“Ambiguity as a System”: Music as Order and Chaos

Mann not only calls music “calculated order,” but also “chaos-breeding irrationality,” however. And indeed, Doctor Faustus is a novel about a devil’s pact, and thus in part about the question of how chaos and evil creep into the systems dreamed up by musical theorists and utopian philosophers. Mann first hints at this in chapter VII, in which an as yet musically untutored Leverkühn discovers the importance of the perfect fifth for Western tonality, the relationship between the tonic and the dominant, and the importance of this relationship for harmonic modulation. Reflecting on his discoveries, Leverkühn calls music “ambiguity as a system” (51/74), a phrasing that neatly encapsulates the central duality that also underlies Mann’s own definition of “calculated order and chaos-breeding irrationality.” Leerkühn here refers to the principle of enharmonic equivalence. The pitches of a Western chromatic scale can be mathematically derived by taking a string and repeatedly shortening it by a ratio of 2:3. Doing so will create a series of rising tones that are each separated by a perfect fifth from the previous one. This process, however, will never yield at a frequency that exactly matches one that one might have created had one shortened the string by a ratio of 1:2, thus creating a series of octaves.2Mathematically speaking, the equation (1/2)m = (2/3)n has no natural solution, for the denominator on the left will always be an even number, the one on the right always an odd one. The “circle of fifths” does not naturally close, and if one stacks (for example) a series of fifths onto a starting tone of C, one will arrive, successively, at G–D–A–E–B–F♯–C♯–G♯–D♯–A♯–F, but never return to a C several octaves higher, only to a pitch fairly close to it. The gap that remains between the ideal and the actual is known as a “Pythagorean comma,” and creates all kinds of problems for Western music (Fig. 10). Among other things, the asymmetry that it introduces means that if one were to perform the same operation in the other direction (lengthening strings to create a series of descending tones), one would arrive at pitches that are close to, but not entirely equal to, the ones from the original sequence. F♯, in such a system, is not the same frequency as G♭. The resulting divergence makes it extremely hard to modulate between different keys, i.e., between scales built on different base tones.

One solution to this problem is to spread out the Pythagorean comma between all twelve fifths, so that none is entirely perfect anymore. If one does this, symmetry is created, the circle of fifths is closed, and F♯ becomes the same as G♭. This is what Leverkühn calls “ambiguity as a system.” Music theorists know it as “equal temperament,” and it has been the basis of Western music since the time of Johann Sebastian Bach. Equal temperament is a powerful example of “calculated order” that nevertheless begets “chaos-breeding irrationality,” for while it makes possible the harmonic masterpieces of the classical musical tradition, it also means that every equally tuned note is actually “irrational,” in the sense that it no longer conforms to any idealized ratio between tones.3For a detailed (and amusingly polemical) introduction to these matters, see Ross W. Duffin, How Equal Temperament Ruined Harmony (and Why You Should Care) (New York: W. W. Norton, 2007).

It would not have been lost upon Thomas Mann that graphic representations of Pythagorean tuning and of the closed circle of fifths bear a striking resemblance to the pentagram, which was used to control dark forces in early modern magic rituals. He would have also been familiar with the passage in Goethe’s Faust I (lines 1505–24) in which Mephistopheles, who finds himself entrapped by just such a pentagram, summons a pack of rats to gnaw through some of the lines traced out on the floor, in the process creating a figure that would have looked very much like a “broken” circle of fifths showing the Pythagorean comma. It’s quite probable that he also knew that a perfect fifth that has been disfigured by the addition of the Pythagorean comma was known to early modern music theorists as a Wolfsquinte—the wolf, like the rat, being commonly associated with the devil. No wonder, then, that he was led to conclude that harmonic theory might be a perfectly fitting preoccupation for a modern-day Faustus figure.4In Carl Weber’s Faust opera Der Freischütz (1821), which Mann alludes to on several occasions in Doctor Faustus, the Satanic figure Samiel similarly resides in a Wolfsschlucht (Wolf’s Glen). In chapter XXV, the devil, discussing music theory with Leverkühn, identifies himself as “Samiel.”

After their initial appearance in chapter VII, such considerations about intervals and the internal relationship between tones largely drop out of Doctor Faustus, to return only briefly in chapter XXXVIII, in the description of Leverkühn’s violin concerto. There, Mann’s avant-garde composer will indulge the rather simple-minded violinist Rudi Schwerdtfeger with a game of stacked fifths. Such music theoretical puzzles were ultimately not Mann’s forte. He did, however, carefully study Paul Bekker’s The Story of Music, which linked the development of equal temperament to the larger history of the West in a manner that would become hugely important for Doctor Faustus. Bekker argues that early modern music, precisely because it does not yet conform to equal temperament, “was based upon the original perception of tone as sung.”5Bekker, The Story of Music, 86. Even when written for musical instruments, in other words, such music is fundamentally “vocal” in nature, in the sense that it confines itself to a relatively limited range of pitches (at least in comparison to nineteenth-century symphonic music) and also that it privileges melody over harmony, horizontal development over vertical complexity. For when tones sound mostly successively, the presence of dissonances is much easier to disguise from the human ear than when they sound simultaneously.

In the eighteenth century, however, the nascent discipline of acoustics set in motion a conceptual shift by which “the resonant forms of air, too, were seen to be subject to the play of physical forces, whence sprang the conception of harmony with all its accompanying characteristics.”6Bekker, The Story of Music, 91. Leverkühn makes personal acquaintance with this “play of physical forces” in chapter III, where his father introduces him to the sound figures first discovered by the pioneer of acoustics, Ernst Chladni. During the time of Bach and of Händel, theorists and practitioners of music alike became interested in the complex effects that resulted from the superimposition of multiple frequencies. They also began to take a more mathematical attitude towards sound in general. The fruits of this were an expansion in expressive range, as well as a shift in emphasis from the horizontal (or melodic) dimensions of music to the vertical (or harmonic) ones. At around the same time, the fortepiano became a staple in every bourgeois household, a victory march only made possible by the adoption of equal temperament. As a result of all this, composers beginning with Haydn and his contemporaries “developed a preference for instrumental tone, because the composer’s imagination [was inspired] by the concord of harmonically ordered tones.”7Bekker, The Story of Music, 88.

Within the logic of Doctor Faustus, then, both equal temperament and harmonically ordered tones are associated with a kind of protective magic, a way of keeping the devil at bay. The fundamentally irrational structure of the natural world, as represented by the Pythagorean comma, is kept in check by the pentagram-like power of the circle of fifths. As we have seen, however, this protective operation introduces irrationalities of its own—among them not only the aforementioned ambiguity of pitches, but also the fact that the chords of the Western harmonic system become subject to an internal hierarchy based on their relationship to the major triad.

Leverkühn’s grand ambition over the course of the novel will be to replace these compromises and hierarchies of the past two hundred years of classical music with a new and mathematically more perfect system: his “strict style.” Wittingly or unwittingly, however, he makes himself the servant of the devil in these attempts. For to smash the circle of fifths also means to break open the pentagram, with potentially devastating consequences.