Pythagorean Longings and Cosmic Symphonies:

The Musical Rhetoric of String Theory and the Sonification of Particle Physics

Peter Pesic

Axel Volmar



In vain does the God of War growl, snarl, roar, and try to interrupt with bombards, trumpets, and his whole tarantantaran.... Let us despise the barbaric neighing which echo through these noble lands and awaken our understanding and longing for the harmonies.


Thus Johannes Kepler expressed in 1620 his deep-felt commitment to the search for Pythagorean harmonies in the planetary spheres above his war-torn Earth, exhorting his readers to join him in the quest. By the time he wrote, that quest had already moved philosophers for two millennia, as it continues to do to this day, a story which goes far beyond the confines of this paper. Here, we will address the “sounds of space” in the search for cosmic harmonies in contemporary physics, specifically in string theory and the sonification of data. Though the focus of this paper will be on the second half of the twentieth century, the nature of the developments we consider require that we contextualize them in the longue durée that stretches back into antiquity and forward to the present day. First conceived in the late 1960s, string theory was increasingly presented with explicit reference to the classic Pythagorean themes, especially in more popular treatments. What exactly does this Pythagorean apotheosis mean and why did it seem so compelling to those who wrote and read it? Why, toward the end of the twentieth century, should those ambitious to promote a cutting-edge “theory of everything” have turned to millennia-old Pythagorean musical imagery rather than more up-to-date “scientific” descriptions and concepts that would seem more congruent to a modern theory? Why should recourse to (perhaps inaudible) cosmic music justify a recondite model of theoretical physics?

For instance, Leonard Susskind, a physicist active in the development of string theory, has critiqued “the myth of uniqueness and elegance” that he traces back to “a mystical mathematical harmony of the universe” he ascribes to Pythagoras and Euclid: “While the connection between music and physics may seem to us naive and even silly, it’s not hard to see in the Pythagorean creed the same love of symmetry and simplicity that so motivates modern physicists” (Susskind 2005: 118–119). His acknowledgment of the “Pythagorean creed” at work in modern physics at the same time raises questions about what he considers its “naive and even silly” quality.

We particularly wish to raise this “Pythagorean paradox” as a significant problem that deserves larger attention; as well-known as the theme of the “music of the spheres” may be, it has drawn surprisingly little interpretation or analysis that goes beyond observing how wide-spread and long-lasting has been its presence in Western cultural history. Even after World War II, physics re-engaged archaic Pythagorean imagery in the service of new theoretical initiatives; at the same time, electronic and film music created new “sounds of space” that contrasted with the “cosmic sublime” invoked by Beethoven, Stravinsky, and Schoenberg. So far, though, the sonifications of high energy physics data sound much more like electronic or computer music than the Romantic or Pythagorean sublime.

The Pythagorean project and its modern exponents


Already by the time of Plato, Pythagoras had become a semi-mythical figure whose history could scarcely be untangled from the mass of stories that grew around him (Martínez 2012). Plato’s Timaeus gave enduring form to the most salient elements of that mythos, namely that both the cosmos and soul are “made out of music,” structured according to the simple ratios that corresponded to the primal musical intervals: octave (2:1), perfect fifth (3:2), perfect fourth (4:3). This was transmitted to the West by Boethius as an essential part of the quadrivium that included arithmetic and geometry along with music and astronomy as the essential content of higher “liberal” education: the same ratios that governed musical consonances were fundamental to astronomy. As music linked these ratios to perceptible sounds, it helped the “new philosophy” of the seventeenth century to bridge physical, changeable phenomena (physis) with mathematical structures, thus enabling the formative steps taken by Descartes and Galileo.


As many aspects of Pythagorean lore were absorbed into Neoplatonic thought, a certain religious undercurrent remained potent. For instance, consider the famous image of the world-monochord published by the English Neoplatonist Robert Fludd (Figure 1a), showing the hand of God tuning the cosmic string. Even though Marin Mersenne criticized Fludd’s mystic numerology and geocentric cosmology, Mersenne reproduced this same image almost identically in his Harmonie Universelle (1637, Figure 1b), next to his arguments for Copernican cosmology. Evidently, the appeal to God as world-musician moved Mersenne more than his many scientific disagreements with Fludd. More generally, the notion of “harmony” remained a powerful aesthetic criterion invoked by mathematicians and scientists to justify Copernican cosmology; Isaac Newton remarked that “Pythagoras’s Musick of the Spheres was gravity” (Westfall 1980: 510n136).


This generalized concept of harmony remained potent into the nineteenth and twentieth centuries, even to the present day. For example, Henri Poincaré argued that “the aim of mathematical physics” was “to reveal the hidden harmony of things,” so that “objective reality [...] can only be the harmony expressed by mathematical laws” (Gray 2013: 75). Albert Einstein expressed his belief “in Spinoza’s God, Who reveals Himself in the lawful harmony of the world” (Einstein 2011: 325). This quest for cosmic harmony, the “cosmic religious experience,” he considers “the strongest and the noblest driving force behind scientific research” (Einstein 2011: 330). Such expressions could be multiplied many times over, not just in Einstein’s writings; indeed, the eminent physicist Frank Wilczek and his co-author Betsy Devine used the epigraph above from Kepler as the dedicatory inscription for their book Longing for the Harmonies: Themes and Varations from Modern Physics (Wilczek and Devine 1988). To be sure, Wilczek’s book does not treat string theory, but seeks the “harmonies” to be found in many other realms of physics, classical and modern, giving evidence of the broad currency of Pythagorean imagery in contemporary writing about physics. Still, the remarkable outpouring of writings about string theory take up these Pythagorean themes with particular intensity.

Figure 1a: Robert Fludd’s diagram of the cosmic monochord, from Utriusque cosmi.... historia (1617). 

Figure 1b: Marin Mersenne’s similar diagram from Harmonie Universelle (1637).

String theory and its original expositors


After World War II, high energy accelerators gradually revealed a new world of subnuclear particles, raising the questions why they were so numerous and what might connect them. During the 1970s, the so-called Standard Model of particle physics provided a synthetic theory that accounted for these particles’ strong, weak, and electromagnetic interactions. String theory is an unproven, yet intriguing, attempt to unify these three forces along with the force of gravity within a single mathematical framework.


In the late 1960s, a number of developments led physicists to consider that certain formulas for particle scattering could be interpreted in terms of elementary strings, rather than particles. Pursuing studies of the so-called dual resonance model, in 1968 Gabriele Veneziano formulated an expression for high-energy particle scattering that used a mathematical function devised centuries before by Leonhard Euler for completely different purposes. The elegance and simplicity of Veneziano’s amplitude formula struck many physicists as a promising new key to understanding the strong interactions it described. Trying to interpret that amplitude, Yoichiro Nambu, Holger Nielsen, and Leonard Susskind in 1969–70 independently suggested that the dual resonance mode described the behavior of relativistic strings (Nambu 2012; Nielsen 2012; Susskind 2012; Susskind 2005: 204–206). Yet their initial papers and subsequent recollections do not use the explicit Pythagorean language of musical ratios and world-harmony, referring only to the mathematical expressions for the energy levels of a simple harmonic oscillator in quantum theory, which itself takes off from the classical description of the vibrational modes of a string.


String theory involves hypothetical massless strings, extremely small and moving at relativistic velocities, which cannot be simply equated to the physical strings we know from ordinary experience. Thus, the discoverers of string theory really worked with an analogy several times removed from the sonic experiences invoked by the Pythagorean myths: an analogy of an analogy of an analogy (Pesic 2014: 279–280). Though one might well ask to what extent this tower of analogies really reflects the “original” Pythagorean story, however distantly, it surely does not invoke it directly. To trace the growth of such more explicit Pythagorean imagery, we need to consider the growing body of writings that accumulated around the newborn theory.


Throughout the 1970s, string theory advanced the claim that the various subnuclear particles were really different modes of vibration of extremely small “strings.” [1] The theory remained questionable for the majority of theoretical physicists until in 1984 Michael Green and John Schwarz achieved a major breakthrough, sometimes called the “first string revolution.” In its aftermath, string theory became more and more popular, especially among younger physicists. Consequently, an increasing number of textbooks on string theory were published.


[1] Thus, nucleons (the proton or neutron) have a characteristic size of about 10–15 m, whereas the proposed strings exist at the Planck length scale, about 10–35 m.


This generalized concept of harmony remained potent into the nineteenth and twentieth centuries, even to the present day. For example, Henri Poincaré argued that “the aim of mathematical physics” was “to reveal the hidden harmony of things,” so that “objective reality [...] can only be the harmony expressed by mathematical laws” (Gray 2013: 75). Albert Einstein expressed his belief “in Spinoza’s God, Who reveals Himself in the lawful harmony of the world” (Einstein 2011: 325). This quest for cosmic harmony, the “cosmic religious experience,” he considers “the strongest and the noblest driving force behind scientific research” (Einstein 2011: 330). Such expressions could be multiplied many times over, not just in Einstein’s writings; indeed, the eminent physicist Frank Wilczek and his co-author Betsy Devine used the epigraph above from Kepler as the dedicatory inscription for their book Longing for the Harmonies: Themes and Varations from Modern Physics (Wilczek and Devine 1988). To be sure, Wilczek’s book does not treat string theory, but seeks the “harmonies” to be found in many other realms of physics, classical and modern, giving evidence of the broad currency of Pythagorean imagery in contemporary writing about physics. Still, the remarkable outpouring of writings about string theory take up these Pythagorean themes with particular intensity.

Figure 2: An early illustration from a popularization (Parker 1987: 249), making the comparison with violin strings, whose “loops” illustrate the modes of vibration of strings.

Musical metaphors and subtexts in string theory


In order to communicate complex mathematical structures to their peers, and especially to wider audiences, physicists frequently make use of analogies, metaphors, and models to establish a common ground of understanding and negotiation. This holds true particularly for introductions and popular accounts on the subject matter, some of which explicitly turned to musical metaphors in order to promote the concept of strings as the “building blocks” of nature.


One of the first popular books that introduced string theory to a wider audience, Barry Parker’s Search for a Supertheory: From Atoms to Superstrings (Parker 1987) placed string theory within the larger history of modern physics. To elucidate the initial work of Nambu, Nielsen, and Susskind, Parker explicitly instanced a violin string:



The best way to think of these vibrational states is to picture them as occurring on a violin string. As you no doubt know, a violin string can vibrate with one, two, or more loops along its length [see Figure 2]. Nambu and his colleagues noticed that these vibrational modes could be related to the hadrons. One of the hadrons, for example, corresponded to a string with a single loop, another to one with two loops, and so on. This is not so strange if you stop for a moment and think about it. Particles in quantum mechanics are described by similar vibrational states. In the case of Nambu’s string theory, though, the strings were very special. They had no mass, were elastic, and their ends moved with the velocity of light. But if they had no mass and represented massive particles, where did the particles’ mass come from? This was taken care of by the tension of the string: the greater the tension, the greater the mass (Parker 1987: 249).

In Parker’s account, the analogy of the musical string functions merely as a pedagogical aid for lay readers. The same year, however, string theorist Michio Kaku used the analogy between a superstring and a violin string to open up larger historical contexts of music as a productive metaphor in cosmology. [2] In their popular book Beyond Einstein: Superstrings and the Quest for the Final Theory (Kaku and Thompson: 1987), Kaku and the journalist Jennifer Thompson applied the string metaphor not only to explain the concept and the behavior of individual superstrings but also to emphasize the potential of string theory as a unified “theory of everything:” 


The superstring theory can produce a coherent and all-inclusive picture of nature similar to the way a violin string can be used to unite all the musical tones and rules of harmony […] Knowing the physics of a violin string, therefore, gives us a comprehensive theory of musical tones and allows us to predict new harmonies and chords. Similarly, in the superstring theory, the fundamental forces and various particles found in nature are nothing more than different modes of vibrating strings. (Kaku and Thompson 1987: 5)


This emphasis on “harmony” evokes reminiscences of the Pythagorean dream of unification. In his 1988 Introduction to Superstrings, Kaku reinforced this link by suggesting a modernized version of the ancient “music of the spheres” in which the unheard cosmic music is not conceptualized in terms of ratios between celestial objects, such as planets, but as the result of the various vibrational modes (or resonant patterns) of the tiny but highly energetic strings:


Superstring theory, crudely speaking, unites the various forces and particles in the same way that a violin string provides a unifying description of the musical tones […] In much the same way, the superstring provides a unifying description of elementary particles and forces. In fact, the “music” created by the superstring is the forces and particles of Nature. (Kaku 1988: 17)


In his best-selling book The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory (Greene 1999), Brian Greene stressed musical metaphors even further by calling the essentials of superstring theory “nothing but music” and by entitling a section of his book “The Cosmic Symphony” as “orchestrated” by strings:


Music has long since provided the metaphors of choice for those puzzling over questions of cosmic concern. From the ancient Pythagorean “music of the spheres” to the “harmonies of nature” that have guided inquiry through the ages, we have collectively sought the song of nature in the gentle wanderings of celestial bodies and the riotous fulminations of subatomic particles. With the discovery of superstring theory, musical metaphors take on a startling reality, for the theory suggests that the microscopic landscape is suffused with tiny strings whose vibrational patterns orchestrate the evolution of the cosmos. (Greene 1999: 135)


In suggesting that musical metaphors take on a “startling reality” according to string theory, Greene does not use these metaphors merely to convey the concept of superstrings and its potential to provide a unified description of nature for an audience of non-experts. His “we” includes general readers along with professional string theorists such as himself. In this collective manifesto, Greene makes explicit the Pythagorean and musical context in which he wishes his metaphors to be understood as valid descriptions of the world at the subatomic level. Also, the strong imagery of the “cosmic symphony” (the title of the third section of his book) in which strings “orchestrate the evolution of the cosmos” (Greene 1999: 135) does not add much explanation to a deeper understanding of the theory but rather capitalizes on a rhetorical appeal to the scientific sublime.


[2] Superstrings incorporated additional ideas of “supersymmetry,” a hypothetical symmetry connecting fermions (such as spin 1/2 particles like electrons or protons) and bosons (such as spin 0 or 1 particles like the pion, photon, or W particle).

The music of the starry sky


These allusions to the sublime draw on larger currents of the eerie sublimity of “space,” such as those considered elsewhere in this issue (Pinch 2014). To understand their late-twentieth-century manifestations, we need to consider the Romantic matrix from which they emerged. Our examples will consider musical evocations of cosmic soundscapes from the nineteenth century forward. Immanuel Kant had famously evoked the sublimity of “the starry sky above us,” to which Beethoven gave sonic expression at the hushed center of the choral finale to his Ninth Symphony. Indeed, Beethoven copied these very words from Kant into his conversation book (Kinderman 1985: 102). True to Rudolf Otto’s classic description of the feeling of holiness as mysterium tremendum experienced in the “still small voice” from the whirlwind, Beethoven has us encounter the godhead in a moment of great quiet, yet profound (though subdued) excitement (Otto 1973). In his symphonic apotheosis, just after “the millions” have fallen to their knees before the “loving Father,” Beethoven allows us to hear their mystic chorus of recognition: “He must dwell beyond the stars” (“Über Sternen muss er wohnen,” Figure 3; SoundObject 1).

The felt effect of this passage stems not just from the amazing beauty and transcendent ordering of the visible stars (as Kant indicates) but from a new phenomenology of awe and wonder. Pianissimo tremolos in the strings subtly tug against a different triplet rhythm in the winds, evoking the twinkling starlight; above all, the voices join the orchestra in a slowly unfolding, shimmering dominant ninth chord that synthesizes the extremes of dissonance (A-C#-E-G-B-flat, including pitches a semitone apart, A and B-flat) that nonetheless point to an unheard, but unequivocal, tonic (D). As he “sonifies” Kant, Beethoven adds the the eerie stillness and cosmis distance of the stars a subtle vibration that indicates the human frisson in the face of the infinite (Treitler 1982: 168-169).

Figure 3 and SoundObject 1: Ludwig von Beethoven, Symphony no. 9 in D minor, fourth movement  (measures 647–654), from  8’10”–8’44”.

In this passage, Beethoven was perhaps the first composer to give sonic expression to the Pascalian awe of infinite space, using sound to instill a kind of direct mystical experience in ways no word or visible image could convey. Indeed, Beethoven indicates the crucial importance of this moment by its placement in his symphonic design, as the essential locus of exalted rapture whose mysterious power ignites the ensuing Dionysiac praise of Joy in the remainder of the final movement. Yet the still center of this work is not the “loving Father,” nor even the stars beyond which He dwells, but the eerie strains of space itself, for the listener perceives more the sense of infinite distance and strangeness than any kind of distinct object: a mysterious twinkling without visible stars. Paradoxically, this moment uses harmony to evoke an unheard harmony, the tonic implied (though not stated) as the virtual but invisible pole around which the dominant ninth chord revolves.



As influential as this moment was for Beethoven’s successors, for our purposes here we consider its resonance for a composer notably critical of Beethoven: Stravinsky. At the heart of his Sacre du printemps (1913), in the opening of its second, climactic part, Stravinsky also creates a parallel moment of eerie cosmic harmonies, which dramatically invokes the night sky under which the human sacrifice proceeds (Figure 4; SoundObject 2). Though surely his aesthetic is far from Beethoven’s, Stravinsky chooses comparable harmonic and textural means to evoke the vast emptiness of space rather than any comforting encounter with the stars. The overarching harmony of D minor (strangely the same key as Beethoven’s Ninth Symphony) vibrates against conflicting dissonances bracketing it by semitones (sixth chords on D# and C#). As with Beethoven, the moment is hushed and awe-struck, pianissimo; the orchestration uses unfamiliar and unearthly combinations of low pedal-tones in the horns, undulating figures in the winds, and the spectral whistling of string harmonics (which are in fact direct demonstrations of Pythagorean ratios).

Figure 4 and SoundObject 2: Igor Stravinsky, Le sacre du printemps, opening of Second Part (rehearsal number 79–84), from 0'20"–2'18".


The scene is cold and inhuman, though exalted; the sense of its sublimity (as Kant had realized) depends on deep-lying alienation and even fear on the part of the awestruck hearer. Even more than in Beethoven, Stravinsky here evokes a sense of space as infinite, more-than-human, a godhead to which human sacrifice is offered to bring the spring. As Pascal thought, “the eternal silence of these infinite spaces fills me with dread [m’effraie]” (Pascal 1995: 66) and also with a new kind of wonder. For his part, Arnold Schoenberg took up this sense of wonder in his setting of Stefan George’s poem “Entrückung” as the final movement of his String Quartet No. 2 in F# minor, op. 10 (1908); the inclusion of a human voice surely looks back to Beethoven’s choral finale. George’s text “Ich fühle Luft von anderem Planeten (I feel the air of another planet)” explicitly evokes space as the “air of another planet” that engulfs the singer in “unfathomable thanks and unnamed praise” (Figure 5; SoundObject 3).

Figure 5 and SoundObject 3: Arnold Schoenberg, String Quartet No. 2, op. 10, fourth movement (one bar after rehearsal number 20), from 2'36"–3'00".

Schoenberg’s hushed and eerie strains, mostly pianissimo, eventually lead to a mystic apotheosis: “Ich bin ein Funcke nur vom heiligen Feuer / Ich bin ein Dröhnen nur der heiligen Stimme (I am only a spark of the holy fire / I am only a whisper of the holy voice).” This blend of self-abnegation and cosmic identification distills “the air of other planets” as it passes through human sensibility via music.

VideoObject 3: What is Superstring Theory? Talk Nerdy To Me, HuffPost Science 2012, 00’06” to 01’38”.

Hearing the “cosmic symphony”


Whether or not the works just considered consciously informed the sonic metaphors of string theorists, their “cosmic symphony” can usefully be situated within that larger music-historical context. By rendering the universe a sort of divine “masterpiece” and thereby linking string theory to the Genieästhetik (aesthetics of genius) of Western classical music, Greene simultaneously ennobles the assertions of string theorists as comparable to works of art. Yet a large part of Greene’s rhetorical success depends on conveying that he and other string theorists find this cosmic symphony a compelling theme, both in its musical and philosophical senses.


After the success of Greene’s book, authors of popular writings on string theory have increasingly utilized musical metaphors. One of the newer accounts, George Musser’s The Complete Idiot’s Guide to String Theory, goes so far as proclaiming that the “music of the strings” represents “the ultimate symphony” (Musser 2008: 3). In recent documentaries and web videos on string theory, the aesthetics of the scientific sublime constitutes a common feature usually accomplished by the use of spectacular visual effects, computer animations, and sound effects as well as suggestive metaphors. For instance, in The Elegant Universe, a TV documentary on Greene’s book, a solo cellist performed the prelude from the G major suite by Johann Sebastian Bach (BWV 1007) while the narrator (Greene) described strings as performing together the “grand and beautiful symphony that is our universe.” Here the aura conveyed by sonic performance contributes largely to authorizing Greene's extravagant musical rhetoric:

According to a video by Kaku on the popular YouTube education channel Big Think, the “universe is a symphony of vibrating strings,” so that even human beings are “nothing but melodies, nothing but cosmic music played out on vibrating strings and membranes,” implying that his viewers are themselves part of the cosmic symphony:


Today the musical string narrative is commonplace, as in this video clip from The Huffington Post on their YouTube channel HuffPost Science, in which a physicist compares superstrings to a guitar string:

VideoObject 1: The Elegant Universe, Part 3: Welcome to the Eleventh Dimension, aired in July 2012 on PBS, 16’06” to 17’27”. Strings are illustrated by a cello and accompanied by synthesizer sound effects.

VideoObject 2: Michio Kaku, The Universe Is a Symphony of Vibrating Strings, from Big Think 2011, 00’24” to 01’27”.

Selling the strings


As these examples demonstrate, some string physicists go far in trying to “advertise” or “sell” their theory. In this arena, physicists like Greene and Kaku take advantage of popular media; their rhetorical strategies seem to be their own, not imposed by media producers or science writers. Indeed, these and other high-profile scientists often have become their own producers and writers, in effect. The majority of textbooks and technical introductions to string theory, however, do not use musical metaphors at all or only allude to them in passing—usually only in the opening passages of the book, which tend to offer a motivational overview encouraging the reader to persevere through difficult mathematics (Zwiebach 2009: 7 and 10). But why exactly this sustained rhetorical campaign using musical metaphors?


One practical, if not cynical, answer involves plain economic factors and the fear of budget cuts. After the thorough confirmation of the Standard Model, particle physics in the 1980s entered a kind of waiting period, searching in vain for clues that would lead to an even more general “grand unified theory” or “Theory of Everything,” as it was termed. Increasingly, a large part of the theoretical community gravitated toward string theory as “the only game in town,” the most promising source of new ideas that would justify the ever-more-expensive experimental programs, such as the Large Hadron Collider (LHC) at CERN that finally (in 2012) confirmed the existence of the last remaining particle of the Standard Model, the Higgs particle (Woit 2006: 221–236). These recent successes of particle physics, however, were preceded by notable reverses in US high-energy physics. Though the storage ring ISABELLE was begun at Brookhaven National Laboratory to detect the W and Z bosons of the Standard Model, before it was completed, physicists at CERN had already found both bosons. Two years later, the American project was cancelled. Out of these experiences, plans for a Superconducting Super Collider (SSC) in Texas were conceived in the early 1980s. After $2 billion had already been dispensed for buildings and almost 15 miles of tunnels, the US Congress cut the funds in 1993 (Mason 2009).

“Selling the strings” thus formed part of a larger initiative to maintain the prestige and urgency of high-energy physics (both theoretical and experimental) in increasingly adverse times, especially in the US, during which their larger audience at times felt growing alienation and impatience with these grand (and expensive) projects. For the public understanding of science, musical metaphors seemed to offer opportunities to provide the imagination of the lay public with affective experiences of abstract mathematical theories. Furthermore, string theorists faced considerable opposition even among other physicists (not to speak of scientists in quite other fields) who not only competed with them for forever scarcer funding but also questioned the grandiose assertions on which string theory tried to make its case. These economic issues form the background against which, whether consciously or not, Kaku projected himself as a white-maned Einstein redivivus and Greene a hip yet ever-articulate purveyor of the wonders of string theory and its “elegant universe.” For both, as we have seen, the Pythagorean project provided ideal material. Returning now to our opening question about the “Pythagorean paradox,” why should exhuming hoary stories of world-harmony hold any claim to the attention of a modern, disenchanted, even cynical audience, all too used to advertising?

Re-enchanting the world through music


However skeptical or even suspicious of the claims of science the public may be, the disenchantment (Entzauberung) of the modern world famously diagnosed by Max Weber is surprisingly superficial (Weber 1964: 270). Under the surface acknowledgment of the mechanization of the world, many (including the physicists themselves) harbor “longings for the harmonies” that suggest barely veiled religious feelings not far underneath the surface of rationality. Recall that Einstein’s “cosmic religiosity” was expressed in a search for world harmony. Indeed, some modern scholars view Pythagoras himself to have been primarily a religious leader, the founder of a secretive brotherhood that venerated number and harmony, not merely studied them dispassionately (Burkert 1972; Martínez 2012).


In his polemic for string theory, Kaku unabashedly enlists religious feeling: “The mind of God we believe is cosmic music, the music of strings resonating through 11 dimensional hyperspace. That is the mind of God” (Kaku 2014, see also VideoObject 2). Kaku’s startling assertion seems to be directed to the public’s barely suppressed desire for the re-enchantment of the world, if not a wholesale return to magical and traditional religious views. In more subtle ways, a striking number of scientists seem to have profound sympathies with archaic Pythagorean longings, which indeed had accompanied and shaped the development of science from ancient through modern times (Pesic 2014). That quest at every point confronts a world that seems in many ways not unified but fractured, disunified. The deep underlying principles of unity must be sought behind a facade of broken symmetries, discrepant near-equalities. Thus, modern science seeks the laws of nature as essentially hidden, not manifest; since the sixteenth century, science has considered its task to be finding those hidden laws through careful scrutiny and experimental trial of the manifest surface of phenomena and the scientific data thereby elicited (Pesic 2000).

As has been emphasized by its critics, string theory as yet lacks any contact with experimental confirmation, either at present or in the near future. With his usual directness, Richard Feynman asserted that “string theorists don’t make predictions, they make excuses” (Woit 2006: 175). Worse still, there remains deep controversy about what exactly string theory is, especially how it should be interpreted in terms of observable phenomena. Edward Witten, one of the most celebrated figures associated with the development of string theory, observed in 1983  that “what is really unsatisfactory at the moment about string theory is that it isn’t yet a theory,” a problem that remains to this day (Woit 2006: 175). String theory (and its proposed generalizations to superstring theory and M-theory) remains a collection of intriguing fragments, an enticing “hunch” (as Gerard ’t Hooft put it) rather than a fully formed theory comparable to the Standard Model of particle physics (Woit 2006: 176). As such, even within technical circles, arguments for string theory tend to stress its “beautiful” or “elegant” features, which can best be expressed to a general audience in terms of Pythagorean tropes of harmony.

Beauty leads the way


This longing also points to a certain belief in aesthetics, even a stylized aestheticism in science, such as Paul Dirac invoked as a criterion for new initiatives in theoretical physics: “it is more important to have beauty in one’s equations than to have them fit experiment,” as with his own eponymous equation that, through symmetry, predicted the existence of the antielectron (Dirac 1963). But what constitutes beauty in the exact sciences and especially in physics? In the Pythagorean world, as we have seen, beauty equals harmony. In modern physics, however, the concept of harmony has largely been substituted by the aesthetic principle of symmetry. Kaku, for instance, makes direct connections between the two in order to explain the meaning of beauty in physics:


To a musician, beauty might be a harmonious, symphonic piece that can stir up great passion. To an artist, beauty might be a painting that captures the essence of scene from nature or symbolizes a romantic concept. To a physicist, however, beauty means symmetry. (Kaku and Thompson 1987: 99)


Thus, proponents of superstring theory, such as Greene, emphasize what they consider the “elegance” of its equations as one of the most attractive features of string theory as an avenue of research, despite the lack of experimental confirmation. In this respect, mathematical beauty represents a factor internal to scientific research, not solely an external criterion that might justify a theory to the wider public. In this view, string theorists may clearly be understood as the heirs of Dirac’s longing and especially of his mathematical aestheticism. Those who follow Dirac feel that elegance and beauty represent signposts to truth, if indeed equations might describe the complexities of nature via symmetry principles yielding aesthetic experiences comparable to those afforded by music or art. As Kaku notes, these symmetry principles are the correlates to harmony in the sonic realm; the interplay between musical consonance and dissonance translates into the various symmetries (perfect or broken) embodied in the equations of physical theory.

Articulating scientific practice and self-conception


In popular writings, however, scientists do not only intend to convey scientific truths to larger audiences, as discussed above, but also seek to share their own enthusiasm for research and its aesthetic underpinnings. Yet not many non-experts in mathematics are able to relate to such experiences that come along with abstract theoretical work. This seems to be especially true since the rise of quantum theory, which rendered analogies such as the planetary model of the subatomic world (conceived by Ernest Rutherford and refined by Niels Bohr) completely meaningless and obsolete. Analogies between scientific and artistic practice offer hope in filling the gap. In this respect, the metaphor of the musical string not only represents a stand-in for the superstring but also evokes the image of the physicist as an artist. Kaku fosters such associations when he states that “physics is nothing but the laws of harmony that you can write on vibrating strings.” (see VideoObject 2, 00’09” to 00’14”)

Yet the complexity and turbulence of the cosmos suggests a much noisier and more avant-garde “symphony” than Bach or the other “harmonious, symphonic pieces” Kaku instances. Even if the possible vibrational states of the primal strings generate the properties of all the observable particles, their combined sound would be largely dissonant and aleatoric, more like the contemporary works of John Cage (such as Music of Changes, Cage 1961; SoundObject 4), Karlheinz Stockhausen (Klavierstück XI, Stockhausen 1957; SoundObject 5)or the “stochastic music” developed by Iannis Xenakis (Pithoprakta, Xenakis 1967; SoundObject 6). In place of Fludd and Mersenne’s cosmic monochord tuned by God, consider instead a “cosmic synthesizer” or computer that generates waveforms through virtual digital “instruments,” not actual vibrating bodies. Paradoxically, though string theory emerged during the heyday of analog synthesizers in popular music as well as in avant-garde composition, advocates of string theory never envisaged such a “cosmic synthesizer” and preferred archaic Pythagorean references (Pinch and Trocco 2002). Indeed, the frequent references to Western classical music in popular writings and videos on string theory reveal a strong leaning toward aesthetic conservatism.

SoundObject 4: John Cage, Music of Changes (Book I), excerpt from

SoundObject 5: Karlheinz Stockhausen, Klavierstück XI, excerpt from

SoundObject 6: Iannis Xenakis, Pithoprakta, excerpt from


The most obvious explanation for this conservatism would be that advocates of string theory, wanting to evoke the most widely-held and prestigious emblems of harmony, referenced the classical masters and the Pythagorean harmonía with its ancient aura. Still, this does not explain why they themselves seem so conservative in their own tastes, as expressed by the criteria of “elegance” that suggests something more out of the eighteenth century than the twentieth; compare Ludwig Boltzmann’s joke that “if you are out to describe the truth, leave elegance to the tailor” (Einstein 1995: v). Even worse, those not sympathetic to string theory judge it markedly inelegant, unaesthetic in its convoluted abstract mathematics and use of multiple dimensions: the whole aesthetic criterion is thus controversial. But even those critics seem to judge according to an austere and conservative canon of beauty whose provenance seems to come from the oldest levels of Pythagorean tradition, in which simple ratios are “more harmonious” than complex ones. Even among themselves, physicists seem (probably unknowingly) to recapitulate and rehearse the latest version of an archaic vision of world-harmony through “simple” mathematics, except that the mathematics of string theory is far from simple.

Figure 6a: Higgs boson data (Credit: ATLAS collaboration; copyright CERN), showing the number of events observed plotted against the invariant mass in GeV. The lower register shows the background as a straight line, highlighting the appearance of the “bump” at 126.5 GeV, interpreted as the invariant mass of the observed Higgs particle.

Sonifying high energy physics


Quite apart from the theoretical “symphonies” of string theory, some experimental high energy physicists have begun to apply “data sonification” as an alternative to data visualization or purely abstract analysis. Just as digital computers offered the opportunity to introduce computer simulation to physics, analog-to-digital converters and digital signal processing tools paved the way for new modes of displaying and analyzing scientific data.


In 1960, the psychoacoustician Sheridan D. Speeth of Bell Telephone Laboratories designed the first digital auditory display to analyze seismological data. Speeth digitized seismograms of earthquakes and underground nuclear explosions, shifted the spectrum of the signals into the range of human audition, and subsequently trained musicians to discriminate the physical origin of the signals by listening (Speeth 1961; Volmar 2013a; for analog auditory displays and the history of scientific listening, see Volmar 2013b). The dissemination of personal computers, signal processing software, and Sound Blaster technology in the 1980s led to further explorations into auditory data representation and analysis, resulting in the foundation of the International Community for Auditory Display (ICAD) in 1992. To support empirical research, sonifications render abstract data sets into synthetic sound events that in turn can be analyzed via trained listening (Kramer 1994; Kramer et al. 1999; Hermann, Hunt and Neuhoff 2011; Supper 2012; Volmar 2013a). The use of sonification has grown markedly over the past few decades, though only rather recently applied to experimental high energy physics. We will consider two such applications that (consciously or not) seem to sound more like contemporary avant-garde music than the Romantic “sounds of space.”


These applications concern the search at the LHC for the Higgs particle, the last remaining particle of the Standard Model that long eluded detection, though the rest of that model had been abundantly confirmed in many other experiments. Indeed, the search for the Higgs proceeded for decades, exploring ever higher energy ranges to find this elusive particle, which had been proposed to break symmetries so as to cause other observed particles to be massive rather than massless.

In the wake of the discovery announced in July 2012 of a Higgs-like particle, the LHC Open Symphony group announced the first sonification of the relevant data (see LHC Open Symphony website). The text accompanying this sonification claims that it is “following some of the basic principles which guided Pythagoras and many other musician/scientists: harmonies in natural phenomena are related to harmonies in music,” resulting in the score shown (with the data that generated it) in Figure 6.

Figure 6b: Sonification by Domenico Vicinanza, from LHC Open Symphony.

SoundObject 7: Sonification of Higgs data for piano solo, from LHC Open Symphony.

SoundObject 8: Sonification of Higgs data for piano and marimba, from LHC Open Symphony.

SoundObject 9: Sonification of Higgs data, “concert version,” from LHC Open Symphony. [3]

In general, sonification of non-auditory data requires prior choices about how to render the data points in terms of audible parameters so that those somewhat arbitrary choices may allow the human ear to discern patterns that would otherwise have remained hidden to visual inspection of graphs or plots, or available only abstractly through various correlation functions (Supper 2012; Sterne and Akiyama 2012; Vickers 2012). Special training may well be required to learn the auditory signs that may thus be disclosed, which might not be apparent at first hearing. In this case, the sonifiers claim that “intervals between values in the original data set are mapped to intervals between notes in the melody. The same numerical value was associated to the same note. As the values increased or diminished, the pitch of the notes grew or diminished accordingly” (see LHC Open Symphony website). Yet their sonification criteria are disturbingly arbitrary: for instance, the notes #6 (G) and #8 (C) in fig. 6(b) are a perfect fifth apart but represent neighboring lines in the graph 6(a); note #12 is higher than #8, which represents a lower line in the graph. [4] Then too, their criteria impose a diatonic scale on the data (mapping a data value of 25 to C, 26 to D, 28 to F, etc.). The resulting diatonic melody is a pure artifact of this initial choice.


Comparing data to sonification, the critical area is found around invariant mass 126.5 GeV, corresponding in the data plot to a slight bump above the expected background and in the sonification to the highest notes F–C–E, just at (and before) the third beat of measure 2, comprising three in a stream of sixteenth notes. Given the disjunct nature of the musical line throughout, one might at first not notice these three notes in particular, though they are the highest pitches. Were one trained to look for just such pitch high points, the Higgs “peak” would draw one’s attention. In the example, that peak occurs near the center of the passage, as if it were a “climax” of the melodic line, its dramatic high point.


To be sure, that excerpt is carefully chosen precisely to “center” the Higgs event, which functions as its climax only given that prior artistic shaping. Much depends also on the choice of the metronome mark, metric framework, and subdivision (sixteenth notes in common time at quarter note = 60), as well as the pitch range chosen to be easily representable in the treble clef. Then too, different orchestrations bring out different aspects. SoundObject 7 renders all the notes equally in the timbre of a piano. SoundObject 8 superimposes a Latin beat on these pitches by adding a dominant-tonic marimba bass that turns the passage into a catchy rhumba (emphasized at one point by rattles), whose repetition further underlines the dance structure. To this, SoundObject 9 adds flute and xylophone timbres, repeating the “theme” with varied orchestration to form an A section; a B section (in classic dance form) modulates slightly to give a contrasting tonality and feeling, returning to the original material (A′). These manipulations in the sonification designedly bring out the crucial high notes of the “Higgs peak.”


The degree of artistic interpretation applied to the raw data raises questions about the exact status and purpose of this sonification as a representation of the data, rather than a highly underlined presentation of it, however charming. Still, no representation of data can claim neutrality or simple “objectivity”; all involve different emphases resulting from the chosen scale, medium, or units (Daston and Galison 2007). This sonification claims a certain faithfulness to the data by being “covariant” (borrowing a term from mathematical physics), defined here to mean always associating the same data point with the same pitch. But nothing in the data itself mandates the choice of pitches. At the very least, the claim that this passage “follows” Pythagorean principles should be read as meaning that those principles were used to construct the sound-design, rather than simply emerging from it. Sonification was not used in the original analysis that located the mass of the Higgs particle, for which the visual graph (and even more the tests of statistical significance applied directly to the data) was determinative. Thus, the sonification was applied after the fact to popularize and disseminate the experimental results in a new way. Indeed, as we have noted, the LHC Open Symphony sonifications are greatly influenced by the practical musical traditions of the West: the choice of C major tonality and sixteenth notes have more to do with common Western harmonic patterns than with cosmic harmonies.


Already in 2010, a group of particle physicists, composers, software developers, and artists led by the British CERN physicist Lily Asquith launched the LHCsound project to “introduce particle physicists to the possibility of using sonification as an analysis technique and to begin to establish the methods available for doing this” (see LHCsound website). The sonifications presented were made from both real and simulated data from the ATLAS detector at the LHC. The quite different sonifications prepared by the LHCsound project offer a much more avant-garde sound and also seem to employ a more sophisticated mapping compared to the Open LHC Symphony (though details are not available on their website).

SoundObject 10: EventMonitor sonification, from LHCsound.

SoundObject 11: HiggsJetSimple sonification, from LHCsound.

SoundObject 12: Top quark jet sonification, from LHCsound. [5]

According to Asquith, SoundObject 10 was one of their first attempts “to tackle real-time data,” sonifying random events that passed a certain trigger in the detector. Strikingly, this sonification driving an electronic piano sounds very much like a “total serialism” work of the 1950s, in particular Pierre Boulez’s Structures I for two pianos (Boulez 1955). SoundObjects 11 and 12 are reminiscent of electronic music of the 1950s, with its characteristic fragmentary quality and reliance on unfamiliar timbres and textures, the “sounds of space” Pinch discusses in this issue, widely used in many science fiction soundtracks (particularly the famous sound track to the 1956 space adventure, Forbidden Planet, composed by Louis and Bebe Barron). In line with these mid-twentieth-century works, LHCsound took a very different approach than Beethoven, Stravinsky, and Schoenberg in their “cosmic symphonies”, or the LHC Open Symphony with their diatonicism.


Indeed, those who made the LHCsound sonifications surely were exposed to the kinds of music that arguably form a context for these examples, even if they were not consciously imitating those compositional genres. It should be added that the Higgs events sound very much like those made from top quark data, their sonifications scarcely distinguishable, at least to our unpracticed ears. If so, there is no evidence that the sonifications offer any perceptible help in differentiating between these very different kinds of events; rather, the sonifications seem to function more as autonomous “compositions” or musique concrète, jumping-off points that allow sound artists to use this these data in their creations, rather than as empirical tools for working scientists. The question remains whether this sonification program goes beyond offering attractive gestures to avant-garde music to convey any other substantive content.


[3] SoundObjects 7 to 9 by the courtesy of Domenico Vicinanca, LHC Open Symphony. Sonification run on the GEANT network through EGI. Sonification support: Mariapaolo Sorrentino, Giuseppe La Rocca (INFN-CT).

[4] Note that the sonification seems to begin with the third data point in fig. 6(a), reading from the left.

[5] SoundObjects 10 to 12 by the courtesy of Lily Asquith and LHCSound.

In Asquith's account, the “epistemic culture” (Knorr-Cetina 1999) of high energy physics is primarily concerned with “graph making” and visualization rather than with listening. Indeed, the analysis of the “unfathomable quantities” of data accumulated by modern experimental systems like the LHC represents one of the major challenges of twenty-first century science. Sonification, therefore, indicates the fear of getting lost in vast dataverses and the hope of finding new ways of making sense of them. Whether or not sonification will be able to offer a solution to this problem remains to be seen.

Though string theorists have so far invoked a rather conservative aesthetic, ironically the actual sonifications of high energy data sound more like contemporary music than hieratic archaic harmonies or Romantic and early-twentieth–century “sounds of space.” This should serve as a caution, lest in these sonifications we mistake our own echo for a primordial Pythagorean signal. We should recall that Kepler also confronted the tension between his longing for cosmic harmonies and the almost constant dissonance his investigations revealed (Pesic 2014: 78–88). As Kepler struggled to make sense of this disturbing tension, contemporary physics manifests a similar inner conflict between theoretical harmonies and the strange yet contemporary music the data discloses.



As Alexandra Supper has argued, sonifications are often constructed in order to evoke sublime listening experiences that are “emotionally loaded and visceral, perhaps even mythical and spiritual, and certainly awe-inspiring and enthralling,” a practice that “works in favor of the public popularity of sonification, but at the same time undermines sonification’s claims to being accepted as a scientific method” (Supper 2014). The sonification of scientific data may therefore be seen more as a strategy to involve audiences in the “adventure” of modern science than as an epistemic tool to facilitate scientific research. For the public understanding of science, sonifications serve much the same purpose as the “cosmic symphony” does in popularizing string theory, except that actual sonifications offer additional sensual experiences, rather than musical metaphors, and encourage new alliances between science and the arts.


Apart from that, the sonification approach conceived by the LHCsound group also points in another direction. In a talk presented at the TEDxZurich in 2013, Asquith describes the purpose of the project as follows:

Physicists aren’t necessarily very good at thinking about things differently. We’re very good at some things: making graphs, interpreting graphs quickly. We’re specialized. And the problem with specialism is that it encourages narrow mindedness. I’m not sure that’s always a very good thing for us. Others quite easily see the beauty in the theory and the playfulness in the experiment […] We’re in a time when data is being collected in unfathomable quantities. Most of it remains completely unusable to almost all of us. This was a step in trying to change that. (TEDx Talks 2013, 16’10” to 17’17”)



Boulez, Pierre (1955). Structures: Premier Livre. Vienna: Universal Edition.


Burkert, Walter (1972). Lore and Science in Ancient Pythagoreanism. Cambridge, MA: Harvard University Press.


Cage, John (1961). Music of Changes; [for] Piano [1951]. New York: Peters.


Daston, Lorraine and Peter Galison (2007). Objectivity. New York: Zone Books.


Dirac, Paul A. M. (1963). “The Evolution of the Physicist’s Picture of Nature.” Scientific American 208/5: 45–53.


Einstein, Albert (1995). Relativity: The Special and the General Theory. Second edition. New York: Crown Trade Paperbacks.


Einstein, Albert (2011). The Ultimate Quotable Einstein. Ed. Alice Calaprice. Princeton: Princeton University Press.


Gray, Jeremy (2013). Henri Poincaré: A Scientific Biography. Princeton: Princeton University Press.


Greene, Brian (1999). The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. New York: Vintage Books.


Hermann, Thomas, Andy Hunt and John G. Neuhoff (eds.) (2011). The Sonification Handbook. Berlin: Logos.


Kaku, Michio (1988). Introduction to Superstrings. New York: Springer US.


Kaku, Michio (2012).Math Is the Mind of God | Dr. Kaku’s Universe.” Big Think.


Kaku, Michio and Jennifer Trainer Thompson (1987). Beyond Einstein: The Cosmic Quest for the Theory of the Universe. New York: Bantam Books.


Kinderman, William (1985). “Beethoven’s Symbol for the Deity in the ‘Missa Solemnis’ and the Ninth Symphony.” 19th-Century Music 9/2: 102–18.


Knorr-Cetina, Karin (1999). Epistemic Cultures: How Scientists Make Sense. Chicago: Indiana University Press.


Kramer, Gregory (ed.) (1994). Auditory Display. Sonification, Audification, and Auditory Interfaces. Reading, Mass.: Addison-Wesley.


Kramer, Gregory, Bruce Walker, Terri Bonebright, Perry Cook, John H. Flowers, Nadine Miner and John Neuhoff (1999). Sonification Report: Status of the Field and Research Agenda. Santa Fe, NM: International Community for Auditory Display (ICAD).


Martínez, Alberto A. (2012). The Cult of Pythagoras: Math and Myths. Pittsburgh, PA: University of Pittsburgh Press.


Mason, Betsy (2009). “Last Days of Big American Physics: One More Triumph, or Just Another Heartbreak?Wired Science. September 9.


Musser, George (2008). The Complete Idiot’s Guide to String Theory. New York: Alpha/Penguin Group (USA).


Nambu, Yoichiro (2012). “From the S-Matrix to String Theory.” In Andrea Cappelli, Elena Castellani, Fillip Colomo, and Paolo Di Vecchia (eds.), The Birth of String Theory (pp. 275–82). Cambridge: Cambridge University Press.


Nielsen, Holger G. (2012). “The String Picture of the Veneziano Model.” In Andrea Cappelli, Elena Castellani, Fillip Colomo, and Paolo Di Vecchia (eds.), The Birth of String Theory (pp. 266–74). Cambridge: Cambridge University Press


Otto, Rudolf (1973). The Idea of the Holy: An Inquiry into the Non-Rational Factor in the Idea of the Divine and Its Relation to the Rational. London: Oxford University Press.


Parker, Barry R. (1987). Search for a Supertheory: From Atoms to Superstrings. New York: Plenum Press.


Pascal, Blaise (1995). Penseés. Trans. A. J. Krailsheimer. New York: Penguin Books.


Pesic, Peter (2000). Labyrinth: A Search for the Hidden Meaning of Science. Cambridge, Mass.: MIT Press.


Pesic, Peter (2014). Music and the Making of Modern Science. Cambridge, MA: MIT Press.


Pinch, Trevor J. and Frank Trocco (2002). Analog Days: The Invention and Impact of the Moog Synthesizer. Cambridge, Mass.: Harvard University Press.


Pinch, Trevor J. (2014). “Space is the Place: The Electronic Sounds of Inner and Outer Space.” The Journal of Sonic Studies 8.


Speeth, Sheridan D. (1961). “Seismometer Sounds.” The Journal of the Acoustical Society of America 33: 909–16.


Sterne, Jonathan and Mitchell Akiyama (2012). “The Recording That Never Wanted to Be Heard and Other Stories of Sonification.” In Trevor Pinch and Karin Bijsterveld (eds.), The Oxford Handbook of Sound Studies (pp. 544–60). Oxford: Oxford University Press, USA.


Stockhausen, Karlheinz (1957). Nr. 7: Klavierstück XI. London: Universal Edition.


Supper, Alexandra (2012). Lobbying for the Ear. The Public Fascination with and Academic Legitimacy of the Sonification of Scientific Data. Maastricht: Universitaire Pers Maastricht.


Supper, Alexandra (2014). “Sublime Frequencies: The Construction of Sublime Listening Experiences in the Sonification of Scientific Data.” Social Studies of Science 44/1: 34–58.


Susskind, Leonard (2005). Cosmic Landscape: String Theory and the Illusion of Intelligent Design. New York: Little, Brown and Co.


Susskind, Leonard (2012). “The First String Theory: Personal Recollections.” In Andrea Cappelli, Elena Castellani, Fillip Colomo, and Paolo Di Vecchia (eds.), The Birth of String Theory (pp. 262–65). Cambridge: Cambridge University Press.


TEDx Talks (2013).Listening to Data from the Large Hadron Collider: Lily Asquith at TEDxZurich.”


Treitler, Leo (1982). “‘To Worship That Celestial Sound’ Motives for Analysis.” The Journal of Musicology 1/2: 153–70.


Vicinianza, Domenico (2014). “The First Higgs Boson Data Sonification!LHC Open Symphony.


Vickers, Paul (2012). “Ways of Listening and Modes of Being: Electroacoustic Auditory Display.The Journal of Sonic Studies 2.


Volmar, Axel (2013a). “Listening to the Cold War: The Nuclear Test Ban Negotiations, Seismology, and Psychoacoustics, 1958–1963.Osiris 28: 80–102.


Volmar, Axel (2013b). “Sonic Facts for Sound Arguments: Medicine, Experimental Physiology, and the Auditory Construction of Knowledge in the 19th Century.The Journal of Sonic Studies 4.


Weber, Max (1964). The Sociology of Religion. Boston: Beacon Press.


Westfall, Richard S. (1980). Never at Rest: A Biography of Isaac Newton. Cambridge: Cambridge University Press.


Wilczek, Frank and Betsy Devine (1988). Longing for the Harmonies: Themes and Variations from Modern Physics. New York: Norton.


Woit, Peter (2006). Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law. New York: Basic Books.


Xenakis, Iannis (1967). Pithoprakta. London: Boosey & Hawkes.


Zwiebach, Barton (2009). A First Course in String Theory. Second edition. Cambridge: Cambridge University Press.