Musical Work and Its Semantics
I.
First, let me clarify the question of what a musical work really is and of what kind of entity it is.*) I am following on from the work Bernard Bolzano, the logician, philosopher and mathematician, who formulated in his Wisseschaftslehre ([BOLZANO 1837], §19) principle which might be called Bolzano's principle. It is as follows: what is common to written, spoken, and thought sentences is the "sentence as such" (Satz an sich). The sentence as such is an abstract thing; it is that entity which some lines on paper, or a certain sequence of sounds, should express. It exists in an abstract way, independently of the real world; nor is it temporarily independent.1) Charles Sanders Peirce's type/token distinction should also be mentioned here (see [PEIRCE 1930], §246, §301). A token is something written, spoken or thought, it is an entity of the physical world; in contrast to this, type is abstract, being rather something that is common to many token(-s).
Analogous to the defining of the sentence as such - there is the sound of a musical work and the score of a musical work and an imagined (thought) musical work. What is common to sound and score and imagination is the musical work, composition. (Expressions "musical work" and "composition" will be considered as equivalent, as will "sound", "audial expression" and "performance", similarly: "note record", "notes" and "score") There can be a multitude of sound and scores (and imaginations as well) of one musical work; there is however only one composition. Composition exists in an abstract way. Some composition can be "unknown" when we do not have the score or the audial expression; in spite of that the composition - in an abstract (objectual) way exists. We can also have scores but no sound, or the sound(s) but no scores.
Composition, as well as all others abstract objects, are discovered by us - not invented. Composers fantasize compositions, record them in notes. The composer as the creator picks out just one from all of these many compositions which already exist in the realm of abstract entities (apparently, while doing this he or she is seeking to find the best one).
Generally speaking: every work of art is, in contrast to its concrete expressions (realizations), abstract. Sculpture, architecture, painting, seem to be, perhaps, exceptions. But as a matter of fact, there are the expressions (realizations) of the works on the one side and the works themselves on the other side. If a painter paints a painting, and then paints another one, but entirely the same painting without copying the previous one, which of them should be identified as the copy and which as the original? This would also be the case of a sculptor who carved the same sculpture twice, or an architect whose single plan is used to build multiple buildings. The only thing that can be said is that what is temporally the first is the first performance and what is temporally the second is the second performance, and so on; thus there are different performances, realizations (or pertinently, "versions", "attempts to elaborate", or "renderings", or "interceptions") of one work.
Sound, score and imagination could be, perhaps, considered as expressions of musical work, expressions in the logical sense of this word (purposely, I will not use the term "sign" as it is common in esthetic semiotics). Sound, and imagination as well, is no expression, sounds simply come forward, appear one after one (of course, man reconstructs the composition to himself through the medium of notional representation). It is thus only the score what is an expression.
The sphere of expressions in music is not depleted by that. For every concrete performance there exists a Cartesian graph2) where all the musical parameters are completely and precisely written. Such a graph of a composition's sound is more unambiguous than an ordinary score. In the case of a non-differentiating performance, there is only one graph. Hence: the graph of sound (graph of performance) is, like a score, considerable as an expression.3)
Now another differentiation needs to be established. The written word (a crooked ink-blot on paper) is a record of a word, the uttered word (articulated sound) is an utterance of a word. Thus generally: written or spoken expression is a record or an utterance of an expression. Expression is thus, in spite of some of its occurrences, abstract (written or spoken expression are empirical things). For example, the word expression is only one but in the previous sentence it occurs three times in all.
In music: the score of one composition can be, of course, in many "performances", occurrences. In the same way as tones are differentiated from the sound tones, notes should be differentiated (note records as such, scores as such) from the occurrences of notes (particular note records). Analogously, we have the graph as such and the occurrences of a graph.
Consider now what performance as such (sounding as such) is. Performance, the audial expression of a composition, is a certain sequence of tones, sounds, in time. Associated with this concrete, "real", sequence is its abstract correlation, performance as such (this is, as it will be seen later, a specific function). This performance as such can be expressed by a graph.
In connection with this, another problem should be - for the sake of clarity - introduced now. Audial expression should be, as Ludwig Wittgenstein reflected in his Tractatus Logico-philosophicus ([WITTGENSTEIN 1921], 4.014, 4.0141), isomorphic to its score. But the score of a musical work does not register everything. It does not register, for example, partial tones, and the dynamic is only presented in framework, etc. Other distortions of isomorphism are, for example, abbreviated records like trill, and so on, or abbreviations: for example, the sign of iteration says repeat the part of a bar, or repeat the whole bar, and so on. (The reason for this imperfection of scores is the inability of man to comprehend during the play in detail all the extended parameters while playing. The craftsmanship of an expositor lies in his fleshing out what the score of a composition offers.) Thus, the score is not isomorphic to performance.
Performances can be divided in two groups: entirely identical performances (imagine, for example, the repeated playing of a composition from a CD) and performances differentiated from each other. Strictly speaking there is a class of (in details) different performances as such, thus for every particular class there are n possible performances as occurrences.
Now the following question may arise. Is composition rather what is expressed by some ordinary score, or rather that what is expressed by some graphs? (Of course, in either of these two cases, it is not the identity of composition with its expression.) It is natural to think of a score as it is the score of a composition, or that a graph is the graph of a composition; in both these cases, the expression and composition are different. Now there is certain common sense formulation so that performance (as such)1 and performance (as such)2 (etc.) are "specifications" ("realizations"of one composition while their "unspecified" representation is ordinary score (as such), i.e., a score as such which point to a whole class of audial expressions as such (and their respective graphs). It is common to say that there are (not only numerically) different performances of the same composition. Consider further, however, that scores are of distinct degrees of exactness - from broad accuracy of choral records, or of scores registering aleatoric music, or of notation of models, ordinary notation, to the total precision of scores (eventually of graphs) of composition created following a specific performance with help of computer notational programs.4) Thus composition can be now seen in many cases as vague, that is as a function in n-dimensional space where in the same positions or even in whole dimensions (for example composition consisting of arbitrary dynamics) is indicated by a whole class (strictly speaking: by disjunctions) of values.5) Thus the problem of identity is here too (although it will not be dealt here).
Consider: musical composition, musical work, as well as its parts (tones, melodies, etc.) are abstract. With regard to the fact that in the realm of abstract entities does not happen some increasing, nor decreasing, of compositions, all the compositions are still a priori given. Compositions are within all times and all circumstances all.
The logician Pavel Tichý expressed in discussion with Pavel Cmorej ([CMOREJ, TICHÝ 1998], p.154) nearly the same opinion on the abstract nature of musical entities: "'Melody' is like 'symphony' - it is not something that happens but something that can have countless performances of which every one is an event, and not a symphony." (A melody-event is what we call audial expression of composition, performance.) Except for musical composition, score as such is also abstract and an audial expression as such too. Particular scores, as well as particular audial expressions, are, in spite of this, concrete. Differ now composition from the score of the composition as such too, and the composition from the audial expression of composition as such. (For the sake of some clarity, there is the following thought of Tichý ([CMOREJ, TICHÝ 1998], p. 154) will be exposed: "How can we hear abstract entity? To hear melody (or symphony) mean to be exhibited and to apprehend sound waves caused by its performance. This complex which is prompted by sound waves must be by man [through the medium of notional grasping; J.R.] actively reconstructed in his mind. By this the human hearer differs from, for example, cat which is exhibited to the same sound waves."
Now let us compare our conception with the opinions of theoreticians concerned with questions of musical art. The refutation of "platonism" is very common (and inadequate reproducing of Peirce's definition of type as the set of all token's). The motivational reason for this refutation of "musical work as such" is probably the emphasis on musical praxis, performing of compositions. Among authors that can be qualified rather as estheticians, or philosophers, a conception similar to ours may be - however rarely - found. Against the pleiade of nominalistic attitudes stands the opinion of Roman Ingarden (who is in philosophy known as an investigator of literary work) and the claim of his student, Zofia Lissa. Ingarden discussed through the whole his book, Musical Work and the Case of Its Identity ([INGARDEN 1958]), that musical work as such - from the moment, it was compound - does not depend from real world. In accordance with our conception, work is (itself as such) independent of real world (or the environment), and, moreover, it is not influenceable by written-recording (or by repeated performance), or "creation" by demiurge (composer) because it already (abstractly) is. Zofia Lissa, similarly to us, claims in her book About the Substance of Musical Work ([LISSA 1982], p. 49) that "Work is only one, performances can be infinitely many". But - surprisingly from the view point of our conception - writes that work is "connected with its composer" (ibid., p 46). The expression/occurrence of expression distinction and its application to the problem of music is in the field of musicology perhaps entirely unknown.
II.
If a certain small comparison of language with music from the viewpoint of logical semantics will be made, it will be seen that music is not similar to language. Among all arguments, the strongest one is that music is lacking the central point of logic: the entailment. A musical work does not have the semantical properties of language because it is not possible to compile a vocabulary of the type expression/its definition; in music there is nothing that can evoke the sentence, nothing that can represent some possible state of affairs (not the past, not the future), nothing that can be something like a musical (single, complex, compound) proposition (for analysis by propositional calculus), there is no predication of predicate to subject (for analysis by predicate calculus); referring to another than psychic "state of affairs" (like extra-musical meaning, icons) is not a language reference but only a tectonical similarity; there is no truth (the fundamental semantic category), music cannot be then falsified, nor verified (in musical work, there are no contradictions and no tautologies), consequently music is lacking the entailment.
What is can be seen common, however, to music and language from the logical viewpoint are - as will be seen later - semantical relations of denotation and of reference. Notice, first, what is the meaning. In accordance with logical conceptions, I prefer that meaning is something we use to understand expressions. The meaning thus must be something objective and not subjective, it cannot be imagination, nor object; it has to be an abstract (objective) entity. Remember the relation of expression and abstract thing as it was discussed above. As in the main semantic conception (I mean here intensional logics) accepted the meaning (an intensional entity) as intermediate step between an expression and respective denotatum (for example between the expression Mt. Everest and concrete stone massif, i.e., Mt. Everest itself, or: between the expression the highest mountain and Mt. Everest). Transparent intensional logic (see [TICHÝ 1988]) revised this conception in the following way. First, the denotatum as an intensional entity is a function from possible worlds and times (thus it is abstract). Denotatum differs from referentum which is something to which some expression in the end refers in physical world. Consider one example: the expression "the highest mountain" denote a certain function as its denotatum; the meaning must be abstract (has to be objective) and therefore Mt. Everest cannot be a denotatum; to understand the expression "the highest mountain" you need not to know which actual mountain it is; Mt. Everest is rather something to what the expression "the highest mountain" (via its denotatum) refers. Denotation is thus a two-argument relation, reference a three-argument relation. Hence: expression (common language expressions, ciphers) denotes denotatum (for example propositions, numbers, functions) and by means of a denotatum pertinently refers to a referentum (a particular thing named by the same parts of sentences; numbers and functions - generally all mathematical expressions - does not have any referent, they have only denotates).
In the field of intensional logic, the problem of necessity of accepting hyper-intensional entities has arose. I prefer the suggestion of Pavel Tichý, i.e., to put certain important step between the expression and the denotatum (in the case of sentences and by them the denoted propositions, in the case of mathematical expressions and by them the denoted mathematical objects) which Pavel Tichý calls construction (this notion of construction is not psychological, nor intuicionistical). Without a precise defining of constructions (see [TICHÝ 1988], or [Materna 1998] construction can understood as an abstract procedure (instruction) for grasping some object. For example, the expressions "equiangular triangle" and "equilateral triangle" serve to identify the same object but through two distinct ways, two "intellectual" procedures (see [TICHÝ 1998]). The denotatum is not structured, it is, for example, a function, a "flat" object; construction - in spite of this - is a structured entity (nonreducible to a set object). Transparent intensional logic identifies the meaning of expression with just this construction. The most interesting aspect of the notion of construction for us is that more than two constructions can pick out one and the same object. Another example is "New York is bigger than London" and "London is smaller New York". These two sentences are the expressions for two constructions (they have distinct parts: "bigger than" and "smaller than" of one proposition (the fact is only one).
In music we must first differentiate (in contrast to many musicological semiotical theories, epistemologically clearly) four basic ways of presentation for a musical composition. On the one side there are notes (score, eventually graph) and the description of notes (eventually the description of graph), on the other side there are the audial expressions and the description of audial expressions. As it was found out above notes as such and soundings as such have to be also differentiated (and their respective description of notes as such and the description of sounding as such, too). Without any inquiring into the relations among them (note, for example, isomorphisms), or without any investigation of the descriptions of notes or descriptions of tones, the following question will be studied. The questions is directed at the problem of whether between notes as such (score as such) and sounding tones (performance), is a musical work, a musical composition, (i.e., an abstract object we discussed in section I).
Reflect now from what elements is a composition formed. In accordance with common sense, composition is made up from tones. Tones are (probably) understood here not as pure pitches but evidently as basic frequencies with dynamics, timbres, and being of certain lengths.6) As was discussed above, not all values are precisely indicated in all cases. Tone differs also from the audial expressions of tone (it is abstract). Let us differentiated those tones from which a composition is formed, i.e., tones that are not identical with an abstract correlation of sound tones (of parts of a performance as such), tonesS.
Applying our previous claims concerning denotation and reference, it can be stated that notes (as such) have as their denotates tonesS (denotation is a relation between expression as such and the object, i.e., toneS here). (Briefly: notes denote tones.) Sound tones than can be considered as referents of notes.
Sound tones, in spite of notes (as such), have neither denotation, nor reference, they are elements of the realm of ordinary things, rather evokes than denotes. They are what the score of a composition (or graph) refers to.
The denotatum of a score is composition, "musical work an sich". The referentum of the score is then a concrete performance of a composition. To recognize what this is, it can be stated (in accordance with claims introduced above) that it is a certain function (as was discussed above: with not all values entirely, uniquely specified). Composition, musical work (as such) (i.e., denotatum of score) is a function.
Can be Tichý's construction inserted between the score as such and function (although not always definite) of which composition is as a certain step? Tichý himself repeatedly answered this question positively (if in [TICHÝ 1988] Tichý postulates the following claims as metaphor in ([CMOREJ, TICHÝ, 1998] he analyzed this metaphor). He said (quoted from [CMOREJ, TICHÝ 1998]): "Melody is more than the sum of tones which are brought together in it. It is a complex, or structure." (p. 139), "all complexes are constructions in my sense" (p. 148), "But terminology left aside, a melody-event (= the performance of melody) is, of course, a simple entity. Melody alone, as every complex, is an abstract entity." (p. 154). Briefly - melody is a complex, every complex is a construction, hence melody is a construction, generally: composition is a construction.
Consider now the following thought which seemingly supports this thesis with the use of musicological data. There are several notational systems (when speaking about music, a few conceptual theoretical systems) which can, in the same way as their particular segments, designate the same composition. It is also possible to transfer from one system to another using transcription (for example from standard notation to proportional) and thanks to particular synonymies (the equivalence of parts of notational scores) which are in nearly all systems also allow transnotation, the alternative representation of the same thing through the instruments of one system (for example, one quarter note can be recorded by two eights notes with ligature). Of course, sometimes something recordable in one system is not be recordable in another system (but this is only a deficiency of these systems). But these cases are only grammatical, syntactical dissimilarities, while denoted composition is still the same and therefore it does not establish the need for constructions (the denotates are not different, there are only different records of one thing here).
Remember our examples "New York is bigger than London" and "London is smaller than New York". These two sentences express one fact through two constructions. There are two constructions because they contain two different relations "(to be) larger than" and "(to be) smaller than". Such a phenomenon can be never found in composition. Tichý's metaphor thus is not correct (and, moreover, if composition is a complex, then his opinion that all complexes are constructions is also not correct).
Consider: for "the semantics of musical notation" thus the notion of construction does not seem to be needed. (But this does not mean that such an analysis cannot be well embedded into the framework of transparent intensional logic which is able with its apparatus to analyze expressions of natural language as well as to analyze the semantics of mathematical notation7). For description of notes, similarly as for the description of audial expression, the notion of construction is quite surely needed (consider the example: "a1 is under b1" and "b1 is above a1" are expressions for two composition of one fact). Transparent intensional logic, aiming to analyze (whole) natural language, has to contain the analyses of sentences about music.8) It is also evident that without a clear understanding of composition (of its model) and its segments, such an analysis of expressions about music is not possible or at best inadequate.
NOTES
* The first version of this study was read by the author at the conference "Problémy analytickej filozofie [The Problems of Analytical Philosophy]" (1999), held in the Slovak Republic.
The area where such sentences are was once named by Plato as "hyperouranios topos", the over-heaven place. This is an obvious connection with Popper's world3 as well. It has more than two axes, it has, for example the axis of time, the axis of pitches, dynamics; pertinently the composition is fixed by the several different graphs (which are compoundable together), for example a graph of dynamics and a graph of frequencies; I do not take oscilograms which are of two axes into account here.) The problem of the sound-records (gramophone-record, magnetophone-record, digital-record, MIDI data will not be dealt here). The standard score evidently appears as a functional graph with the axes of time, of frequencies, which are marked on the lines by notes, the axis of dynamics, and possibly with the axis of timbre. For closer acquaintance with such understanding of musical composition see author's study "Model skladby na základě pojmu funkce [The Model of Composition on the Basis of the Notion of Function]", 2000, in Ambiguity and Music, J. Haluška (ed.), Bratislava: Seminar Mathematic and Music. See the author's article Model skladby na základě pojmu funkce [The Model of Composition on the Basis of the Notion of Function]. Compare, for example, with Tichý in [ TICHÝ 1988] or in [ TICHÝ 1995] . The author is preparing the study entitled "Speaking about music" on this topic.
REFERENCES
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INGARDEN, Roman: 1958, Utwoŕ muzyczny i sprawa jego toźsamości [Musical Work and the Case of Its Identity], in Studia z estetyki (Vol. 2). Warszawa: Państwowe Wydawnictwo Naukowe, 163-295. Reprinted as Utwór muzyczny i sprawa jego tożsamości, Kraków: Polskie Wydawnictvo Muzyczne, 1973.
LISSA, Zofia: 1982, O podstatě hudebního díla [About the Substance of Musical Work], in Nové studie z hudební estetiky [New Studies from Musical Esthetics], Praha: Supraphon. Originally in Nove szkice z estetyki muzycznej, Kraków: Polskie Wydavnictwo Muzyczne, 1975.
MATERNA, Pavel: 1998, Concepts and Objects, Acta Philosophica Fennica 63, Helsinki.
PEIRCE, Charles Sanders: 1930, Grammatica Speculativa, in Collected Papers of Charles Sanders Peirce, Vol. 2, Cambridge: Harvard University Press.
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