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Musical Irrationality in the Shadow ofPythagorasChristopher MacklinPublished online: 21 Jul 2011.

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Musical Irrationality in the Shadow ofPythagoras

Christopher Macklin

Music has been aptly described as an ontological mutant, neither exhaustively captured

nor fixed by, yet inextricably related to, its manifestation in performances and notation.This characteristic lends itself to endless debates on its identity as well as to myriadcontradictory assessments of its utility and value. While some authors have sought to

attribute this phenomenon to the experimental music of the late twentieth century, thepresent essay points out that impossibility, or rather, its sister concepts irrationality and

incommensurability, have been integral to the conception of music since its inception.The shared language of paradox provides a link from the Pythagorean and neo-Platonic

consonance through to the fixed ambiguities of contemporary composition.

Keywords: Aesthetics; Werktreue; Irrationality; Experimental Music; Milton Babbitt;Pythagoras

American Supreme Court Justice Potter Stewart leapt to legal immortality when, in

his brief concurring with the majority ruling for the 1964 case Jacobellis v. Ohio (378U.S. 184, 197), he grappled with defining the threshold of obscenity in film. He

stated, I shall not today attempt further to define the kinds of material I understandto be embraced within that shorthand description [hard-core pornography]; andperhaps I could never succeed in intelligibly doing so. But I know it when I see it, and

the motion picture involved in this case is not that (see also Gewirtz (1996), pp.10231047). This opinion is often summarized as I cant define it [pornography],

but I know it when I see it. In this respect (and some would say in many others),music composed within what Lydia Goehr (2007) terms the regulative work-concept

bears an uncanny resemblance to the prurient concerns that so exercised JusticeStewart.1 Its status as a so-called ontological mutant (Tormey, 1974, p. 204; Goehr,

2007, p. 2), neither exhaustively captured nor fixed by its manifestation inperformances and notation, yet inextricably related to them, lends itself to endless

debates on its identity as well as to external assessments of its utility and value. Asmusicians and scholars, we too know the exasperating sensation of not being able to

Contemporary Music ReviewVol. 29, No. 4, August 2010, pp. 387393

ISSN 0749-4467 (print)/ISSN 1477-2256 (online) 2010 Taylor & FrancisDOI: 10.1080/07494467.2010.587316

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define the subject of our fascination, but remain steadfastly confident that we know

it when we hear it. Such vagueness can be construed as a problem, especially forcomposers, performers, and indeed consumers of music who champion it in the

increasingly granular artistic and educational marketplace. On what level does itmake sense to discuss music as a whole, and what do we do with the paradoxes that

inevitably emerge when we examine individual facets in greater detail?To imply that a definitive, intellectually rigorous answer will be found within these

pages would be hubris of Aeschylean proportions, crying out for retribution, if not

from the Furies, at least from the scholarly equivalent of a skull-shattering tortoise.With that caveat firmly in place (and overhead), however, I suggest that some of the

discomfort associated with confronting the impossibilities of musical practicewithin an aesthetic of musical works can be assuaged by reframing them in contexts

which make provision for impossibilitys sister virtues, irrationality and paradox.We do not even have to wander too far afield to find such frames, as musics history

has long been bound up with the paradoxical and irrational, manifested as much inthe vagaries of its biographical narratives as in the mathematics of its conceptualunderpinnings. Indeed, classical music as a whole can be read allegorically in the

apprehension of one of its most illustrious proponents, the ancient Greek polymathPythagoras of Samos.

For millennia, students have encountered the name of Pythagoras (fl. c. 6th5th c.BCE) associated with two fundamental concepts. First is the attribution of the famous

right geometric theorem, which, in the formulation of Diogenes Laertius (1925),states that in a right-angled triangle the square on the hypotenuse is equal to the

squares on the sides containing the right angle (p. 331). Although recent scholarshiphas suggested that the principle had been in use for centuries before by the

Babylonians (Aaboe, 1974), the scholars of antiquity commonly credited the Greekphilosopher or those of his close circle with its formulation, and the construction asits first proof, which links the conceptual aethers of numerical abstraction to the

physical reality of visual experience of space and form.2 Pythagoras second greateponymous distinction performs a similar feat within the realm of aural experience.

According to Iamblichus Vita Pythacorica (as cited in Riedweg, 2005, p. 28), whichcontains one of the most common versions of the creation myth of Pythagorean

consonance, the philosopher was struck by the pleasing quality of sounds emanatingfrom a smithy and resolved to ascertain what caused them. He discovered that the

sizes of the anvils of the blacksmiths that sounded harmoniously were related to eachother in simple ratios, and from this deduced that musical consonance was the auralmanifestation of rational order. To Pythagoras, and to the subsequent adherents of

his philosophy, the interdependence of mathematical constancy, physical space, andaural experience had mystical significance, suggesting that a dependence on

numerical ratio, or a fundamental rationality, unites both cognition and sensoryperception. Music, according to this philosophy, thus serves as a conduit to the direct

appreciation of each. The implications for subsequent musical experience have beenreverberated up to the present. First, the Pythagorean situation of music as mediator

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between the realms of intellect and sensory perception parallels, and perhaps

provided a model for, the paradox-rich formulations of aesthetics in the lateeighteenth and early nineteenth centuries. The declaration of Novalis (Georg Phillipp

Friedrich Freiherr von Hardenberg) that [i]nasmuch as I give the lowly a highermeaning, the common a hidden aspect, the known the dignity of the unknown, the

finite an infinite appearance, thus I romanticize them (as cited in Shafer, 1975, p.42), which Goehr (2007) terms an expression of the romantic illusion,3 is a directdescendant of the idea that a finite sensory experience can partake of, and should be

experienced as, an expression of an infinite mathematical reality. As music seems toreadily provide this experience, such a view emphasizes the value of musical

experience over other forms of sensory perception. In this framework, the act ofperceiving musical intervals (or perhaps more precisely, perceiving such intervals as

either consonant or dissonant) is thus seen as the direct sensory experience ofnumerical ratio unmediated by the thinking mind. This idea is central to the

romantic aesthetic of pure music, which Goehr (2007) has argued is central to thedevelopment of the regulative work-concept in western classical musical aesthetics. Inthe memorable words of Friedrich von Schelling, music is the art that is least limited

by physical considerations in that it represents pure motion as such, liberated fromany other object and borne on invisible, almost spiritual wings (as cited in Goehr,

2007, p. 156), leading to Gustav Schillings conclusion that [n]o aesthetic material isbetter suited to the expression of the ineffable than is sound.4

If we take the metaphor even further, one can read Pythagoras as providing amodel of the composer as prophet-scientist, acting as a conduit of universal ideas

that can be expressed and understood by all. The laws of sounding bodies and ofgeometric arrangement existed before Pythagoras; they are eternal. He simply

articulated what was already there, literally evoking them in rational arrangementof words and tones so that they would be intelligible and usable to others. Theimpulse to devise similarly universal formulations of music, and to describe or

exemplify them in both words and sounds outlines a veritable radix Jesse throughmusics theoretical history, as Pythagoras begat Nicomachus and Plato, who begat

Boethius, who begat Descartes, who begat Helmholtz, who begat Schoenberg.Moreover, the creator-as-conduit model wields a significant influence on one of

the dominant models of contemporary classical performance practice, whichemphasizes fidelity to the composers intentions in a variety of guises. Goehr

(2007) has demonstrated how this concept of Werktreue developed in conjunctionwith, and as part of, the nineteenth-century art-music-concert culture, while at thesame time informing views of music written before and after.5 The archetypal

form of the position might be that articulated by Donald J. Grout in his essay OnHistorical Authenticity in the Performance of Old Music (1957), by which an

ideal performance is one that perfectly realizes the composers intentions(p. 341).6

That is all fine and good. However, the irony is that the closer one looks for thehistorical Pythagoras, the less comfortable one feels about any of the above. It is

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impossible to study the man himself apart from the mythology that grew up around

him. As Christoph Riedweg (2005) notes:

Very little of our information about him [Pythagoras] dates from his ownlifetimeroughly, from 570 to 480 B.C.E. The farther on in time we move, thericher the documentation becomes for us (in antiquity, the situation was stillsomewhat different). The only coherent descriptions of Pythagoras life andteaching that have come down to us from antiquity we owe to authors of the thirdand fourth centuries C.E. . . . However, since by the fourth century B.C.E. variousgroups were already claiming the idealized philosopher Pythagoras as one of theirown, considerable caution is needed in dealing with these witnesses (whocontradict one another on many points). (pp. ixx)

Some of the most important accounts of his ideas are in instructional works andpolemics by later theorists, most prominently Boethius, but also including Plato,

Aristotle and Euclid, which have their own well-known problems as historical texts.The process of studying works for performance creates a phantom, or at leastchimeric, composer concept which stands in uncertain relation to the original and

which speaks of the present as much as, if not more than, of the past. This has beenfelt perhaps most strongly in the arena of historically informed performance, a term

which rose to fill the void left by the fall from grace of authenticity. Performance ofearly music, as Daniel Leech-Wilkinson (2002) notes, absolutely requires that one

believe both incompatible things together: we try to do it the way it was, and we knowwe cannot, and we argue for both (p. 55). The situation has become no less

ridiculous even after the invention of the phonograph, where one finds commentssuch as that of Richard Taruskin (1995), that many who have made the comparison

will tell you that Gary Graffmans Prokofiev, for example, sounds more like Prokofievthan Prokofievs (ditto Sandors Bartok or Tacchinos Poulenc) (p. 105).

Although the debates of the 1980s and 1990s have made such open public

declarations of allegiance unpopular (see Fabian, 2001, pp. 5354), the ideas tenacitycan be seen in the near ubiquity of programme notes at classical concerts, which today

are equally likely to be found at performances of medieval and Renaissance music as atthe Romantic symphonic concerts from which they emerged and the musically reified

modern music concerts which they influenced. Such notes, even when written by thecomposers themselves and not by a suitably Delphic scholar, too often serve as little

more than invitations to a seance, encouraging the audience to commune with animagined past, with the musicians as the mediums and the composers as their spiritguides, even if the author of the music is sitting only a few rows away.

Focusing on the paradoxes and inconsistencies (dare I say impossibilities?) inmusic can seem untenable, with something having to give. American composer

Milton Babbitt has over his long career advocated for many such changes, mostnotoriously in his argument against the hegemony of the general audience in the

High Fidelity article printed (against its authors wishes) under the title Who Cares ifYou Listen? For issues relating to Werktreue, the dispensable element may just as

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equally feasibly be the performer as the audience. With the advent of recorded and

synthesized sound, and especially since its increased accessibility from the 1950sonward, composers (including Babbitt) composed music which did away with

performers and created works that existed essentially only as substrates for infalliblemechanical reproduction, on tape (in its wider, colloquial sense). In a 1985 interview

(Swartz & Babbitt, 1985), Milton Babbitt said that I probably would, if I could, restrictmyself to the purely electronic medium, because of the amount of control over themusical results. Babbitt saw value in the fact that [w]hen I am through, the sound goes

onto the tape, and I record it, on tape, directly from the synthesizer. I also hear it onspeakers as I am creating it. And when I walk into that studio with my composition in

my hand, I eventually walk out with the finished performance and a tape under myarm (p. 468). Babbitt reiterated and expanded upon the sentiment in 2000, saying,

When youre walking into the studio up at 125th Street (Columbia) with a piece inyour head and eventually, youre able to walk out with a finished piece and a tape under

your arm, its an unparalleled experience. Youre not dependent on anyone butyourself. Thats a slight exaggeration because you get into the hall and youre dependenton the speakers and the amplifier . . . In the most serious sense, its a matter of just

being the master of everything. Your decisions are your decisions (Gross & Babbitt,2000, paragraphs 4 and 5). Yet Babbitt did not stop writing for conventional

instruments, and his protestations that I simply dont have the time to spend at thestudio (Swartz & Babbitt, 1985, p. 368) seem to fall apart given the twin trajectories of

electronic music and the direction of his compositional energies.7 Nor indeed didavant-garde composers such as John Cage stop writing music attended by the same

audiences as those of the musical establishment. Through his advocacy of the giving upof music and of the desire to control sound (Cage, 1962, pp. 8182), in arguing that

music is everything and not just in concert halls, John Cage has, in the eyes of somesuch as Goehr (2007, p. 261), seemed to purposefully argue himself out of a careerwhich he strives to maintain, while Violinist Jascha Heifetz informed his public that he

occasionally performed works by contemporary composers . . . to discourage thecomposers from writing any more (as cited in Goehr, 2007, p. 251).8 The result is an

odd conspiracy of silence. One is reminded again of the ancient Pythagoreans, who,according to legend, killed (or forced to kill himself) the man who demonstrated that

lurking within their metaphysically pregnant geometric shapes were qualities incapableof being wholly described in ratio.9 The simplest example of this is found in Pythagoras

beloved right-angled triangles, where, when two sides have a measure of exactly 1 unit,whatever that unit may be, the triangle will have a hypotenuse of the square root of 2. Inother words, no matter what size or how finely graded the ruler is used to measure it,

the length will fall between two of the demarcations. Incommensurability is aninevitable product of rational construction, and no amount of wishing otherwise (or of

shouting at naysayers of any camp) will make it so.Of course, one reason the story of the Pythagorean response to irrationality strikes

us as perverse is that we now have an intuitive understanding of the fact thatirrationality and incommensurability at one level do not preclude rationality and

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commensurability at another. Obviously, the square root of two squared is two. You

can even hold irrationality in your hand. Draw a square of 2 units on a piece of paper,and join the midpoints of perpendicular sides. Then cut that square out. That square

has sides that are mathematically impossible to measure precisely, yet it is possible touse it to build larger structures. Analogous results are achieved in music by making

irrationality/irreducibility (or its sister in this metaphor, lack of direct compositionalcontrol) a compositional tool. Whether through the conundrums of the unplayablemusic of New Complexity composers such as Ferneyhough, or through the aleatorics

of composers as different as Luciano Berio and Barry Guy, irrational processesindependent of compositional control are used to create a product understood and

predicted in broad outline by its maker. Such music is at some level indeed irrational,incommensurable, and perhaps indeed impossible to define, yet we, like Justice Potter

Stewart, nevertheless know it when [we] see it.

Notes

[1] As Goehr (2007, pp. 101106) points out, much music of interest of varying antiquity can be,and has been, written with this concept serving as a dominating principle, and throughout thisessay, unless otherwise noted, it should be assumed that I am talking about such music.

[2] Accounts attributing the theorem to Pythagoras are found in Diogenes Laertius, Porphyry ofTyre, and Iamblichus of Chalcis, among others. See Riedweg (2005), pp. 2627. An engagingand accessible introduction to the history of the theorem and its proof can be found in Veljan(2000), pp. 259272.

[3] Goehr (2007, p. 158) defines this illusion as the view that an object, a person, or an experience[has the ability] to exhibit simultaneously the character of the human and the divine, of theconcrete and of the transcendent.

[4] From Schilling, Romantik und Romantisch, Universal Lexikon der Tonkunst (Stuttgart, 18341838), 7 vols., vol. 6, 3437; translated in Le Huray and Day (1988), pp. 319322, at 320.

[5] This is indeed a key part of her thesis, recurring throughout the book, but perhaps articulatedmost clearly in Chapter 4.

[6] Richard Taruskin (1995, p. 92) observed that the title of this piece seems to contain one ofthe earliest uses of the word authenticity in the sense that has since become so widespreadand cultish, and which violently roiled the scholarly waters of the discussion of particularlypre-Romantic music for decades. While the debate in early music has cooled in recent years,similar questions (and terminology) regarding recoverability of a historical past continue tosurface in other arenas concerned with heritage, such as popular music and even wholecultural edifices and ecosystems, ranging from the Parthenon to Cuban coffee plantations.For more illustrative examples, see Parker (2008) and Rossler (2008). Rosslers articleappeared in an issue of the journal of the Association for Preservation TechnologyInternational (Pamela Jerome, guest editor) dedicated entirely to questions of authenticity incultural preservation.

[7] In spite of the dramatic proliferation of tools for the electronic manipulation of sound (at alllevels of time and difficulty), only one of Babbitts 26 works published after 1980 listed in theNew Grove has required electronics. See Barkin & Brody, n.d., Works.

[8] The quote Goehr is referring to, which was subsequently anthologized by Crofton & Fraser(1989, p. 10), originally appeared without context in the Voices section of a 1961 issue of Lifemagazine. See Babbitt (1961), p. 33.

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[9] Though, as with so much else pertaining to Pythagoras and the Pythagoreans, accounts areinconsistent. In a commentary of Book 10 of Euclids Elements, Pappus of Alexandria saidthat the discovery of irrational numbers originated in the Pythagorean school and thediscoverer perished by drowning. Iamblichus Vita Pythagorica, on the other hand, tells anumber of different stories whereby the Pythagorean who articulated the nature of theirrational (whether in the isosceles right triangle or in the construction of the dodecahedronfrom a circle) was drowned as a godless blasphemer. This Pythagorean heretic is most oftenidentified as Hippasus of Metapontum (early fifth century BCE). See also Riedweg (2005),pp. 26 and 109110.

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