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.)
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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
- a particular pitch
- a particular onset time
- a particular duration
In addition, we may also think of a note as having
- a particular loudness
- 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 
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.
May 27, 2008 at 11:41 pm |
Your section on pitch simply doesn’t work, it begins from premises that don’t apply to much tonal repertoire, which began in the meantone era and continued beyond that era to be constrained by features of meatone. Meantone does not divide the pitch space into semitones of equal size and is not limited to a set of twelve pitch classes, but is defined rather by the use of fifths which are tempered so that four consecutive fifths span a (octave equivalent) just major third. All meantone-like tuning systems, including 12tet (which can be understood as an 11th-comma temperament) preserve the relationship that the best major third (under octave equivalence) is the sum of four fifths, a necessary condition for common practice harmony and voice leading.
“Enharmonic” equivalence is a feature of 12tet and other 12 p.c. circulating temperaments, but the capacity to succesfully render most tonal repertoire in other tuning environments, including those without enharmonic equivalence, indeed the historical probability that they were composed in such environments, requires that the use of enharmonic equivalence be identified as a special feature of repertoire in a circulating temperament like 12tet rather than as a general feature of tonal music.
(The common practice _could_ have developed under alternative premises other than those of meantone, for example, those of pythagorean intonation, in which the best third for use is triads is the sum of eight _descending_ fifths (i.e. the diminished fifth C-Fb, which is a schisma away from a just major third) , and in which chromatic semitones are larger than diatonic semitones, but that would necessitate a completely different voice leading regime, probably based upon some alternative to the diatonic collection, and there is little or no evidence of such a regime in real tonal literature, even though 12tet is not only a meantone-like tuning, but an equally good pythagorean-like tuning.)
May 28, 2008 at 1:47 am |
The only historical claim I made in the entire post was
For historical reasons, the pitch-class 9 is called A.
Every other claim was theoretical, not historical, in nature. This means in particular that all “objections” of the form “music history developed in manner X, not in manner Y” are completely irrelevant.
Regarding tuning systems, the only assumption I need to make is that it is possible to understand tonal music in terms of an equally-tempered pitch space. If you deny this proposition, then you must necessarily hold that tonal music cannot be played on a modern piano.
May 28, 2008 at 2:43 pm |
Are you making the claim that a modern piano (or another 12tet) instrument is adequate for all tonal music?
While _some_ tonal music was definitely composed within and can only be realized in a 12tet environment AND it is possible to describe some other tonal music in terms of 12tet, descriptions of repertoire or new compositions in which a distinction in terms of actual pitch as well as notation is made between, diatonic and chromatic semitones (and, consequently, between “enharmonic” pitch pairs), will fail to account for that distinction. (And yes, a performance of Byrd, Handel, or Mozart on a 12tet instrument will fail in these terms). This is, above and beyond the issue of historicity, to my ears, a real loss of information and the adequacy of the description must be called into account. The advantages and appropriateness of a mapping of some music onto a particular tuning — be it 12tet, another circulating temperament, meantone, 31tet (which can be understood as an equal temperament equivalent to quarter-comma meantone), etc. — are, whether for historical or theoretical reasons, not simple issues, but certainly a theory of tonal music should be able to account for the tones used in a more precise way.
As to a distinction between a historical and a theoretic claim for a “theory of tonal music”, I have to ask if you really intend to restrict your theory to speculative theory (theory designed for the purpose of creating new repertoire) and not the description of existing tonal repertoire and its composition.
This matter could probably be solved satisfactorily if one began not with an equal division of the octave into twelve tones, but rather the principle of octave equivalence, a chain of fifths, the length of which varies according to the particular tuning used, an equivalence relationship between the best major third and a number of fifths in the chain, and the definition of the diatonic collection as a segment of seven consecutive tones on the chain.
May 28, 2008 at 4:17 pm |
Are you making the claim that a modern piano (or another 12tet) instrument is adequate for all tonal music?
“Adequate” depends on what one’s purposes are. For my purposes here, the pitch space I described will indeed be adequate, unless and until noted otherwise.
As to a distinction between a historical and a theoretic claim for a “theory of tonal music”, I have to ask if you really intend to restrict your theory to speculative theory (theory designed for the purpose of creating new repertoire) and not the description of existing tonal repertoire and its composition.
See Chapter 1 of ITT. What I mean by a “theory of tonal music” is what Westergaard means (no surprise, since it’s his theory that I am attempting to summarize), which is a “logical framework in terms of which we understand tonal music” (ITT, p.9). The operative words are “we” (as opposed to, say, people who lived centuries ago) and “understand” (as opposed to, say, “compose”, or “describe”).
In essence, what is being done here is a form of psychology. A typical proposition of this type of theory will look like “We think of the F as a way of getting from the G to the E”. It will not look like “In Mozart’s day, it was customary to progress from F to E”. A “theory” whose propositions looked like the latter would not be a theory in this sense, but would merely be an account of musical history.
This matter could probably be solved satisfactorily if one began not with an equal division of the octave into twelve tones, but rather the principle of octave equivalence, a chain of fifths, the length of which varies according to the particular tuning used, an equivalence relationship between the best major third and a number of fifths in the chain, and the definition of the diatonic collection as a segment of seven consecutive tones on the chain.
Again, all I need for the purpose of developing Westergaardian tonal theory as presented in ITT is the familiar equal-tempered system customarily used on modern instruments. Effectively , what I’m doing is taking the piano keyboard and saying, “here are our pitches, and here is how we [shall] conceive of them”. Working out a more general theory of pitch might be an interesting thing to do, but it’s beside the point here.
June 2, 2008 at 4:35 am |
[...] of Westergaardian theory: Lines Last time we discussed notes, the atomic units of musical structure. The topic of today’s installment [...]