem obviously wrong until you’ve had considerable experience with the alternative. This explains why I invariably get reactions like “But…but…of course harmonic theory is correct (or useful) — look how ubiquitous progressions like I-IV-V-I are!”

Yes, and the “Four Elements” are also ubiquitous in the natural world.

For the moment, I will leave it as an exercise to come up with the correct analysis (or at least an analysis of the correct type) of the first two measures of the Air from Bach’s Third Orchestral Suite. Here’s a big hint:

The joy of “pathology”

Recently I had a conversation with a mathematician who had worked on the theory of Banach spaces early in his career, but had since left that particular subject. He explained that he had become disillusioned by the fact that “all the natural conjectures turned out to be false”; indeed, Banach spaces can have some “strange” properties, such as having uncomplemented subspaces, lacking a basis, admitting operators with no invariant subspaces, or admitting almost no operators at all. (Actually, in fact, they can even be unexpectedly well-behaved!) The last straw for this fellow, apparently, had been the Gowers-Maurey space (for which Gowers won the Fields Medal) , which has a whole bunch of “weird” properties.

Disparaging language is used with disturbing frequency by mathematicians to describe mathematical concepts. Examples are labeled “pathological”; objects are described as “badly behaved”; functions are called “nasty”; problems are said to be “ill-posed”. In a library once I encountered a book whose title actually was Differentiable Functions On Bad Domains — where “Bad” here is not the name of a mathematician, but the ordinary English word meaning the opposite of “good”.

You might think this is nothing but picturesque language — like calling a certain group “the Monster” — except that there are plenty of mathematicians who actually seem to think the way the labels suggest they do. The ex-Banach-spacer I mentioned above is only one example; spend some time among mathematicians and you will find many more. Indeed, such aversion to the unexpected has a distinguished historical pedigree, I am sorry to say. Who can forget the dismay with which Weierstrass’s construction of a nondifferentiable continuous function was greeted? And even now there are still some people who are pissed off about Cantor’s discoveries, and who would sooner overthrow the standard axioms of mathematics than confront the “paradoxes” of the infinite. Even Hilbert, who had the good sense to regard infinitary mathematics as paradisal rather than paradoxical, nevertheless reacted with anger (!) to Gödel’s results on the limits of formalization, according to Constance Reid. One is reminded of the discoverer of the irrationality of , who, you will recall, was allegedly thrown overboard by his Pythagorean comrades.

I have never sympathized with this way of thinking. As far as I am concerned, unexpected “pathological” phenomena are a large part of what makes mathematics interesting in the first place. Indeed, this accords with the attitudes of other kinds of scientists with regard to their own fields. You generally don’t find physicists crying in agony about the discovery of black holes, or biologists resenting the existence of extremophiles. Why, then, do so many mathematicians insist on doing the equivalent?

A “human” mistake

Update (3-21): Mysteriously (but fortunately), this particular error has now been corrected.

Or: Why you can’t trust Google Maps.

Consider this. Looks great, doesn’t it? A reasonably-nice-sounding hotel with high-speed wireless internet, located a mere 0.1 miles from the University of Missouri mathematics department.

Um, no. Guess again.

The La Quinta Inn of Columbia, Missouri is indeed located at 901 Conley Road — that much is true. That, however, is not what the map shows. What the map shows is where the La Quinta Inn would be if it were located at 901 Conley Avenue.

Conley Avenue, it turns out, is a street on the university campus. Conley Road, by contrast, is 3.4 miles (and a $15 taxi ride) away from the campus.

This is exactly the kind of mistake that you would expect a human to make, but that a computer should definitely not make. After all, you can easily imagine a human rolling their eyes at such a “pedantic” distinction as that between “road” and “avenue”, or assuming that no town would give two different streets the “same” name. (At least, I can easily imagine this, having tried to explain to calculus students that the domain of the function is not all of despite the equation .) A computer, however, is supposed to be ruthlessly precise — which is why you will (almost) never be given a break when typing web addresses, no matter how “close” you come to typing the correct character sequence.

So does this costly error indicate some sort of progress in the field of artificial intelligence, or should I just be ticked off that somebody at Google deliberately programmed the human “not-caring-about-the-distinction-between-’X road’-and- ‘X avenue’ ” bug into their software?

Oh, and the wireless internet at La Quinta is lousy.