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.)