• You’re Calling *Who* a Cult Leader? — in which Eliezer Yudkowsky (one of my favorite bloggers) points out that that it’s okay to be really enthusiastic about something:
  

  Behold the following, which is my true opinion:

  “Gödel, Escher, Bach” by Douglas R. Hofstadter is the most awesome book that I have ever read. If there is one book that emphasizes the tragedy of Death, it is this book, because it’s terrible that so many people have died without reading it.

  I know people who would never say anything like that, or even think it: admiring anything that much would mean they’d joined a cult (note: Hofstadter does not have a cult)[...]

  But I’m having trouble understanding this phenomenon, because I myself feel no barrier against admiring Gödel, Escher, Bach that highly.

  He continues:

  You know, there might be some other things that I admire highly besides Gödel, Escher, Bach, and I might or might not disagree with some things Douglas Hofstadter once said, but I’m not even going to list them, because GEB doesn’t need that kind of moderation. It is okay for GEB to be awesome. In this world there are people who have created awesome things and it is okay to admire them highly! Let this Earth have at least a little of its pride!

  Yes! As I have noted before, there is an inhibition in our culture against expressing strong feelings. Away with this!

  I’m embarrassed to admit that I still haven’t read GEB, even though everybody raves about it and it’s got both a mathematician and a composer in the title. Well, it’s now (higher) on my to-do list. But anyone who has ever visited this blog will know that I harbor a similar level of enthusiasm for Westergaard’s ITT. And that does not make me some kind of crazed fanatic.

• What’s in a number? — in which a member of the Texas Tech music theory faculty correctly explains the meaning of figured-bass symbols (link added by me):
  

  I often tell my students that figured bass evolved as a shorthand notation for species counterpoint.That is, figured bass actually suggests lines, not chords. Consider the example below:

  

  If you look at those examples without worrying about vertical sonorities, the figured bass makes quite a lot of sense. Once you begin trying to assign Roman numerals, the task becomes a bit muddier. In the first example, we can easily understand the E-F motion in the soprano as some kind of neighbor motion or perhaps as the beginning of a passing motion. I prefer that interpretation to one which says the first chord is a root-position tonic and the second chord is a first-inversion submediant.

  And rightfully so. In fact even to speak of this measure as being composed of two “chords” is a misleading distortion. If there are two entities into which this measure is divisible, they are a second-species line on the one hand, and a complex of three first-species lines on the other. (As a minor quibble, I will point out that “worrying about vertical sonorities”, which species counterpoint does just fine, is not to be confused with “assigning Roman numerals”, the discredited province of harmonic theory.)

  In short: figured bass tells us diatonic intervals above the bass and nothing else. If notes are to be altered, the accidentals will appear in the figured bass. Figured bass is simply a shorthand for linear motion.

  So true! Whether or not it is technically accurate that figured bass evolved in connection with species counterpoint per se, this is much closer to the truth than to suppose, as many still do, that it indicates some sort of preexisting awareness of Rameauvian “harmonic” concepts on the part of Baroque-era musicians. (Of course anyone who thinks that hasn’t read Schenker, but that’s for another time…)

(Not too long ago it finally occurred to me why this confusion exists. The reason for it is that people mistake figured basses, which are a form of musical notation, for some sort of analysis of the music. When you look in a treatise and see a figured bass at the top of a page, say, followed by a realization below, perhaps it’s natural to suppose that the figured bass on top is in some sense a “more primitive” structure, from which the realization is derived. But this is a misunderstanding. Figured bass was a performance practice; it was not the purposes of such treatises to engage in music theory as we know it, of the sort practiced by Westergaard — the subject had not yet come into existence as an explicit discipline. So one is by no means obliged to think of a passage in terms of some underlying figured bass. Quite the contrary, in fact: the figured bass is but a shorthand for the realization, and thus if anything the latter “explains” the former, rather than the other way around.)

(Boulez, for non-musical readers, is pretty much the leading figure — or at any rate the leading European figure — in the art music of our era [i.e. post-WWII]. For mathematical non-musical readers, an approximately equivalent person would be Jean-Pierre Serre.)

The interesting part of the site is, of course, the series of pages devoted to Boulez’s 1984 work Dérive I, which is described as:

an elegant, shimmering and vibrating eight minute work which explores harmony and texture from a chordal starting point using material which “derives” from three earlier pieces, Répons (1980), Messagesquisse (1976) and Éclat (1965).

(Obviously, “harmony” here is to be understood in the sense of vertical pitch collections — nothing to do with harmonic theory! I would probably have used the word “sonority” here instead.)

It seems the excerpts included comprise virtually the entire piece, so go have a listen. The commentary hardly constitutes a detailed analysis, but overall the quality is pretty good for something on the Internet.

(Via Complement.Inversion.Etc.)

You can hear the passage in question (which, of course, features Jesus in agony) beginning at 8:54 or so in this clip:

I admit, this could easily be a misprint for “Allegro agitato” (though the tempo in the above performance doesn’t strike me as quite fast enough for that; unfortunately I don’t remember the other recordings I’ve heard well enough to compare). Still, I can’t resist indulging, at least for a moment, in the thought that Beethoven is seeking some mysterious nuance here. He did after all quite deliberately create a surreal atmosphere by opening the oratorio in the highly unusual (at least in 1802) key of E-flat minor — a stroke that has permanently endeared this piece to me, whatever its flaws.

I saw this one on a chalkboard earlier today; evidently the room was serving as the venue for an analysis class. It’s exactly the sort of elementary exercise that usually takes me a day to solve, if I’m lucky. But this time, I’m happy to report, I managed to figure it out in just a few minutes (while ostensibly listening to a lecture on something else).

Problem: Let be such that every real-valued continuous function on is bounded. Prove that is compact.

Solution: By the Heine-Borel Theorem, it suffices to show that is closed and bounded. Since the norm function is continuous, it is bounded on by assumption; but this is the very definition of what it means for to be a bounded set. To prove that is closed, suppose is adherent to but not in . Then, because , the map is a continuous function on . However, as is adherent to , for every there exists a point such that (so that ). Thus is unbounded, contrary to the hypothesis that every continuous function on is bounded.

Easy, right? Well, it’s certainly not hard to hit upon the idea of constructing an unbounded function on a non-compact set. The difficulty is that we have absolutely no information about the set — so how can we possibly write down a suitable function, one that is unbounded specifically on ? This is where the norm comes in — it’s the one explicit function that comes free with your space, and which you can always reach for when you have nothing else. Add in the Heine-Borel Theorem, which relates compactness to the behavior of the norm, together with the idea that largeness is the reciprocal of smallness (e.g. of distance), and you’re pretty much done.

In response to a request that was put to me on a certain social-networking site that will remain nameless…

Here are 25 propositions that I endorse, but which I expect are capable of provoking argument. They are in no particular order.

***
1.There are no deities; all major religions are mistaken.

2.“Atonal” music is no such thing; it is merely highly complex “tonal” music. Schoenberg‘s early works are music-theoretically more similar to his middle and later works than to the music of the eighteenth century.

3. Rameau‘s theory of “harmony” was wrong from the start; over time it has gotten even wronger.

4. Schenkerian theory, though originally a huge step in the right direction, is now an anachronism; it has been superseded by Westergaardian theory.

5.In exactly the same sense in which there is progress in the sciences, so too is there progress in the arts. The best composers of today know more about music than Beethoven, just as the best mathematicians of today know more about mathematics than Gauss. This doesn’t undermine Beethoven’s greatness any more than it undermines Gauss’s.

6. The role of “natural talent” in the intellectual pursuits is misunderstood and greatly overestimated. You may never learn to paint like Leonardo, but you may indeed come close enough that your previous self couldn’t tell the difference.

7. People who talk about “g factor” with any regularity tend to be jerks.

8. The nature of language has been very poorly understood by philosophers, and only marginally better (on average) by linguists.

9. The error of logical positivism was not the verifiabilty criterion itself, but the fact that it was formulated as a criterion for meaning. What they meant to say was this (and this).

10. The main purpose of studying mathematics is to develop intellectual agility. Mathematicians are (or should be) people who get the same kind of pleasure from manipulating concepts that children do from playing with colored blocks.

11. Intellectuals have a blind spot when it comes to politics. Last year, people who think for a living were capable of arguing that the economy was good under Clinton and bad under Bush, hence you should vote for Obama.

12. Climate science is young and presumably involves some bad-ass partial differential equations; it should not be compared to Darwin’s theory of evolution during policy debates on global warming.

13. Teachers of music theory are perpetually embarrassed by their inability to pinpoint what it is they are trying to teach students. But the test is quite simple: you will know you have done your job when your students can accurately write down music they hear.

14. Most people who wax on about the greatness of Bach‘s fugues wouldn’t be able to tell the difference between a fugue by Bach and a fugue by a mediocre contemporary — let alone one by, say, Handel.

15. Richard Dawkins should be knighted.

16. Max Tegmark basically has the right idea with his Level IV Multiverse. (This is my preferred solution to the problem of cosmological fine-tuning.)
.
17. Course grades should be abolished, especially in graduate school. Oral exams/interviews, work samples, and recommendations are entirely sufficient for academic evaluation. (In short: follow the Princeton model.)

18. The “many-worlds” interpretation of quantum mechanics is the correct one.

19. Semantic externalism is so, so wrong. (See here.)

20. There’s really no downside to signing up for cryonics. (Assuming you can overcome enough inertia to actually do so, of course.)

21. General topology is an example of what the definitive solution to a time-honored philosophical problem (the nature of space and continuity) looks like. It is an abstract subject that should blow your mind. It is emphatically not a mere vocabulary that is useful for analysis or algebraic topology.

22. The argument about the Axiom of Choice is over. AC won. It’s one of the Official Axioms of Mathematics. If you don’t like its consequences, see a therapist. (Or: study alternative systems. But for God’s sake don’t claim that standard mathematics is “wrong”.)

23. Finitism is not only misguided, it’s philistine. (OK, so there are only a finite number of atoms in the universe. So the @#$% what? We’re talking about math here.)

24. Karl Popper missed the point. We don’t need to philosophically distinguish “science” from “pseudoscience”; it suffices to distinguish good theories from bad.

25. The phrase “classical music” should be banned. The term is “art music”.

***

Here is an analysis of the first two measures of the B-flat major Prelude from WTC I.

Warning: this analysis breaks some of the rules (well, one in particular) of strict Westergaardian theory as expounded in ITT. In fact, it does so twice (at two distinct stages). Exercise: see if you can identify the rule that is broken, and give a convincing rationale for relaxing it.

1. The basic structure:

2. Segment the final two beats of the first span with: an incomplete neighbor in the bass, a complete neighbor in the soprano, a borrowing from the bass in the alto, and a rearticulated suspension in the tenor:

3. Anticipate the G in the soprano:

4. Borrow from these structural lines to create the texture of the passage:

5. Delay the fourth half-note:

6. Elaborate further (the operations being, I hope, clear):

7. Elaborate still further to obtain the passage as Bach gave it to us:

Happy New Year!

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.)

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.

[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.)

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.