Making sense in music
An enquiry into the formal pragmatics of art (part 12)

by Jos Kunst (1978)



1 This formulation is borrowed from Hintikka: however, he is not to be held responsible for the way we are using it.

Some theoretical aspects

1 Pragmatics is concerned with utterances-in-context. and specifies the effects produced by their use. This can be done in terms of changes in situations. It presupposes knowledge of semantics, which specifies truth conditions of sentences, in terms of possible models satisfying them. In its turn, this presupposes the sentences' well-formedness, which is governed by rules specified by syntax.

2 Where, theoretically, the worst comes to the worst, there is always Nattiez' 1975 — Fondements d'une sémiologie de la musique — a book that deplorably fails to do what one might reasonably expect from any enterprise that terms itself 'foundational', and instead presents its reader with what one can only call theoretical putty. By way of an example, we refer to the justification of the author's most crucial commitment: the assumption of a "neutral level" between the composers' and the listeners' activities. If the "definitions" on pp 54 and 407 are meant to show how this makes sense, they are really wildly question-begging. Moreover, when, as is proper, one requires them to do only an honest definition's work, it proves absolutely impossible to decide what is discriminated by them from what else. Finally, one would wish that the musicologists' linguistic patter would stop at last.

3 Those of my readers who are apt to be scared off by the (possible) ontological weight of possible worlds will be comforted by the fact that for the purposes of our theory the possible worlds will come to represent just the moments of a (real or reconstructed) past which provide a perspective in which music can be understood (see below).

4 It will be wise, we think, in view of the simplicity of theorizing, not to assume differences (such as theological, or theologically tainted, ones) between humans and (higher) animals, that we can manage to avoid. Piaget's works provide ample evidence for the reality of non-verbal (preverbal) computing, inferring, conceptualizing etc. in children. That is why we will easily be tempted to construe the animals' "perceiving" in the same conceptual, computational etc. way as the humans'. (For the moment, we willingly yield to this temptation.)

5 Mathematical concepts involving our spatial imagination often become really operative only after we have "seen through" and thus freed ourselves of the verbal formulation in which they first came to us. Insight is gained only by the spatial concepts becoming operative; and it seems at least possible that for the mathematician who first invented them the nonverbal form preceded the verbal one.

6 The notion of well-formedness originates from the theory of formal languages. There the class of all well-formed formulas (or, traditionally, wffs) is uniformly defined by a set of recursive rules in such a way that for each string of symbols it is in a very simple manner (i.e. algorithmically) effectively decidable whether the string belongs to the class or not. Only for wffs can semantical problems be relevantly posed; the truth-conditions of an ill-formed formula like

  1. "p(∧"

(in the usual interpretation of the symbols) cannot even be searched for. Moreover, whereas

  1. "p∧¬p"

cannot ever be true, not being satisfiable by any model, it can be part of a satisfiable formula like

  1. "q∨(p∧¬p)"

In contrast, (1), as a string of symbols, can never occur in any satisfiable formula:

  1. "q∨(p(∧)"

is just as nonsensical as (1), and not even "not satisfiable".

Linguistic theory has transposed the notion of well-formedness to the domain of natural languages. Rules for well-formedness have there to be found experimentally, by using native speakers' competence to test hypotheses. Now, psychologically speaking, if I am such a native speaker and if I must decide of a sentence proposed to me whether it is well-formed or not, I try to understand it, i.e. to find its truth-conditions. If these truth-conditions cannot be found (not even in the form of the empty set!), I reject the sentence as ill-formed. As Anderson 1976 has it, "A syntactic characterization of the language emerges as a by-product, only, of the map acquired between sentence and meaning. Ungrammatical sentences are those which do not properly map into an interpretation" (p. 506).

It is clear, then, that in the case of natural languages well-formedness rules have to be related to specific aspects of sentences ("I'd like a c-cup of c-coffee" is not ill-formed in all its aspects) and to context ("Colourless green ideas sleep furiously" could be well-formed in poetry only; but in this last case no set of rules, and therefore no "generative grammar", is forthcoming, or even really thinkable).

The foregoing suggests that whereas in the case of formal languages questions of well-formedness can be answered independently of semantical ones, this is a practical impossibility for natural languages, and a theoretical (future) possibility only for a limited subset of their manifestations. If we now take the step of extending the notion of well-formedness to perception generally, we must, and can, continue to rely on semantical and even pragmatical strategies in deciding whether a given percept is well-formed or not. Complete sets of a priori rules seem theoretically conceivable only for pragmatically limited sets of percepts, among which certainly is not the set of percepts called "music", nor any interesting subset of it.

7 Fodor 1975 (p.34) tries to give examples of learning which ("very probably") are not concept learning, and mentions rote learning and "sensory" learning. We are unconvinced. In his words. in these cases "what is remembered of an experience typically exhausts what is learned from that experience. Whereas concept learning somehow "goes beyond" the experiential data" . Offhand, I would say that any retrievable perceptual memory goes beyond the experiential data, and precisely in the sense that it constitutes an at least once successful hypothesis. But I will not argue these points here. I only can't help suspecting that the rather radically non-verbal nature of the "sensory" concepts he mentions (learning what a steak tastes like — learning what a middle C sounds like on an oboe) has again been messing up things.

8 We are definitely not implying that the "language of thought" could ever make do with just images, moving or not. But we are saying that the discussion is far too often being conducted on an altogether too primitive level.

Our contribution to the muddle would be that the Hintikkan way of constructing "intentionality" (see Hintikka 1975 p. l92 ssqq), if applied to suitably interrelated (bunches of) perceptual "possible worlds", would provide a way out of at least the vast majority of Fodor's problems. (One could then make do with very little in the way of discursive description.) In the beginning of the book, he seems not to reject such constructions (cf p. 32), and in modeling a "language of thought" they ought at least to be given a try.

9 We owe the term to Laske 1975A. It was coined by R. Murray Schafer (see his: The Tuning of the World, N.Y. 1977 (Knopf)). It is perhaps important to stress here that our use of the term may diverge from theirs in one important respect: just as we consider music exclusively and systematically as music heard, we consider the soundscape always as someone's, or some community's, soundscape, and therefore not as a set of sonic "objects" (whatever may be the ontological status of those) but as that part of the conceptualized and structured environment that can be auditorily perceived (processed) by the humans inhabiting it. It seems to us difficult to consistently maintain any other use of the term, but we confess to not being always sure about how Laske and Schafer think on these matters.

10 Now a way of solving this problem is to say, instead of A∧¬A or A∨¬A, something like: "A at one point and not-A at some other point". This could suggest the use of a many-valued logic. Normally these logics are interpreted by matrices of truth-values, and (with the notable exception of Post algebras) these matrices are, more often than not, interpretable as describing product logics. This is a possibility that at a given moment we have ceased to explore, chiefly because of the much greater clarity and directness achieved by the model-theoretic interpretation of modal logics, which are also "many-point" logics (the Lewis systems all dealing with a possibly infinite number of them) and which in axiomatics and model theory can be made to reflect some important intuitive notions people have about music. Cf below, 1.2.

11 This subjectivity is not hopelessly unassailable. It depends on differences between personal pasts, and in as far as these pasts are musical ones, we may be able to provide approximative models for them. If we can formalize musics, we can also formalize what are probably the most relevant aspects of a "personal" past.

12 The reader will easily understand our not having much patience with things like R. Scruton's discussion (in his 1974) of the question if "sadness" is a concept rightfully applicable to a Schubert sonata movement. Applicable or not, the word actually says next to nothing about what this particular Schubert piece may do to a listener.

Some mathematical aspects

1 The treatment given here will be extremely loose and aims only at providing an intuitively clear presentation of our models for understanding music. The best introductions are Hughes & Cresswell 1968 (who give even within a space of about 20 pages, a sufficient treatment of propositional logic as the reader will need it here) and Snyder 1971. Our notation stems from Feys 1965.

2 And, with it, most of the enlightenment provided by the explication of the notion of possibility. But this is a common feature of many "possible worlds" semantics: cf, however. the interesting and elegant treatment of this problem by Goble 1973 and Dunn 1973.

3 In case this is not so, we are talking of new aspects arising for the first time in the listener's mind: ways in which he had not yet heard it, and which (co)guarantee its uniqueness (always in the listener's mind). I do not think we must exclude interesting musical situations.

The Bivalence Function

1 Such trivialities as for p: we have low tones, and for q: they are played before noon, or: they are written in 4/4, are (at least in most cases) ruled out by the requirement that the new perspective provide a convincing background, a plausible "possible past", helpful for understanding music, thus by requirements imposed by our intended interpretation of w10. For some considerations of a heuristical nature, cf 3.1.3. Cf also 4.2 and 4.3.

2 The chief function of the "ceiling" b2 is, in this interpretation, to direct more surely our attention to the fact that, suddenly, the music gets higher and softer at the same time.

Chopin op. 28 nr 7

1 Our acceptance, for all worlds, of p∨¬p as an axiom must be carefully kept separate from the decidability, by any given subject (listener), whether, in some given case and for any proposition p, p or not. The capacity of consciousness being limited as it is. it is best to consider our worlds, in the Hintikkan way, as specifying partial descriptions (constructions), or, alternatively, as determining the sets of all worlds compatible with what is asserted in them (cf 0.3.3).

2 Many crucial formulas occurring in our Figs 22-28 are commented upon and translated in the text; in addition, with a view to helping along those readers who are, as yet, only beginning to familiarize themselves with logical formulas, we will provide, in footnotes, a number of rough-and-ready translations into verbal language. Still, we must warn the reader that the figuring out of the formulas themselves, in view of the precise understanding of their relations within the modal frames, is essential to any adequate understanding of our enterprise. Starting out from the A-worlds, two things must be done: 1) verify the relation between A-world and piece: does the A-world describe a percept the piece may give rise to? and 2) verify the relation between each A-world and the other worlds making up the BivF it occurs in: does the BivF specify an UNLL process the percept may give rise to?...

w01(1): □a
"In this music there is only melody"
w11(1): ¬b⇒a
"Whenever there is motion of an appropriate sort (in this case: whenever, within a statement, the music 'goes from one pitch to another') this motion is melodic"
w16(2): d⇒(¬b→a)
"The foregoing law is restricted to the first half of each statement"
w32(2): ¬b⇔d
"Motionlessness through two or more attacks never occurs in the first half of each statement; outside it, it is always there"
w36(1): (b↔d)⇒¬(a∨(b∨d))
"If the foregoing law is broken, this only happens outside the melody, and by motion outside the first half of a statement"
w56(2): a⇒(¬b↔d)
"It (the law mentioned in the last item but one) continues to hold within the melody"

3 Let us note in passing the interesting possible parallels occurring in two fields superficially unrelated to ours. viz., the philosophy of science and genetic epistemology. The analogies are simple but suggestive.

  1. In the philosophy of science the "incommensurability problem" of theories is well-known. Now if I use a certain (rationalized) concept of melody, and if I try to use it in communicating with persons like the above-mentioned "grandmothers", either I totally fail, or I change their concept of melody (just as the above-specified way of listening to the Chopin piece is liable to do). In either case, I am unable to communicate with my original "grandmother": and it might well be the case — in fact I want to suggest it — that it is even a theoretical impossibility. Something like a "paradigm switch" is at issue: it is even doubtful whether she herself has her original concept still available after I have spoken to her — if I have been successful, that is.
  2. Also analogous is, we take it, the way in which children, in assessing, e.g.,length, at some stage of their development, change over from ordinal notions to metrical ones (cf Piaget 1961 p. 394).

4 Our models have one (not always trivial) "built-in" way of limiting concepts' ill-definedness, viz., the one given by the access relation between possible worlds, in the following manner. In all worlds to which some given world has access, we must have it that concepts used in those worlds and referred to by the same letters are the same concepts. One aspect of ill-definedness is then obviated, viz., the fluidity, the variability of concepts' applicability, albeit in a very limited micro-perspective only.

5 As a sequel to note 2 (3.2.3), we give:

w111(1): j⇒k
"Any harmonic progression involves only the degrees V and I"

6 We again give a rough-and-ready translation:

w66(2): ¬(b∨d)⇒j
"Any motion not taking place in the first half of a statement involves harmonic progression"

7 Again, some rough-and-ready translations:

w162(1): c⇒¬d
"In the first half of each statement no two successive attacks have the same pitches"
w172(1): ¬c⇒d
"Whenever there is maximum mobility (each attack bringing new pitches) it is in the first half of a statement"
w172(2): c⇔¬d
the foregoing two conjoined
w174(3): (c∧d)⇒d
"If in the first half of a statement two successive notes are the same, this occurs in the melody"
w227(3): ¬(c∨d)⇒¬a
"Whenever outside the first half of a statement each attack brings new pitches, this occurs outside the melody"

8 One rough-and-ready translation:

w308(2): m⇒((a∧d)→(p→¬c))
"If all statements remain rhythmically uniform, then, if the first half of the melody changes its direction more than once, it will have no successive identical pitches"

9 Our last instalment of rough-and-ready translations:

w322(2): (a∧¬e)⇒(¬f→g)
"Not counting the second half of each statement, the melody, if it has not consistently (i.e. for at least two successive attacks) one note per attack, has two notes per attack"
w332(2): a⇒(¬f→g)
"The melody (anywhere!), if it has not consistently one note per attack, has two notes per attack"
w342(2): (a∧e)⇒(¬f→(g∨h))
"In the second half of each statement, the melody has either consistently one, or two, or more than two notes per attack"

Winding up

1 Cf Thomas Pynchon, The Crying of Lot 49, N.Y. 1966 (quoted after the Bantam edition, 1967, p. 105:

"It's extraordinary," said Mucho, "everything's been — wait. Listen." She heard nothing unusual. "There are seventeen violins on that cut," Mucho said, "and one of them — I can't tell where he was because it's monaural here, damn." It dawned on her that he was talking about the Muzak. It had been seeping in, in its subliminal, unidentifiable way since they had entered the place, all strings, reeds, muted brass.
"What is it," she said, feeling anxious.
"His E string," Mucho said. "it's a few cycles sharp. He can't be a studio musician. (...) "

A non-musical UNLL process! cf 4.3.1.

2 On this last topic cf notably Lewis 1969 and Schiffer 1972. Let us note in passing that both are fields Lévi-Strauss is scarcely interested in and that therefore they cannot have inspired the remark we criticized in the foregoing paragraph.

3 Remark that, through the stipulation of equiprobability, it is uncomfortably akin to the case of the dice-thrower wondering which one of two equiprobable results he is going to get.

4 Let us stress once again that lack of choice now is no argument against the conventionality of our behaviour: conventions represent coordination problems solved. Once Hume's rowers have established their rhythm, it is imposed on each rower as his sole choice. Once English has become an established language, English phrases have their determinate meaning. Conventions. once established, are out of our hands, — at least in a great many cases.

5 It is difficult to decide whether we ought not to say "possible precedent", thereby widening the class of possible mappings. The problem is that then no bounds seem to exist any more, whereas practical experience teaches us that our supply of "possible pasts" for any given music is limited, and certainly bound up with the class of real musics of the past. But we do, equally certainly, admit, in the way of "possible pasts" for a music, parts (aspects) of past musics or combinations of them. And that, one is tempted to say, pretty well exhausts the possibilities any one is entitled to reckon with if he wants to ensure communication. We will stick to "precedent" without qualification, but bear in mind what we have said here.

6 Precedent: cf 4.2.3, note; cf also the fact that any more or less extended work provides precedent for itself (cf 3.2.4: 1, and 1.2.3: the M case).

7 This fact co-determined, we take it, our construal of possible worlds not as sets of individuals, but as points in an internalized space-time at which propositions are true or false.

8 Philosophical approaches we know of confine themselves to languages fit for philosophical use, and try to determine from statements, without reference to addressees' knowledge states, what they are about. I think that these approaches, in being decidedly non-pragmatical, preclude themselves from ever capturing a great many non-negligible niceties in verbal practice, in literature and elsewhere. After all, most films are about something.

Nevertheless, it will not be amiss to try and specify, if only perfunctorily, the 'virtue' as well as the limits of our line of explication.

Firstly, its discriminatory power rests not only on the condition of a given object's changing concepts, but also on its using them, and this is, to say the least, a somewhat loose way of speaking: to say that a book (or any other thing), in being " about " some concept, uses it, seems to presuppose some sort of intention on its part. Now there is some (metaphorical or perhaps metonymical) sense in which artefacts which carry meaning may be said to use some concepts and not to use others. A natural object such as a mountain may change concepts in one who views it; it does not use them in the sense we mean. Kafka's Castle may be said to use the concepts of, e.g., a certain village, or exclusion from a closed world, but it does not use, in our sense, the concept of Kafka's Castle, or the (theoretical) concept of allegory, or such like.

Secondly. when is a concept changed? Theoretically. any change affects all concepts. But many will tend to think that the Castle does not change. e.g., the general concept /village/, but that it does change the concept of some specific village, if only by creating it (a change if there ever was one) (In the same way. the concept of, e.g. ultraproducts may be created in me by a book on model theory; we will then say, surely, that the book is about ultraproducts.)

The question of how we can achieve a practical grip on concepts' use-and-change as a criterion for aboutness we will have to leave open here, and this constitutes, for the moment, a severe (but perhaps, in the long run, not irremediable) limitation of our explication's usefulness. Its chief 'virtue' lies, we take it, precisely in its lending itself to generalization, e.g., the way we do it in the text. It also easily accounts for such cases as Schiffer 1972 p. 56:

(3a) A: "Let's play squash."
       S: Holds up his bandaged leg.

Surely, S's (non-verbal) utterance is "about" (about) his playing squash.

9 We cannot here go into his theory of myth itself; but the reader may understand, in the context of the present book, that it seems unnecessarily risky to us.

10 Meyer 1956 talks of motor responses, and we use the term motor concepts instead. Meyer's problems are solved on our approach if we may invoke the fact that the activation of motor concepts is frequently accompanied by overt motor behaviour. Thus if I see someone being run over by a car, and if I am at all capable of "relating to" that event, of understanding it, I construe the situation with myself in it (cf Mandler 1975 p. 236/7), and it is probable that I will have a violent startle reaction. Analogously explainable overt motor behaviour can be observed in onlookers of spectator sports.