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Making sense in music
An enquiry into the formal pragmatics of art (part 7)

by Jos Kunst (1978)

Appendix I

S7 as a discriminatory system

Let us say that if I understand a sound by reference to natural law, I take into account all possible worlds I can imagine which instantiate the workings of the natural law I refer to, be they in the past or in the future. ln terms of the alternativeness relation defined on our models this means that it will be symmetrical as well as reflexive and transitive: all worlds have access to all worlds, and the resulting modal logic is S5.

Let us represent any bunch of such worlds by the following shorthand schema:

(1)

and let the "present case" in which I invoke some natural law for the purpose of "explaining" (understanding) some given sound, be graphically singled out from this bunch in the form of a world w₁:

(2)

The assertion of some natural law will then, given S5, be logically stronger than our conventional laws could be In S5 all □ operators are equivalent to strings of infinitely many of them, whereas in all models used in this book up till now, everywhere, even if □p was true, □□p was false (cf 1.2.3). This logically stronger sense of law-likeness we will denote by writing "□ n times repeated, for any n", as follows:

(3)

and this will be taken to say: "at w₁, I explain to myself the occurrence of a given sound by reference to natural law". Clearly, if this is the only way of understanding relevant to a given sound, it is not part of any music.

The reason why we, nevertheless, brought up this subject resides in the theoretical and practical importance of the following deviant BivF structure:

(4)

which represents the cognitive frame within which a changeover may take place from the natural law understanding of a sound (event or activity heard) to a musical understanding of it.

One obvious domain of application for this structure is the genetical one - be it in one's first music learning or in the prehistorical "birth of music" - but also, if one considers, as we do, the genesis of music as an ongoing process, this deviant BivF has a specific and interesting role to play. We refer to cases of ambiguity whether some activity or event heard (or aspect of it) is part of the music or not. Many readers will now think of (quasi)surrealist, and/or happening-like situations, but I think we must accept that this has been, and will be, an important aspect of all musics at all stages of history. It is bound to occur in any non-idealized listening process; it is part and parcel of all musics with built-in transmission insecurities of any sort: I would say, of all real-world musics.

By way of a sufficiently crude example, let us take the case of a musician performing a piece in a tempo that I find totally wrong. That aspect of his playing I then refuse to understand musically. Instead, I come up with some psychological or other natural law explanation for his behaviour (he is ashamed, or technically unable, to take the right tempo, he has been going to the wrong teachers, or, more sketchily, he must be mad, etc.). (In most cases we just suppose that there must be some such explanation, and leave it at that.) A proposition to that effect, specifying (aspect of) sound plus origin (as we had it in 0.6.5) then fills the P place.

Something may then make me revise my ideas, and provide me with a proper musical interpretation (understanding) of his tempo. P is then negated, and I gain access to the right hand column in the customary way. But I may also reserve judgement; not know for sure if P has been negated or not,and, for the time being, entertain two mutually incompatible hypotheses at once. In that case I occupy two "positions" at once in our deviant BivF, viz., w⁰₁ and w¹₂ (cf 0.7.4 and also the case mentioned in 3.3.4).

Lastly, let us return to our different modal systems, and show at which worlds in our deviant structure the axioms of which systems hold:

(5)

It is seen that all and only those worlds which have access to the musical perspective (the right column past) are S7.

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