LOGICS AS MODELS OF MUSICPeter Gibbins

I

Music curries sometimes speak of 'musical logic'. Certain sorts of musicalappreciation involve an evaluation of the extent to which a piece of musichas structure, and adheres to certain rules. Frederick E. Taylor sees merit intaking this critical usage literally. For in his 'Music and its Logic'1 he argues,first, that there are in music analogues of the logical notions of axiom,,implication, rule of inference and consistency; secondly, that musical workshave a structure which is appropriately analysed in terms of the analoguesof these notions; and thirdly, that musical works are intelligible in virtue ofhaving a structure which may be exhibited by reference to these logicalanalogues. His purposes in articulating this logical model of music are toattack Meyer's relativistic theory of music, which accounts for a listener'sperception of musical structure wholly in terms of his conditioned prepared-ness and responses, and to defend the view that there is something universalin music, namely its logic, which accounts for this perceived structure.

Taylor develops his model in the following way. He first assumes that. . . music is a temporal arrangement of sounds in which it may be felt that groups ofsounds follow naturally from others.2

A partial or complete representation of a unified set of sounds is called asound-segment.3 There are two kinds of relations between sounds which aresufficiently analogous to the logical relation of implication to suggest theidea of musical rules of inference. The first of these relations involves expect-ation.

A particular group or sequence of sounds implies another when the feeling of expect-ation or anticipation is induced in the listener.4

The second rests upon similiarity of sound. One sound segment impliesanother if they share a musical aspect (melody, harmony, rhythm).

In a piece of music there are sounds or groups of sounds which are notderived by means of rules of inference from others. These sounds or groupsof sounds are called axioms, and are of two types.

The first, general or external axioms, are abstract relations of sounds found in manyworks of a particular style or period which when coupled with a musical rule ofinference imply a proceeding set of sounds.6

The second class of axioms, particular or internal axioms . . . comprise the subject

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matter of themes, motifs, tone rows and rhythms. . . . Unlike general axioms they areseldom used in more than one work.. .6

Pieces of music have a further property, analogous to the logical propertyof consistency. A sequence of sound-segments taken randomly may bedescribed as inconsistent if, as will very likely be the case, some sound-segmentsdo not bear the usual relation of expectedness or similarity to theirimmediate neighbours. A sequence of sounds is consistent if and only if it isnot inconsistent. Musical works are usually consistent in this sense.7

Such is Taylor's logical model of music in which a place is found forfeatures of music which correspond to axioms, implication, rule of referenceand inconsistency. The model does some useful work. It explains musicalambiguity (when a sound-segment is derivable in several ways), it suggests adefence of musical universalism ('what is culturally determined and learnedis not the logic of music but only its axioms'8), and it gives a precise sense toexpressions like 'musical logic*.

Unfortunately Taylor's logical model of music suffers from serious defects,which are of two sorts. For one might argue that it does not follow thatsurprise-free music is always accompanied by & feeling of expectation in thelistener (though there are cases in which sequences of sounds are expected,and in which expectation amounts to a feeling). One might argue that evenif the rule of inference based on expectation were a reasonable one, expect-ations might be culturally determined and that this possibility is damagingto the universalist case, since it makes musical logic culture-dependent. Onemight argue that a model of music which makes so much of rules and sur-prise-free structures is at best parochial and restricted.

But it is not my purpose to make criticisms of this first, external sort. Forwe may ignore such straightforward difficulties and consider logic as a purelysyntactical model for musical works, assuming that some formal model ofmusic might in fact be adequate to capture those formal aspects of musicwhich we wish to capture.

II

If logic is an appropriate model for musical syntax, then it seems that thefollowing should be the case:

First, there are operations on musical objects that behave like logicalconnectives which operate on wffs (well formed formulae).

Secondly, there is an operation corresponding to implication which hassimilar properties to the conditional in logic.

Thirdly, there is an operation corresponding to negation which has theusual properties of a negation, and which allows us to define consistency inthe usual way.

In Taylor's model there are musical objects, 'sounds', which are represented

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PETER GIBBINS 159

by 'sound-segments'. The sound-segments correspond to well formedformulae (wffs). The logical connectives take the set of wffs, or product setsof wffs into the set of wffs. They make new wffs from previous wfls. Thereare of course many logical connectives with interesting properties, e.g. con-junction, disjunction, implication, negation, etc. In music there appears to beonly one sort of operation, namely concatenation. For suppose there is anoperation 'O' such that if 'x' and 'y' are sound-segments then 'xOy' is asound-segment. Then 'O ' can hardly express relations between sound-seg-ments such as 'implies' since 'x implies y' is a proposition and is not itself asound-segment. The only operation on (say) a pair of sound-segments whichresults in a new sound-segment seems to be 'horizontal' or 'vertical'concatenation.

There is no connective in Taylor's logical model of music which behaveslike implication, for 'musical implication' is not a connective in a languagein which sound-segments behave like wffs, but is rather a metalinguisticterm which behaves like semantic entailment. Musical implication as well asbeing unlike the conditional in logic is unlike both entailment and the con-ditional in failing to be transitive in 'non-counterfactual' cases. For considerhe three successive sound-segments x, y, z. If y is expected after x and z aftery, it need not be the case that z is expected after x. And if x is similar to yand y to z, x need not be similar to z. Implication derived from expectation orsimilarity does not have the properties of the logical conditional, or theproperties of an entailment. In addition similarity is usually thought to besymmetric. If x is similar to y (in virtue of sharing a musical aspect), then y issimilar to x. It follows that musical implication derived from similarity is ananalogue of equivalence if it is an analogue of the conditional.

Finally, consistency is usually defined in terms of negation. A system isconsistent if and only if it contains no theorem such that its negation is also atheorem. It is difficult to see what the negation of a sound-segment could be,for it would have to be some other sound segment. If we argue that con-sistency is the primitive notion and that a negation of a sound-segment is asecond sound-segment inconsistent with the first, we are led to the resultthat y may be a negation of x while x may not be a negation of y. A 'negationwith this property is at best a peculiar sort of negation. For a sound-segmentmay have two 'negations' which are also 'negations' of one another.

UI

Music, as described by Taylor, does not map well on to logical syntax.There are troubles for the logical model of music. This is not to say that noformal model is adequate for the description of a piece of music, nor that ifthere were such a model that it would fail to give us a deeper understandingof certain aspects of certain sorts of music. But devotees of modelling should

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look elsewhere than to logic for their models. For example, modellers mightfind that group theory suggests a way of describing the (harmonic, melodic)transformations within a piece, of music allowed by certain styles and periods(though I confess it seems to me unlikely that any very general results wouldbe obtainable by means of this approach, simply because 'music' embracessuch a wide variety of human activities and products).

REFERENCES1 Frederick E. Taylor, 'Music and its Logic', 4 Ibid., p. 216.

in British Journal of Aesthetics, voL 14,1974, 6 Ibid., p. 217.pp. 214-30. * Ibid., p. 218.

2 Ibid., p. 214. 7 Ibid., p. 219.8 Ibid., p. 215. 8 Ibid., p. 227.

CORRESPONDENCE

Dear Sir,

It is perhaps unfortunate that your reviewer of The Ideal City, whatever his philosophicalviews, is so little acquainted with the subject. To quote only one example: Mr Scrutonassumes that 'few of those conceptions and theories (of the 18th century) have been able totranslate themselves into visible reality'. Itis of course well known that neither the NapoleonicEmpire style nor the so-called Prussian style could have taken their lasting form withouttheir Neo-Classical French predecessors. The situation was similar in Russia.

Yours faithfully,H. Rosenau

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