Euler, Cowell, Polyhedra and the Music Genome
By the 18th century, Europe was fervently attempting to classify and dissect the world through the rediscovered power of the natural sciences; independent in many spheres from the intellectual tyranny of medieval theology, philosophers were emerging critically questioning the human institutions incubated under feudalism and theocracy during the European Dark Ages; rediscovered from Classical Greece, the perspective and dramatic arts, rhetoric, empiricism, mathematics, and many other fields of thought contained in the Trivium and Quadrivium returned to preeminence under broader and freer education and printed text. At the epicenter of this burgeoning enlightenment, Leonard Euler (1707 – 1783) emerged as an unquestionably colossal intellect, determined to break the shackles of superstition and discover truth in his cosmos. He is most well-known for his pursuits in mathematics, but Euler attempted paradigm shifts in numerous fields including physics, astronomy, and music. While his work has been critical in the continual development of mathematics since the 18th century, his work in music has been relatively unconsidered in its respective field until the late 20th century, when Neo-Riemannian Theorists found precedence for their theories in his musical treatise from 1739, Tentamen novae theoriae musicae (Attempt at a New Theory of Music). Even today, the full implications of Euler’s theories are only modestly finding meaning in the broader musical world, beyond the pages of relatively unapplied academic music theory articles and graduate texts.
What Euler was attempting in his theory was not a theory of harmony and rhythm made in service to only understanding the music of his day, but a theory of music generally, which could be employed for analysis as well as composition and categorization. An analogy can be drawn between Euler’s attempt at a paradigm shift in the understanding, composition, and classification of music and his similar work in polyhedra and graph theory, wherein he created a system by which to generally classify all polyhedra and graphs, connecting discrete structures of polyhedra and graphs to the continuous structures of manifolds. We will return to this comparison shortly.
The most significant aspect of Euler’s music theory, which would be unmatched until innovations by composer-theorists like Arnold Schoenberg and Henry Cowell in the 20th century, is its radical abnegation of the historical model of music theory. Rather than try to understand music by expanding traditional models advanced since the end of the Middle Ages, Euler decided to apply the same rigor to musical dissection and classification as he was employing in mathematics. He was attempting to find common patterns in the general and abstracted structures of sound – pitch and duration – and find broader meaning in such patterns. Rather than search for musical truth in music as practiced, he attempted to find musical truth in the components of sound, which then could be applied to better understand music as practiced, either past, present, or future.
The first break with nearly a millennia of music theory Euler established was to classify all harmonies (and consequently durations) as consonant, albeit with variable degrees, rather than the traditional dichotomous arrangement of sounds as either consonant (perfect and imperfect) or dissonance. He argued that by presenting sounds, as done in music since the Ancient Greeks, as durations of rational ratios, all sounds should be considered consonant, since all ratios are variably consonant in their denoted periodicity; no ratio is technically dissonant, since all ratios are rational. Euler was radical in proposing such an understanding of ratio in music, which had decreed for nearly 2000 years that only a few select ratios should be considered consonant and, consequently, musically superior. Applying a musical theory similar to his concept of the “Euler characteristic” of polyhedra, Euler devised a method, in which he proposed to give a single mathematical formula for all general phenomena of music; the musical formula was specifically designed to address harmony, successive pitches (melody), rhythm, and changing rhythm (meter and tempo), however Euler implied that this method could be expanded to encompass all musical dimensions (dynamic, timber, instrumentation, form, etc.).
To this end, Euler used the principle of the "exponent" when comparing two similar musical structures (pitches in harmonies, rhythmic values in rhythmic successions, etc.) to propose a derivation of what he termed the “gradus suavitatis” or “degree of agreeableness” between these structures. This degree could then be used to compare seemingly similar or dissimilar structures in music (different harmonies, rhythms, melodies, meters, forms, etc.); through this mathematically derived comparison, deep similarities could be uncovered that might not lie at the surface of the music or which might be obscured due to historical aesthetic bias. For example, one might not immediately think that the music of Johann Sebastian Bach (1685 – 1750), Philip Glass (b 1937), and Francesco Landini (c. 1325 or 1335 – 1397) are similar, given their different systems of pitch hierarchy, harmonic progression [1], and instrumentation (not to mention their temporal difference by centuries); however, by abstracting their propensity for rhythmic motorism under metrical systems of mostly dyadic and triadic rational hierarchies, one can group these seemingly disparate composers and music styles into a group based on at least one dimension of similarity or “degree of agreeableness.”
To demonstrate this concept in anther, perhaps more visual way, let us return to Euler’s concept of polyhedra classification by the use of his “Euler characteristic.” In mathematics (more specifically in algebraic topology and polyhedral combinatorics), the Euler characteristic (also called the Euler number or Euler–Poincaré characteristic) as defined by its eponymous discoverer, is a number intrinsically characteristic to a space (e.g. a polyhedra) that describes that space's structure, regardless of any deformations (bends, folds, holes, etc.) in that space. Through this characteristic number, one can group polyhedra, graphs, or topological spaces in general into groups with similarity, even if a similarity does not seem intuitively evident under our limited human perceptions.
To exemplify, consider all of the Platonic Solids (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, all of which are polyhedra), all discovered and generally well-understood by the Ancient Greeks (see Figure 1). Euler discovered that while these solids appear to be quite different shapes, they share a fundamental, singularly reducible property, which they will not share with some other shapes: their “Euler characteristic.” This characteristic was defined by Euler according to the surfaces of polyhedra by the formula: X = V – E + F, where V, E, and F are the numbers of vertices (corners), edges (sides), and faces (the constituent polygons) respectively in the given polyhedron. Let us simply calculate the Euler characteristic for each platonic solid:
Tetrahedron: vertices = 4; edges = 6; faces = 4 4 – 6 + 4 = 2
Cube: vertices = 8; edges = 12; faces = 6 8 – 12 + 6 = 2
Octahedron: vertices = 6; edges = 12; faces = 8 6 – 12 + 8 = 2
Dodecahedron vertices = 20; edges = 30; faces = 12 20 – 30 + 12 = 2
Icosahedron: vertices = 12; edges = 30; faces = 20 12 – 30 + 20 = 2
Figure 1: The Platonic Solids – These are special three-dimensional polyhedra since they are the only ones in 3D-space that can be constructed using only congruent (identical in shape and size) regular (all angles equal and all sides equal) polygonal faces (triangles, squares, and pentagons exclusively) with the same number of faces meeting at each vertex. Only these five solids meet these stringent criteria.
The platonic solids with Euler characteristic 2, while being very prescriptive in their constructive nature, actually, by their Euler characteristic, fit into a much larger set of seemingly unrelated solids, these solids having other, non-Platonic aspects: convex polyhedra (or spherical polyhedra), which include larger sets of polyhedra, such as the Archimedean solids, which are not Platonic. Thus, through this characteristic number, we can take a seemingly singular set of objects that appear among themselves to be different and which by their constructive nature are not related to other sets of polyhedra (like the Archimedean solids), and show how they are intrinsically similar to other polyhedra under a broader or generalizing category of “convex.”
As a counter example, let’s consider the Tetrahemihexahedron, which looks relatively similar to the octahedron but with certain faces made to be concave (see Figure 2). Let us calculate the Euler characteristic for the Octahedron and Tetrahemihexahedron.
Octahedron: vertices = 6; edges = 12; faces = 8 6 – 12 + 8 = 2
Tetrahemihexahedron: vertices = 6; edges = 12; faces = 7 6 – 12 + 7 = 1
Figure 2: The Tetrahemihexahedron and the Octahedron
While perhaps the Tetrahemihexahedron and the Octahedron appear relatively similar, the Tetrahemihexahedron, in fact, fits more characteristically into another class of solids, known variously as the nonconvex polyhedra.
Euler’s theory of music operates in a very similar manner to the above example with polyhedra. He uses a similar formula that aims to take the dimensions of music and abstract them from cultural, historical, human-based perceptive bias in order to find an irreducible characteristic in the intrinsic nature of various transformations of a particular dimension (pitch, rhythm, timbre, etc.). Thus, Euler’s concept of “degree of agreeableness” in musical structures is similar to the “Euler characteristic” in geometric structures. Consequently, at its ultimate end, Euler’s music theory attempts, both in a pitch and rhythmic space (with suggestions to apply this theory to other dimensions as well) to create a system by which to locate similarities between disparate musics and to classify groups of similar musics that might seem different or have similarities that are obfuscated by some level of perceived complexity but nevertheless have similarities at a fundamental level.
The cornerstone and first radical change Euler’s theory proposed in understanding music was the use of frequencies - and their ratios when combined (both in frequency’s pitch and rhythmic domain of audibility) - to compute his “degrees of agreeability” (“gradus suavitatis”). Since all frequencies under consideration were integers or terminating decimal numbers, all ratios between two frequencies would be rational numbers, and consequently could reasonably be considered consonant to some degree [2]. In these theories, the more two frequencies coincide (whether these be 2:3 or 313:983), or the lower the common multiple between them is (6 in the case of 2:3 and 307,679 in the case of 313:983), the more consonant they are. Unlike his predecessors, Euler did not refer to historical notions of “consonance” and “dissonance” carried from the Middle Ages; rather, he revolutionarily set no limit on the sound structures he could consider “consonant.” In his theory, consonance is a continuum with groups of similarity as per a structure’s “degrees of agreeability.” Even more profoundly, Euler was suggesting not just a method to understand music that already existed but to write new music unlike any ever heard before. Unfortunately, Euler’s ideas would not be used to advance music in such a radical manner in the 19th century. Not until the early 20th century in the works of composers like Charles Ives, Henry Cowell [3], Johanna Beyer [4], and Paul Hindemith [5] (among many others) were ideas similar to Euler’s (though not directly inspired by him) seriously discussed and put into extensive practice.
As Henry Cowell would realize and implement in his own compositional practice nearly two centuries later, Euler realized that harmonic ratios were nothing more than relationships between the rates of events. In this, an elegant parallel can be seen between harmony and rhythm; in harmony, events are vibrational periodicities; in rhythm, events are successive periodic durations. Thus, Euler’s theory of harmonic consonance and “degree of agreeability” can be compressed to encompass rhythm. Harmonies like 400Hz : 500Hz (or a perfect fourth, or the sound of a C and the F above it on the piano) can be understood equally as a polyrhythm (4 equal beats in the same time as 5 equal beats). Taking this idea further, one could make a “harmonic” progression of rhythms, or rather, a progression of rhythms which imply a harmonic progression. Henry Cowell would be the first to overtly attempt such a compositional method in his Quartet Euphometric from 1919, 180 years after the first publication of Euler’s Tentamen novae theoriae musicae in 1739. Only after Cowell’s and Ives’ bold experiments in music, would the musical ideas, first systemically proposed by Euler, find an audience in the mid-20th century in the works by composers like Conlon Nancarrow, Pierre Boulez, Edgar Varese, György Ligeti, and later Milton Babbitt, Eliot Carter, Brian Ferneyhough, Clarence Barlow, and Jonathan Dawe.
Euler’s theory even has commercial implications today with projects like the Music Genome Project [6]. Now managed by Pandora, the Music Genome Project is an effort to “capture the essence of music at the most fundamental level” using over 450 selected attributes to categorize and describe all genres of music. This has been done through feeding data, produced heavily from human input information (questions posed to human’s and answers given by them, rather than musical data abstracted from human biased) into a complex mathematical algorithm, which then organizes vast amounts of music in a database according to similar characteristics. The Music Genome Project currently has only 5 “genomes”: Pop/Rock, Hip-Hop/Electronica, Jazz, World Music, and Classical, which then have refined sub-categories. Perhaps, it is not difficult to see how Euler’s “degrees of agreeability,” particularly when applied to many musical dimensions simultaneously, could be a powerful tool to identify broad similarities within and across these prescribed “genomes.” While one might hesitate to suggest similarities between Rap and 14th century Trecento polyphony; between pop music a la Lady Gaga, Katy Perry, and Britney Spears and operas by Mozart, Rossini, and Meyerbeer; between Jazz, the Medieval Ars Subtilior, and Conlon Nancarrow; Euler might gladly force our hand, ultimately to discover profound connections between seemingly disparate categories of art, opening our ears to a broader listening experience.
END NOTES
[1] I mean “harmonic progression” only in the most general sense, not in the Common Practice Tonality sense.
[2] Only in the 20th century, would ratios of irrational numbers come into question, such as in the music on Conlon Nancarrow with rhythmic and tempo ratios such as or . Given such musical suggestions, Euler might have conceived of these tempo ratios as “dissonant” rather than consonant, since there is no rational expression (or ratio of integers) that will reduce these numbers.
[3] See Cowell’s book New Musical Resources and his works Quartet Romantic and Quartet Euphometric.
[4] See Johanna Beyer’s piece Dissonant Counterpoint, inspired by Cowell’s book, New Musical resources.
[5] See Paul Hindemith’s book The Craft of Music Composition, wherein he outlines a new theory of consonance and dissonance similar to Euler’s. Also see his work, Ludus Tonalis, to hear his consonance/dissonance, theories in practice.
[6] This project has not, to my knowledge used Euler’s mathematical method and has rather relied on massive human-input derived data; however, this project could perhaps benefit from recognizing Euler’s attempt at doing a similar universal classification of music and adopting modified versions of Euler’s methods, rather than relying heavily upon human input, which might be biased and not as classificationally powerful on a broad level.
