Although music theorists formulate theories of tonality for a vari-ety of reasons, three seem to be especially important. The first is to develop a vocabulary of concepts for describing what relationships count as tonal or, more specifically, functionally tonal. Concepts are terms we use to categorize our observations into broad types.
According to ‘The Classical Theory of Concepts,’ defining a con-cept involves establishing a set of necessary and sufficient condi-tions that something must satisfy if it is to fall under that concept.1 Music theorists have traditionally expended considerable effort on developing concepts to describe a wide range of tonal relationships.
Many of these concepts allow us to describe how notes behave lin-early. For example, when describing the tonal properties of the music shown in figure I.1a (Beethoven, Six Variations, WoO 70), we might observe that the thirty-second note B in the treble clef,
m. 13, passes between A and C. This account presupposes that the concept ‘passing tone’ can be defined as an unaccented dissonance that moves by step between two consonances a third apart. Such a definition conveys the necessary condition that passing tones move by step and is sufficiently precise to differentiate passing tones from other dissonances, such as neighbor tones, cambiatas, and suspen-sions. Alternatively, we might use quite different concepts to describe how notes behave harmonically. In the case of figure I.1a, we might observe that the passage begins and ends on a G triad with the root in the bass. This description presumes that the con-cept ‘triad’ can be defined as a chord with a root and two other members a third and a fifth above. The latter definition conveys the necessary condition that triads are built from thirds and fifths, and yet is sufficiently broad to encompass major, minor, augmented, and diminished triads.
It goes without saying that both of these descriptions tell us sig-nificant things about the tonal properties of figure I.1a. And yet,
Figure I.1. Explaining tonality.
a. Beethoven, Six Variations, WoO 70, mm. 13–16.
b. Beethoven, Piano Sonata, Op. 22, 4th movement, mm. 19–21.
neither one actually explains why Beethoven’s music is ‘in’ G major.
The problem is that to explain why the passage is tonal, it is not enough to describe what melodic tones and triads are present; we must also say why they are related to each other in some ways and not others. To do so, music theorists invoke various laws of harmony and voice leading. These laws are general claims about the ways in which melodic tones and triads usually behave in tonal contexts.2 They are often, but not always, expressed in a conditional form: if X occurs in context Y, then Z will happen. For example, to explain why the passage in figure I.1a establishes the key of G major, we might invoke the following law: “If the leading tone appears in tonal contexts, then it normally ascends by half step onto the tonic.” This law explains why the melody rises from F to G in mm. 13–14, why the alto part follows suit in mm. 14–15, and even why the same theme is in B-flat major when Beethoven transposes it up a minor third in his piano sonata, Op. 22 (see figure I.1b). Having said this, it is important to note that the law governing leading tones is not true all of the time. In figure I.1a, for example, the alto F in mm.
15–16 moves to B3 and not G4, presumably to avoid doubling the soprano part. Since most laws of tonality are generally but not uni-versally true, they are best classified as law-like generalizations.
Besides introducing concepts to describe what relationships are tonal and invoking law-like generalizations to explain why tonal music behaves in some ways and not others, theorists often have yet another important goal: to explain how specific tonal relationships are produced. This task requires them to develop a set of procedures.
Procedures consist of strings of commands that are usually expressed in conditional form: to produce X, do Y, then Z, and so on. Over the centuries, tonal theorists have developed procedures for accomplish-ing a variety of tasks from harmonizaccomplish-ing a scale to composaccomplish-ing a prel-ude from a given figured bass. Take, for example, figure I.2 (A procedure for composing typical tonal melodies). This procedure has six basic steps. First, pick a final tonic for the melody as a whole (fig-ure I.2a). Second, begin the melody on a member of the tonic triad and end with a stepwise descent onto the tonic (figure I.2b). Third, pick a climax note midway through the melody and not more than an octave above the tonic (figure I.2c). Fourth, reinforce the tonic at the opening (figure I.2d). Fifth, join the opening to the climax and
the climax to the cadence (figure I.2e). Sixth, fill in any details and check to see that the melody has a good overall shape and satisfies any general laws of tonal voice leading, for example, that leading tones normally ascend by half step onto the tonic (figure I.2f).
The preceding discussion has highlighted the central role con-cepts, laws, and procedures have traditionally played in tonal theory, but it is important to realize that these components are a lot more difficult to deal with than we might initially suppose. Take, for example, concepts. While it is certainly possible to find necessary and sufficient conditions for many concepts, cognitive scientists have found that certain concepts cannot be defined in this manner.
Instead, they tend to define such concepts by appealing to the notion of prototypes.3As Alvin Goldman explains:
Concepts are represented in terms of properties that need not be strictly necessary but are frequently present in instances of the concept. These
a. Pick a final tonic for the melody as a whole.
c. Pick a climax about two thirds through the melody and not more than an octave above the tonic.
Climax Cadence
3 4 2 1
d. Reinforce the tonic at the opening.
Climax Cadence
3 4 7 1 4 2 1
e. Join the opening to the climax and the climax to the cadence.
Climax Cadence
3 4 7 1 2 3 4 2 1
f. Fill in any details and check to see that the melody has a good over-all shape and that it satisfies any general laws of melodic motion.
Climax Cadence
3 2 3 4 6 7 1 2 3 4 21 Figure I.2. A procedure for composing typical tonal melodies.
properties are weighted by their frequency or by their perceptual salience.
A collection of such properties is called a prototype.4
He adds: “Under the prototype view, an object is categorized as an instance of a concept if it is sufficiently similar to the prototype, similarity being determined (in part) by the number of properties in the prototype possessed by the instance and by the sum of their weights.”5Although Goldman does not explicitly say so, the
‘perfect’ prototype may not actually ‘exist’ in the world at all; it may be an idealization that combines features from many different individuals. We can illustrate these points by reconsidering our definition of passing tones (see figure I.3, Five forms of passing tone). Although we defined passing tones as unaccented dissonances that move by step between two consonances a third apart, some passing tones do not satisfy this definition. In figure I.3a, for exam-ple, the pitch B in m. 1 seems to behave as a passing tone, even though it is consonant, and in figure I.3b the notes F and E in m. 1 are both dissonant and seem to connect two consonances a fourth, not a third, apart. More remarkably, figure I.3c contains an accented passing tone, figure I.3d includes a chromatic passing tone, and, if you believe Schenker, figure I.3e contains a leaping passing tone (or springender Durchgang)!6 In other words, it is much easier to think about passing tones in terms of prototypes and variants, than it is to provide a necessary and sufficient definition that works in all cases.7
Laws, too, pose their own problems.8To begin with, not all gen-eralizations are law-like.9Take, for example, the claim that all pieces in G major have a key signature of one sharp. Even if true, which it is not, this generalization does not stand up as a law because it does not explain why there is any connection between having a signature of one sharp and establishing the key of G. To ensure that particular
Figure I.3. Five forms of passing tone.
generalizations are law-like, Nelson Goodman and others have suggested they should support so-called counterfactual condition-als.10 Counterfactual conditionals are hypothetical statements that suggest what would have been the case had things occurred differ-ently (see figure I.4, Counterfactual conditionals). For example, when explaining why the cadence in figure I.4a establishes the key of G major, we might invoke our law governing leading tones. In this case, the F in the first chord ascends by half step to the G in the final sonority. We might support this law-like generalization by not-ing that if the phrase had been in F major, then the F in the first chord would have descended to E, before moving back onto F for the final chord (see figure I.4b). Since we know that the piece in question is in G major, our remarks about what might have hap-pened if the piece were in F major are known as counterfactual con-ditionals. While counterfactual conditionals have proved very useful in helping us determine whether a particular generalization is indeed law-like, they nonetheless raise their own sets of questions; it
Figure I.4. Counterfactual conditionals.
is unclear not only how to ensure that they are relevant in any given context, but also that they can be used to support all law-like gener-alizations.
It is also debatable whether law-like generalizations are always necessary and sufficient for explanations. Certainly, many experts believe that scientific research is fundamentally law seeking or nomo-thetic.11 This prompted Carl Hempel and Paul Oppenheim to advance ‘The Covering-Law Model’ of explanation.12According to them, explanations are arguments in which the premises are sets of covering laws and initial conditions, and the conclusion is some statement about the phenomena to be explained (see figure I.5,
‘The Covering-Law Model’). If the laws are universal and the argu-ments are deductively valid, then the result fits ‘The Deductive-Nomological Model,’ and if the laws are not universal and the arguments are only inductively valid, then they conform to ‘The Inductive-Statistical Model.’ Figure I.6 (Explaining suspensions) illustrates what Hempel and Oppenheim had in mind. Suppose, for example, that we want to explain why a particular suspension C resolves by step to B (see figure I.6a). We might do so by invoking a simple law of tonal voice leading: namely, that suspensions normally resolve down by step onto consonances (see figure I.6b). Given the initial conditions that the seventh C–D on the down beat of m. 2 is dissonant and that the dissonance is a suspension, this law-like gen-eralization allows us to deduce that the dissonant tone C on the down beat of m. 2 will resolve down by step onto the consonant tone B in m. 2. This is a perfectly acceptable explanation.
Although ‘The Covering-Law Model’ certainly produces acceptable explanations, it is unclear whether covering laws are absolutely necessary for all plausible explanations. In particular,
‘The Covering-Law Model’
critics have suggested that some types of explanation, such as so-called functional explanations used in biology, or narrative explanations found in history, do not necessarily involve covering laws, at least not in any explicit way.13 Functional explanations explain how particular parts of a complex system help to reinforce the system as a whole. For example, when explaining the tonal motion of Beethoven’s “Waldstein” Sonata, we might note that, in the first movement, the function of the recapitulation is to recom-pose the tonal motion of the exposition, that the function of the exposition is to modulate from the first key (C major) to the second key (E major), and that the function of this modulation is to create a pattern of tonal tension, and so on. Though this explanation seems plausible enough, it is unclear what covering laws it uses.
Historical narratives often proceed on similar lines. For example, when explaining why the climax of a given aria appears on a high
‘C’ we might note that this aria was written for a particular tenor to sing at La Scala, and that high ‘C’ was his top note. Although this explanation seems to make historical sense, we would never sug-gest that, as a general rule, composers always make sure that the climax of an aria necessarily corresponds to the highest note in the singer’s register.14
Figure I.6. Explaining suspensions.
b. c.
Initial The seventh C-D on the down The sixth B-D on the weak Conditions beat of m. 2 is dissonant beat of m. 2 is consonant
This dissonance is a suspension This consonance is a resolution.
Covering Suspensions generally resolve Suspensions generally resolve Laws down by step consonances down by step onto onto consonances Explanation Resolution on weak beat Suspension on down beat
is a consonant sixth is a dissonant seventh
By the same token, Sylvain Bromberger and others have raised doubts about whether covering laws are sufficient for all explana-tions.15 We can paraphrase their point by comparing the explana-tion given in figure I.6b with the one shown in figure I.6c. In figure I.6b, we explained why the dissonant note C on the down beat of m. 2 resolved down by step to a consonant note B by invoking the law-like generalization that suspensions normally resolve down by step. However, we can also use the same covering law in a quite dif-ferent way. This time we might start with the initial conditions that the sixth B–D on the weak beat of m. 2 is consonant and that it resolves a suspension on the preceding down beat. Since our cover-ing law states that suspensions generally resolve down by step, we can deduce that the consonant note B is preceded by a dissonant note C. Although this argument is logically consistent, it does not carry the same weight as the argument given in figure I.6b. The problem is that the argument in figure I.6b explains the causal con-nections between the suspension and the resolution, whereas the one in figure I.6c does not. This, in turn, suggests that it is not enough to invoke covering laws in our explanations; our explana-tions must also be able to explain how one event causes another.
One way to guarantee such casual connections is by reformat-ting our covering laws in procedural form.16As mentioned earlier, procedures are strings of commands that we express in conditional form: to produce X, do Y, then Z, and so on. This point is illustrated in figure I.7, (A procedure for generating 7–6 suspensions). This procedure involves three distinct steps:
1) take an upper voice that descends by step C to B (figure I.7a);
2) add a lower voice that moves in parallel sixths below, E to D (figure I.7b); and
3) displace the first note of the upper voice over the second note of the lower voice (figure I.7c). This step produces the 7–6 suspension.
Significantly, this procedure implies all of the same knowledge as the explanation given in figure I.6b. In particular, it implies that suspensions generally resolve down by step. And yet, the procedure adds something extra: it also indicates that suspensions are caused by displacing the upper voice over the lower voice.
Whatever advantages procedural explanations may give us, they can, however, be slippery things to deal with. Reconsider, for a moment, the procedure given in figure I.2. This strategy identified six basic steps for composing a typical tonal melody. In order for us to determine whether the procedure is successful or not, we must decide whether or not our new melody resembles the melody in fig-ure I.1. But what does it mean to say that two melodies resemble one another? To answer this question we must invoke some notion of similarity, but it is by no means obvious what this step involves.
As the philosophers Quine and Ullian explain: “Everything is sim-ilar to everything in some respect. Any two things share as many traits as any other two, if we are undiscriminating about what to call a trait; things can be grouped in no end of arbitrary ways.”17In other words, we can judge similarity in different ways, depending upon what examples we take as prototypical and on how we decide to measure similarity.
It seems, then, that in describing what relationships are tonal, explaining why these relationships create music that behaves in some ways and not others, and explaining how to produce specific tonal relationships, music theorists draw on a rich assortment of
Figure I.7. A procedure for generating 7–6 suspensions.
a. Take an upper voice that descends by step from C to B.
b. Add a lower voice that moves in parallel sixths below, E to D.
c. Displace the first note of the upper voice over the second note of the lower voice to cre-ate a 7–6 suspension.
concepts, laws, and procedures. Although each component raises its own methodological issues, a given theory of tonality takes a specific cluster of concepts, laws, and procedures, and structures it in a par-ticular way. If these theories are successful, then we normally expect that these knowledge structures will give us explanations and pre-dictions that are coherent, reliable, and capable of being tested empirically by other theorists. And they should work for all and only all tonal music, or at least all and only all functional pieces. But how do we go about building such a theory? How, in fact, do we check to see that it actually covers all and only all music that we classify as tonal? Let us see how we might answer these new questions.