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Making sense in music
An enquiry into the formal pragmatics of art (part 5)

by Jos Kunst (1978)

3. Chopin op. 28 nr. 7

3.0 Two modes of being of the prelude

3.0.0

Chopin's op 28 nr 7 may be said to exist in two very different forms: one is the historically "genetic" one, the one Chopin might be taken to be responsible for: I refer to its appearance as nr 7 of a series of 24 different (sometimes radically so) pieces; in that form the presence of the other 23 pieces is essential, to guarantee the specific effect of "strangeness", of highly artificial "simplicity" produced by nr 7's relation to its proper context.

The other form is the drawing-room music form. Anthologized in albums, in collections as the "les Sylphides" one, it has, from a historical point of view, been seriously falsified, but it is certainly in that form that it has become best known, at least to a number of generations of boarding-school piano pupils and the like, in short, to our grand-mothers. Thus a secondary historical authenticity has grown upon the piece in its isolated version.

3.0.1

This whole problem is certainly a very interesting one, but we are not going to tackle it here. For the moment it will be wise to avoid complications: therefore we will address ourselves to the piece in its simpler form, the drawing-room one. What we are doing in this essay is just ground-work, but, on the other hand, certainly meant to enable us, at some later stage, to tackle problems like the above one in a really serious way.

3.1 Proposition letters, formulas, BivFs and networks

3.1.0

The groundwork mentioned in the preceding paragraph will consist mainly in a demonstration of the ways in which BivFs may come to interrelate and form networks. As a heuristical starting-point, Chopin's little prelude is used: our aim is thus very definitely not to contribute to Chopin studies, but to try out our BivF networks as a way of systematizing, of capturing within a theoretical frame, an analysis of listening. In order to do that, we will propose a number of BivFs,certainly not wholly arbitrary ones (we will want them to be adequately related to the Chopin piece), but meant, before anything else, to show the different ways in which whole networks can be generated out of them.

3.1.1

Obviously, there can be no algorithm, and at this stage of our enquiry even no practicable rule-of-thumb method of deriving BivFs from any given music.

First of all, there is the problem of establishing some list of basic concepts. It shall never be 'complete' in an unqualified sense. It can at most satisfy some adequacy requirement defined relative to a) some (group of) listener(s), and b) some verification procedure for the (each) listener (and/or the group).

Moreover, there is the problem of precision. In our case, verbal versions of basic concepts (cf the right-hand column of our reference lists) are only meant to ensure their unambiguous workability within the context provided by the given piece. They need not be precise in any more general sense.

Lastly, the relation between concepts actually used and 'basic', 'atomic' ones may vary in complexity. Thus, a listener's using "(a∧b) →c" does not necessarily imply his having any use for, e.g., "b" taken separately.

3.1.2

Each formula or set of formulas making up a world wⁱ_j is meant to characterize, at least in part, one and only one conceptualization of a given musical process/event. In our reference lists we will associate with each letter a verbal expression specifying a musical concept, generally in the form of an attribute or property. When used in our models, letters will therefore come to mean, roughly, something like

"p"

"I conceptualize this event as (being) p", or "I hear p-ness";

"¬p"

"I conceptualize this event as (being) not p", or "I hear 'not-p'-ness";

"p∧q"

"I conceptualize this event as (being) at once both p and q" or "I hear 'both-p-and-q'-ness";

"p→q"

"I conceptualize this event as (being) such that if it is p, then it is q" or "The way I hear it, p-ness implies q-ness"

and so on, for all PC connectives and well-formed combinations thereof. Moreover, let us stress again that, as argued in Ch 0, "I hear p" is to be understood as "I (re)construct p", in the sense of "(re)construct" specified there (cf 0.3, 0.4). [1]

3.1.3

BivFs will be constructed by substitution in Figs 6 and 9, always keeping in mind the intended interpretation. Thus, intuitively, one is justified in expecting that

• for p will be substituted an expression such that □p may represent a musical law to be broken, a listener's orientation to be 'unlearned' in the UNLL process at issue;
• for Fig 6's q (Fig 9's r) will be substituted an expression such that □q (□r) may represent musical laws to be preserved or established, listener's orientations to be 'learned' in the UNLL process at issue;
• for Fig 9's q will be substituted an expression such that □ (q∧r) may serve, in the reappraisal of the past, as a sort of substitute law, possibly 'explaining', in a sense, one's previous adherence to □p as an orientation in the given music (cf also 4.3).

3.1.4

The rules A through G, specified in 1.2.3, and which govern the construction of our models, will now be leading to a larger-scale structure. The labels of possible worlds will, accordingly, not correspond to those of Figs 6 and 9, but (cf Fig 11's precedent!) to the columns and rows of a structure numbering some thirty-odd columns and nine rows. Rows 1-8 will in a very simple way be related to the respective eight "statements" making up our piano piece, as specified in Appendix II. In addition, there will be a row zero, in which will occur n.n. worlds only, viz., those embodying the M' case described in 1.2.3. As for the number of the columns, it depends essentially on the number of BivFs constructed. And on the set of BivFs admitted in our total structure the same remark applies as on the set of basic concepts used in constructing them (cf 3.1.1): it will at most attain some specifiable relativized form of completeness.

Fig 20 shows some ways in which a BivF may be embedded in the total structure, which we will come to call (4.4.6) the piece's "pragmatic map". (This listing is far from exhaustive!)

(cf Figs 6 and 9, Fig 8)

Fig 20

3.2 The prelude: naive listeners

3.2.0

There are good listeners and bad listeners, and the last category gives rise to theoretical problems of its own. Take, by way of limiting case, the excessively bad listeners, the not-listeners in the eyes of some. One might propose to capture their listening-process in the following minimal model:

Fig 21

Something is recognized, and that is all there is to it. We write a capital S; theoretically, anything may serve as a conceptforming background.

3.2.1

But let us not think of ourselves as "good listeners" in a too naive way. It seems certain, at least to me, that the Fig 21 schema goes to a certain extent for all of us, especially in as far as we limit ourselves to what could be called style-recognition: the style then being a set of characteristics a music borrows from a set of predecessors, and implements without deviation. It is plausible that, for many characteristics, we do just that; but it could also become a quasi-total phenomenon. The fact that many non-naive users of music for sometimes long stretches of time have absolutely no use for certain works they "just know too well" could be explained by the possibility that precisely the way of undergoing the musical learning-processes which interest us here, and which are based on the formation of concepts and the production of insights, is, for a certain time at least, precluded to them.

In this context it is important to note that concepts, once formed, insights, once gained, are not thereafter automatically and always available. Concept learning is not always a once-and-forall one way process. Most concepts, to become at last readily available, have to be re-discovered, re-learned many times (cf also 2.3.1, Remark). For all that, pieces still may 'wear out'.

3.2.2

One might perhaps say that our piece, Chopin's op 28 nr 7, is characteristic for the taste of at least one category of notoriously bad listeners: I refer to the grand-mothers alluded to in 3.0.0, fond of simple, noble and lyrical melodies.

Now we could avoid some rather difficult problems concerning the concepts they use, especially the problems centered upon the relation between musical experience and verbal comments on it, by simply laying down a set of clear-cut definitions for musical concepts, and starting out from them. But this would run counter to the central motivation of our enterprise. In the first place, what we are interested in is people's listening strategies. If then we think that some people are naive, that naiveté itself has to be accounted for. In the second place, I want to suggest that even such sophisticated listeners as my reader and myself actually are using, alongside sophisticated listening strategies, many less sophisticated and even very naive ones. (I am not claiming that we use all of those in parallel at any given listening session.) What is then interesting is whether and how these are interrelated – and here UNLL may have an important role to play, for 1) it brings together conceptually related strategies of different degrees of sophistication, and 2) it is a process provoked by the music heard, and thus, inasmuch as we may be said to hear the same music, we might have (part of) it in common, which makes it important for the music's social functioning. (As argued above (3.2.1) our need to re-learn and re-discover concepts helps ensuring that, at least for a number of times, a given music (as characterized by the UNLL processes provoked by it) may remain substantially "the same" for each given listener.)

A network of interrelated BivFs provides us with a whole gamut of listening strategies, ranging from the naive to the sophisticated, and linked together by the "dynamics" of UNLL processes. Moreover, the logic underlying the network fully allows (as we shall see later) for even conflicting strategies to be coordinated in it (cf 3.3.2, and also 0.7.4). Let us therefore, and without qualms, admit our grandmother's case.

3.2.3

Bearing this in mind, we concatenate three BivFs

1. BivF(a, ¬b→a, (d∧¬b)→a) (cf 2.2.3 end)
2. BivF(¬b→a, ¬b→d, d→¬b)
3. BivF(¬b↔d, a→(¬b↔d)) (cf 2.1.3 end)

delete tautologies throughout, and obtain Fig 22. We give the following reference list [2]:

a ... belonging to the melody
b ... motionless (as for pitch)
d ... belonging to the first half of each statement: 

Fig 22a

Fig 22b

3.2.4

Several comments are in order.

1. nn worlds come in two positions: on rank 0 and on rank 1. As implied by 3.1.4, rank 0 cannot refer to any statement in the piece, and rank 1 refers to the first statement. It is clear that, in this way, nn worlds positioned on rank 0 refer to "external" musical pasts used as background for understanding, as in the M' case of 1.2.3; on the other hand, nn worlds positioned on rank 1 refer to aspects of the piece corresponding to 1.2.3's M case. An example: if we want to say: ¬b⇔d ("All statements have motion in their first half only and immobility only elsewhere"), there is no past, no special musical tradition or genre, preceding this particular piece, that anyone might think of in order to confer law-likeness to the first statement as it is in this respect. But, equally clearly, the second, third, fourth and fifth statements may be conferred law-likeness in this respect by reference to the first.
2. In interpreting the tableau of Fig 22 and the ones that are to follow, we will do wise, in general, not to require any special path, or set of paths, to be followed. Listeners are understood to begin where they like, to take any combination of paths, to move back and forth as they see fit, etc. From each wⁱ_j we require the listeners to "know of" the wⁱ_k of lower rank and belonging to the same column. If that column is empty (as in the case of w²₆ and w⁵₆) then its place is taken by the next accessible one to the left (here 1 and 4, respectively). All other accessible w_j are optional; one "knows of them", or not (cf the possibility of remaining stuck at w⁰₂ of the BivF paradigm (2.0.1)).

3.2.5

3. All this follows simply from our interpretation of BivFs combined with our ways of writing down their concatenations. Two cases arise in Fig 22.
   1. BivF 2 is concatenated with BivF 1, but 1 is also complete in itself, it has a law-formulating w¹₆. Thus, we will have w²₆ as a different option leading away from w¹₁, but w¹₀ and w¹₁ will be w²₆'s proper background in BivF 2.
   2. BivF 3 is concatenated to BivF 2 in essentially the same way. Nevertheless, we have also written in w⁴₁ and w⁴₂, and the arrows leading to them. This has been done for the following reasons.

   In each world the conjunction of the formulas contained in it is true. Therefore one may argue that w³₁(1) and w³₂(1) are vacuous, in the sense that they add nothing to the truths asserted in w³₁ and w³₂; and that is why these two worlds can be part of BivF 3: what is asserted in them is exactly what is asserted in w⁴₁ and w⁴₂. Why then write in these last two also? As the reader may recall, in 2.2.2 we adopted the policy of admitting in the right column of the 3-place BivF the in the above sense vacuous formula (1) p→(q∧r), because it represents a part of the past a) meant to survive, and b) emphasizing the link with the left-hand column. The specific role it has to play is thus determined by its being viewed, in a way, "through" the A-world, and its presence marks the column in which it occurs as a right-hand one, representing a retrospectively reconstructed possible past. Now it seems unwise to exclude the case of the listener orienting himself on the concept ¬b↔d long before the sixth statement; that is why w⁴₁ and w⁴₂ are given as well.

3.2.6

4. Finally, the interpretation of Fig 22's formulas. It presents only one special problem, bound up with the concepts of melody and pitch mobility: both undergo very definite shifts of meaning, in some sense of that term.

   I do not think that the listener of column 0 presents an extremely unlikely case. I even think that for many listeners who do not go beyond Fig 21, "melody" would be the best catchword for the present piece, and thus a lower-case "a" could be substituted for the capital "S".

   What does this mean? Are the notes not belonging to the melody simply ignored ("not heard") or are they regarded as forming a purely accessory element? Undeniably Chopin's "accompaniment" notes behave for a relatively long time very discreetly indeed. And one could also argue that the distinction between not hearing something, and hearing it in a purely ancillary role, is, in the case of a superficial listening, entirely spurious.

   A listener may get stuck at any world (in the meantime, he may move forward unchecked along other paths) but Fig 22 explicitly mentions the possibility of getting over the Fig 21 stage by listening to the material offered by the sixth statement. I think it is wise to analyze this situation as a change of meaning of the concept a. First it was used "imperialistically", and encompassed all of the music, later, in columns 1 and later ones, it admits of the simultaneous existence of other musical material.

   The case of the concept "b" is similar. In column 1, where it is first introduced, its use implies throughout that the notes for the pianist's left hand, for all statements occurring before the sixth, exhibit no independent motion at all. Thus "motion" is, in column 1, equated with quasi-melodic motion, and the music is not felt to "go", with the player's left hand, from the single low register note to the middle register chords. The low note simply is the basis of the chords and the whole thing is motionless. In this way, the concept of motionlessness is clearly specialized, so as to reflect specific listeners' intuitions: we ought, therefore, to speak systematically of "relevant motionlessness", and provide specifications. Such specifications would have to come in terms of some basic definition of pitch motionlessness in a suitably weakest sense. As such, we propose what is going to be our concept c (cf 3.4.2): we have (weak) motionlessness iff, in a chain of two successive attacks occuring within a statement, there is one which has the same pitches as the foregoing. Then, by way of an in this context useful concept of "relevant" motionlessness, we will define b, viz., as (weak) motionlessness through at least two successive attacks (which thus implies the existence of at least three consecutive identical attacks!). This last concept will give us, in the melody or out of it, until the sixth statement, ¬b↔d.

   Thus, Fig 22 seems to move, from left to right, from more naive ways of listening towards more analytic ones. But we must stress, I think, that whereas, e.g., column 0 excludes any analytic listening, column 4 does not exclude, e.g., the relatively non-analytic listening that orients itself on the regular alternation of motion-motionlessness – motion-motionlessness etc. that determines, before the sixth statement, the "shape" of the piece [3].

3.3 Handling ill-defined concepts

3.3.0

I think that the subject matter of 3.2.6 is also important in that it confronts us with the problem of how to handle ill-defined concepts. Not always, however, is there any analytic, and therefore scientifically useful, definition at hand, or ad hoc devisable, capable of making precise a naive concept [4].

3.3.1

Consider, e.g., the following item:

i ... (part of a) "Stück im Volkston"
or, alternatively, "we have (part of) a "Stück im Volkston".

The term dates back to XIXth century German music, which, in the wake of an XVIIIth century ideology idealizing the common (rural) people, their sense of justice, the simple rightness of their feelings, etc., has cultivated it as a new "genre", characterized by an artful naiveté, a touching simplicity (Schumann, Brahms), in later cases like the Mahler one, often loaded with soulful and heavy ideology, in short, "depth".

3.3.2

Now, putting aside the ideology, and placing ourselves on the relatively chaste viewpoint of history, does this piece, or does it not, belong to the aforementioned "genre"? It is difficult to settle such a question. The way each of us tends to settle it will tell us as much of his idea of a "Stück im Volkston" as of the piece. And precisely this might be a means of resolving somewhat the concept's vagueness.

I take it that the concept i belongs to the class of concepts exemplified by Wittgenstein's "Spiel"-concepts that can only be paradigmatically defined. According to Stegmüller's analysis (cf his 1973 pp.196/7) only necessary conditions can be specified for something's belonging to the set corresponding with it – sufficient conditions remain irremediably vague. Each newly encountered case must have a "considerable" or "significant" number of properties in common with all, or "almost all" members of some basic and extensionally given set.

I want to suggest that such concepts are always global ones, and that (in terms of a listener's strategy) the role they fulfil is essentially that of Minsky's "frames" (cf his 1974), in which the "top levels" (corresponding to the above-mentioned necessary conditions) are fixed (they "represent things that are always true about the supposed situation") and in which "the lower levels have many slots that must be filled by specific instances or data" (p. 1). In our terms, Minsky frames are, I take it, just global and in the above sense ill-defined concepts. (Both Stegmüller and Minsky, one as a philosopher and the other as a theoretical psychologist, are, in the passages we are quoting, referring to Thomas Kuhn's paradigm notion.) In a listening strategy a frame will be "tried out" on a given music, perhaps to be exchanged later by another. This is precisely what happens, as we have seen, to many other (i.e. small-scale and welldefined) concepts also; the ease of in the above sense ill-defined global ones is just a special one.

But there is another complication. In a field as riddled with nonverbal conceptualization as that of music, even the necessary conditions associated with the concept evoked by a verbal expression like "Stück im Volkston", however well-determined for each listener, may vary from one listener to another: it may well be the case that, given their different personal pasts (and the lack of effective verbal communication on music) one listener's "basic and extensionally given set" of a concept's instantiations (see above) is relevantly different from another's.

In this way, we could envision the case of a listener asserting that i→l, where

l ... having syntactical symmetry,

the syntactical symmetry in question being the 2x2x2 parallel relationship between the statements, outlined in Appendix II: the second statement parallels the first, the pair 3,4 parallels the pair 1,2 and, finally, the second half of the piece parallels the first half.

The only problem we then face is that in any piece which always has l, i→l, and likewise even i⇒l, are vacuous expressions. Therefore, in order to know whether some listener has a use for the concept i, we have to offer him three options. If he has no use for it, he simply has l everywhere. If he uses i (and thinks that i→l) he will have i∧l everywhere. If he reserves judgment, he simply follows both tracks at once (cf 0.7.4).

Now what does this amount to? It means only that some listener associates the concept i with a number of better-defined concepts used in understanding the piece, and thus, in certain of his nn worlds allows conjunctions like i∧l. Only the requirement that these nn worlds represent admissible "possible pasts", here as in other cases (cf 2.1.1, note), excludes trivialities. Thus, only after a sufficient number of putative i-pieces have been formalized can any reasonable attempt be undertaken of defining what i means to the listener whose competence is being formally represented.

3.3.3

Accordingly, we propose the following four BivFs. We continue to number them as we began to do it in 3.2.3. All take Fig 22's column 4 as left-hand column; BivFs 5 and 7 are somewhat stronger than BivFs 4 and 6, respectively.

4. BivF(¬b↔d, l∧(¬b↔d), a→(¬b↔d))
5. BivF(¬b↔d, (i∧l)∧(¬b↔d), a→(¬b↔d))
6. BivF(¬b↔d, l∧(a→(¬b↔d)))
7. BivF(¬b↔d, (i∧l)∧(a→(¬b↔d)))

As can be verified in Fig 23, w⁸₆, by its (1) and (3) conjoined, only asserts that in as far as the melody is concerned, the piece remains a symmetry-bound "Stück im Volkston"; column 10 says that it remains that anywhere. Both contain the information of column 5. (N.B.: we are saving column 6 for later: cf 3.4.1).

Fig 23a

Fig 23b

3.3.4

One may argue that a conjunction like i∧l, if systematically used, does not enable us to "separate" the two conjuncts: the model, in that case, just does not exclude the possibility of their being strictly equivalent. Now there is a case in which even one little piece like the present one may give us more, and that is the case in which a musical behaviour associated with i undergoes a radical change, thus yielding a sort of testcase. For either in changing its behaviour the music loses its i-quality also, and then i implies the original behaviour, or the i-quality survives alone, and then it is shown that i does not imply it. Again, one model is capable of accomodating both possibilities, just as our thinking and/or "feeling" is.

3.3.5

The here proposed, and to i associated, character is j→k, wherein

j ... producing harmonic progression, and
k ... involving only the degrees V and I, one for each statement.

We write out in Fig 24:

8. BivF(j→k, i, i→(j→k))
9. BivF(j→k, i→(j→k), i)

two BivFs having in common their left columns, and part of their right ones as well [5].

Always to the same column 11 as a common left column, in order to reflect the temporary link between j→k and 1, we add:

10. BivF(j→k, l)
11. BivF(j→k, i∧l)

Fig 24a

Fig 24b

3.3.6

What we have in Figs 23 and 24 is a way of connecting BivFs different from the one used in Fig 22. If we call the result of concatenating BivFs (2.2.3; 3.2.5) a BivF chain, we might call the structure of Figs 23 and 24 a BivF cluster.

We will define them as follows.

C is a BivF chain iff C is an ordered sequence of BivFs each of whose right columns is the left one of the next in the sequence.

K is a BivF cluster iff K is a set of BivFs which have in common their left column, and/or part of their right column.

In all cases treated so far, A-worlds are on the same rank.

3.4 Musical effects not bound up with the sixth statement only

3.4.0

There is still a third way of connecting BivFs to form larger structures; the resulting structures we will call BivF sequences. A-worlds are here situated on different ranks.

S is a BivF sequence iff S is an ordered sequence of BivFs in which the terminal claim (the total claim asserted in the world corresponding to Fig 6 and 9's w¹₂) of each BivF contains the initial claim (the claim asserted in the world corresponding to Fig 6 and 9's w⁰₁) of the next in the sequence.

3.4.1

We present a first example. (The fact that both BivFs involved in it are degenerate is obviously immaterial.) We begin by adding one element to the BivF cluster of Fig 23, viz.

12. BivF(¬b↔d, ¬(b∨d)→j)

and then hook onto this BivF the following:

13. BivF( ¬(b∨d)→j, T)

in the way shown in Fig 25 6.

BivF 13 belongs to a class of BivFs treated by us in 2.0.1 and 2.1.3 (cf Fig 5). Fig 5's w⁰₀ is here represented by w⁴₁, Fig 5's w⁰₁ by w⁶₆. In both cases it is clear that the worlds we need are contained in the worlds we have, for

(¬b↔d)→(¬(b∨d)→j), and (w⁶₆(1)∧w⁶₆(2))→w⁶₆(2).

We have decided against writing them down separately, because of the fact that BivF 13 will hardly be got into by any listener unless he is also aware of w⁶₆(1), and thus is, or becomes, aware of BivF 12.

More precisely, we take BivF 13's terminal claim to be plausible only in a situation which is preceded by a listening situation characterized by BivF 12's initial claim.

Fig 25

3.4.2

In 3.2.6 we defined a concept b of "relevant" motionlessness. We did this by first defining the concept c of "weak" motionlessness:

c ... "weakly" motionless

(c is true iff, in a chain of two successive attacks occurring within a statement, there is one which has the same pitches as the foregoing) and then requiring "weak" motionlessness through at least two successive attacks. We will now show that the concept c is useful in its own right. By definition, we have b→c (and therefore ¬c→¬b, by PC).

(Remark. In asserting implications which are true "by definition", as we do here and in the two following paragraphs, we do not mean to claim that subjects "think through" all logical conclusions of what they know, and therefore all logical relations between concepts they may have. Things like the degree of nestedness of operators may well be limiting factors. The three cases which concern us here I take to be unproblematic however forbidding may seem at first glance, e.g., the second's verbalization (cf 0.7.5).)

3.4.3

Similarly, if we define

m ... producing rhythmical uniformity between statements (this uniformity being a symmetrical (equivalence) relation)

n ... producing rhythmical conformity of all attacks in a statement with corresponding attacks in another statement (no symmetry implied)

we will have m→n, by definition.

3.4.4

Similarly again, if we define

o ... changing its direction (at least) once

and

p ... changing its direction more than once

where "changing its direction" means, within one statement, either going downward after having gone upward, or going upward after having gone downward, we will have p→o, by definition.

3.4.5

In relation to the Chopin piece, the concepts mentioned in 3.4.2 give rise, in our proposal, to the following BivFs:

14. BivF(c→¬d, c↔¬d, (c∧d)→a)
15. BivF(c↔¬d, (b↔¬d)∧(¬c→d))
16. BivF((b↔¬d)∧(¬c→d), (b→¬d)∧(¬c→d))
17. BivF(¬c→d, (b→¬d )∧(¬c→d), ¬(c∨d)→¬a)
18. BivF((b→¬d)∧(¬c→d), (b→¬d)∧(b→c))

which interlock in the way specified by Fig 26 [7].

BivFs 14-15 form a chain; so do BivFs 17-18. BivFs 15-16-17 and BivFs 15-16-18 form sequences. The sequences show a step-by-step weakening of the formula w¹⁹₂(1), which implies all of the following:

b→c : holds true
b→¬d : holds true
¬c→d : lost at the 7th statement
¬d→b : lost at the 6th statement
c→¬d : lost at the 4th statement
c→b : lost at the 4th statement.

Remark

Let i/j abbreviate wⁱ_j. Then the following list may help the reader disentangle the BivFs from Fig 26.

3.4.6

The concepts mentioned in 3.4.3. and 3.4.4 we propose to incorporate in the following:

19. BivF(m, l→m, (l∨m)→n)
20. BivF(m, (i∧l)→m, ((i∧l)∨m)→n)
21. BivF(a→(o∧¬p), a→¬p)
22. BivF(a→¬p, (a∧d)→(p→¬c), m→((a∧d)→(p→¬c)))
23. BivF(m, (a∧d)→(p→¬c), T)

whose interconnections are shown in Fig 27 [8].

The sole function of BivF 23 is to produce the arrow w²⁵₈ w³⁰₈A (not counting the vacuous w³⁰₁(1) and w³⁰₂(1)). It presents the limiting case, possibly theoretically interesting, of a BivF part of two clusters at once: BivFs 19-20-23 form a cluster, and so do BivFs 22-23.

BivFs 21-22 form a sequence.

Remark

as in the Remark to Fig 26.

3.4.7

Finally, we define

e ... belonging to the second half of each statement:

f ... belonging to a sequence of at least two attacks of one note each

g ... belonging to a sequence of at least two attacks of two notes each

h ... belonging to a sequence of at least two attacks of more than two notes each

we propose

24. BivF(a→f, (a∧¬e)→(¬f→g))
25. BivF(a→f,a→(¬f→g))
26. BivF(a→f, (a∧e)→(¬f→(g∨h)))
27. BivF((a∧¬e)→(¬f→g), T)
28. BivF(a→ (¬f→g), T);

and we obtain [9] Fig 28.

BivFs 24, 25 and 26 form a cluster; 24-27 and 25-28 form sequences.

Fig 26a

Fig 26b

Fig 26c

Fig 26d

Fig 26e

Fig 27a

Fig 27b

Fig 27c

Fig 28

3.5 "Higher order" BivFs

3.5.0

There is a class of UNLL processes which a) are quite interesting in their own right (and therefore ought to be touched upon) b) can be represented by BivFs as easily as all those we have reviewed up till now (and therefore are, in principle, open to our methods) but c) present the complication that the resulting BivFs cannot be fitted into the network we have been building up. Let us explain what is at issue.

3.5.1

The argument-places of the Bivalence Function may be filled by perceptual propositions as complex as you please; the only restriction imposed on them is that their combinations as occurring in nn worlds represent musical precedents to which the music can be understood as conforming. In any case, they thus reflect "musical behaviour". Now, since chapter 0 we have been emphasizing that we systematically read "music" as "music-heard": as a listener's construction. We now want to start out from the fact that among the things a listener may recognize (and to which thus, according to our definition of them, correspond concepts (cf 0.3.1)) may count his own past UNLL processes as representable by BivFs. These concepts, which thus involve pragmatical descriptions of his past, may be used in his present semantical and pragmatical processing, and give rise to what we, with due caution (as expressed by scare quotes), will call "higher order" BivFs. To summarize:

• concepts are used in UNLL processes representable by BivFs
• these UNLL processes, being remembered, give rise to a special class of concepts
• these last concepts in their turn are used in UNLL processes representable by BivFs.

Let us represent these by Fig 29, which, obviously, is just Fig 9's substitution instance BivF(U,V,W).

Just like Fig 21's S, the capital letters U, V, W stand for a restricted range of concepts, viz., those which, in the given combination, involve the cognitive situation just described

BivF(U, V, W)
Fig 29

3.5.2

I think that the adequate understanding of the simple traditional ABA form involves, notably in the returning of the A section, UNLL processes that have become fairly well-anchored in our minds, and that we have come to expect (cf the pleasure of the re-discovery and re-learning, 2.3.1, 3.2.1, 3.2.2, and 4.1.2). These UNLL processes the Chopin piece in some respects simply implements, in other respects (partly) frustrates, thereby initiating the special UNLL processes now at issue. Thus, if we take U, roughly, as: after a music has undergone some change, its first state will be reattained sooner or later, W as: from its second state, it " starts back " towards the first, and V as: it effectively reaches it, we will have at least three UNLL processes representable by Fig 29's BivF.

1. The motionlessness of the second half of the statements, disturbed in the sixth, seems to come back in the eighth, but just fails to make it (cf BivF 13).
2. The one-part melody of the first statement just fails to reappear in the fifth, where the piece's symmetry would have it (BivFs 27 and 28).
3. Also, the second half of the piece begins to reproduce the melodic curves of the first half, but at the 7th statement the " only climbing " variant already appears, and the 8th statement only preserves the 4th's climbing character by producing a wholly new variant (cf BivF 22). Schematically, the first half has:
   
   and the second half:
   

Clearly, in assigning U, V and W the meanings described in 2) and 3) we interpret the piece's symmetry as a timing-device for "a music's re-attaining its first state". (There are, we think, some good reasons for doing so.)

3.5.3

Now the fact that, in all three cases, very new variants arise, brought about, so to speak, by the "conflicting interests" of -V and W, strongly indicates that the bending-of-the-rules characteristic for UNLL processes is at work here. In 1): mobility restricted to a place and realized in a way which are new in the piece; in 2): "inconsistent" behaviour of a "second voice" note; in 3): the appearance of a type of curve more complex than any of the foregoing ones.

3.5.4

It seems difficult, without having assessed a sufficient number of cases, to formulate hypotheses about the exact mathematical relationships obtaining between what we tentatively called "higher order" BivFs and the "first order" ones given in Figs 22-28. Tentatively, it could perhaps be assumed that Fig 29's formula "¬V∧W" is, in some loose sense, specified by the content of certain columns. For case 1) this would be column 6, for case 2) columns 32/33, and for case 3) column 29. In all these cases columns terminating on pws having only "¬□A∧◊□A" formulas.

Here, certainly, are important avenues to be explored.

3.6 Tableau

3.6.0

Figs 22-28 must be thought of, we think, as combined in a synoptic tableau like the one given in Fig 30. It is based on the principles formulated in 3.1.4. "A" stands for A-world; all lines linking points w and/or A represent arrows, all pointing upwards.

Fig 30

3.6.1

In presenting Fig 30 as a rendering of a listener's possible activities as he orients himself in the Chopin piece, we seem to invite two kinds of criticism. The first kind would suggest that it is incomplete, or wrong; the second that it could not possibly be made complete, or right, with respect to some given listener.

The first criticism is concerned with the adequacy of our analysis of the music's potentialities. Either certain paths are deemed unacceptable, or certain necessary paths are found absent, or both. This is a criticism that certainly has its interest, but that, in itself, will not be essentially damaging to our enterprise. Firstly, as stated in 3.1.0, we have not really been working on Chopin, but on a BivF network. Secondly, in principle, columns may be added: Fig 30, as it stands, in no way pretends to be "complete".

The second kind of criticism would be more serious. Until counterexamples are produced, we defend it as our claim that our general way of representing listeners' cognitive behaviour is indeed adequate, in the sense that any way of listening proposed by any reader we will need to be able to write down and integrate into Fig 30 by adding appropriate columns.

3.6.2

This claim may seem stronger than it really is. Columns may be just specifications of the Fig 21 schema (such as: this is piano music), but, as we will see in 4.1, it is at least highly likely that more problematic aspects, aspects reflecting this music's specific pragmatical effect on a listener, this music's possible meaning in the sense of 0.0.2, require specific (conscious) attention on his part. And it is our claim that these aspects are representable by BivFs. If we are right, we have also a specification of the "meaning-content" of the music: after all, each BivF's right column does specify both the "BELIEF" and the "truth-supporting reason" we need – just as we promised our reader in 0.0.3.

Most of the next chapter will therefore be devoted to investigating in a more general perspective the Bivalence Function concept itself.

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