Chaos and Harmony

“Music Is Math,” explained the Boards Of Canada with the title of the second track on “Geogaddi.” Music, for that matter, is full of nods to numbers and mathematical imagery: from the classic nursery rhyme about counting cats in a row to the paradoxical “2+2=5” that opens Hail To The Thief by Radiohead, basic arithmetic and its universality have been a source of inspiration for countless artists.

From the Nineties onward, “math-” has even become one of the recurring prefixes that music critics use to christen new subgenres: terms like math-rock, math-metal and mathcore have entered the shared vocabulary of journalists and fans of alternative music, and are associated with styles perceived as especially intricate, technical and calculated.

On a deeper level - and in a far more far-reaching way - musical scales, from Pythagoras onward, are built on arithmetic ratios, and as anyone who made it through school with passing grades knows, the rhythmic notation of a piece is customarily expressed in fractions: 4/4, 3/4, 6/8...

But beyond the obvious observations, the superficial references and the merely evocative parallels, drawing more specific connections between music and mathematics immediately raises a question. If all music is, in its own way, mathematical, how do we single out certain styles or tracks as more “mathematical” than others?

There's no single answer, and it can range from marveling at the fact that in a track in 5/4 the numerator isn't a multiple of two, to pointing out real or supposed arithmetic patterns in well-known songs - with a particular fondness for the Fibonacci sequence, especially after the success of “The Da Vinci Code.” These are seemingly intriguing angles, but on closer inspection decidedly reductive ones (it's a rather thin notion of a discipline, one that imagines fourth-grade math as its pinnacle!), when they're not simply the result of arbitrary choices and forced readings: there are thousands of tracks in odd time signatures - are all their authors great mathematicians? And how is it that artists themselves are often completely unaware of the “Fibonacci-like” nature of their own compositions?

And yet, some works have openly adopted schemes, algorithms or logical structures with an awareness that goes well beyond simply counting up to a given number. There are albums that have turned mathematics into not just an inevitable backdrop or an invisible tool, but an explicit compositional principle, one that has enriched artists' palettes with combinatorial systems, Euclidean algorithms, Fourier transforms and algorithmic generation tools.

"Mathematical" music, from this perspective, doesn't merely acknowledge that every note has its own frequency and every rhythm corresponds to a numerical ratio; rather, it creates a syncretism between mathematical ideas and musical expression, letting them become a creative engine, almost an aesthetic manifesto.

The ten examples that follow explore exactly this: pop, rock, jazz and electronic albums that have deliberately incorporated mathematical structures into their compositional methods, sound manipulation, or even live performance. No classically-derived music – the literature on that is already vast – though the occasional mention will be unavoidable.

Through album analysis and brief explanatory forays into the theoretical concepts involved, each stop on this journey will show not only the variety of possible connections between numbers and sound, but also how mathematics can open up unexpected expressive spaces. For those wanting to dig deeper, each section will close with a few suggestions for further exploration. It will be a gradual path – from the most accessible forms to combinations that are fairly hardcore, both mathematically and musically. There's no obligation to proceed in order: readers are instead invited to pick whatever piques their curiosity most and explore beyond the tip of the iceberg as they see fit.

Singing about mathematics:

The Klein Four Group – Musical Fruitcake (2005)

Born out of a talent show within Northwestern University's mathematics department, the vocal quartet Klein Four became, via YouTube, a minor celebrity among science nerds. Their distinctive trait lies in choosing mathematics itself as the subject matter of their songs: a first level of musical "mathematization" that works not on the formal structures of composition, but on lyrical content. On their sole album "Musical Fruitcake", concepts from abstract algebra, logic, and category theory become material for witty lyrics and a cappella songs. Tracks like the micro-hit "Finite Simple Group (of Order Two)" playfully transform definitions and theorems into romantic metaphors, weaving identities, inverse functors, and subsets into the vocabulary of love (in the band members' own words: "human stories in the language of advanced mathematics").

Mathematical undertones

At its core is the deliberate use of abstract algebra's language as expressive material. The band's name recalls the Klein group, or Viergroup, one of the best-known objects in that field. Even the title "Finite Simple Group (Of Order Two)" alludes to the classification of finite simple groups, one of the great mathematical achievements of the 20th century: in the metaphor, the properties of a group made up of just two elements become a symbol of the uniqueness of a couple's bond. The technical language isn't an obstacle but a code that generates double meanings and allusions: for those versed in the subject the joke is plain to see, for everyone else there's still the pleasure of curiosity. It's a fine example of how mathematics can take centre stage without sacrificing either conceptual precision or playfulness, poetry, and creativity.

Connections and examples

The idea of making mathematics the very heart of the message isn't unique to songwriting. The musical "Fermat's Last Tango" turned the proof of the famous Last Theorem into a theatrical story. In pop, Kate Bush in "Pi" recites the digits of the irrational number in an almost incantatory tone, while Van der Graaf Generator included the track "Mathematics" on the album "A Grounding In Numbers", released on Pi Day. All examples in which numbers and formulas stop being an implicit backdrop and become narrative and poetic protagonists in their own right.

Number Stations, where sound meets cryptography:

The Conet Project: Recordings of Shortwave Number Stations (1997)

It can't really be considered a proper musical project, but the compilation "The Conet Project" has nonetheless become a cult item among fans of unconventional sound: 171 recordings of shortwave radio stations broadcasting cryptic messages made up of sequences of numbers or letters, spoken by human or synthetic voices. Although the creators have never been identified with certainty, these signals are widely believed to be a product of the Cold War: secret communications aimed at government spies from various countries. The album, assembled by sound artist Akin Fernandez starting in 1992, documents across four CDs the sonic enigma of the Number Stations, conjuring a liminal world of hypothetical military secrets and hidden codes: a frontier listening experience capable of combining tension, repetitiveness, and unintentionally musical rhythmic-melodic patterns.

Mathematical undertones

The allure of the Number Stations is closely tied to that of cryptography, the mathematical discipline that encodes and decodes information to ensure its security and authenticity. The encrypted messages in the recordings – understood only by their intended recipients – are, realistically, the product of a one-time pad, one of the simplest and most reliable encryption systems. Sender and receiver share a key, a list of characters that, once converted into numbers (A → 1, B → 2, etc.) and subtracted from the received message, allows the meaning of the communication to be reconstructed (after the appropriate reverse conversion: 1 → A, 2 → B, etc.).

From a theoretical standpoint, if the key is completely random and as long as the message itself, the communication can withstand any type of attack (short of the outright theft of the key!).

Connections and examples

"The Conet Project" is a unique document, sitting somewhere between experimental ambient music and historical record. The spectral quality of many of the audio fragments that make it up has inspired numerous artists: Jóhann Jóhannsson, Stereolab, Flying Lotus, Porcupine Tree, and Boards Of Canada have all sampled the project in their tracks. Even the title of Wilco's landmark album "Yankee Hotel Foxtrot" comes from one of the Number Stations in the collection, whose metallic, disembodied voice appears in the long outro of the penultimate track, "Poor Places".

Phase inversion, or mathematics as a tool:

Amerigo Gazaway – Yasiin Gaye: The Return (Side Two) (2014)

When it comes to effects processing and sound processing, mathematics is more or less everywhere – and therefore nowhere in particular. The case of DJ and producer Amerigo Gazaway (yes, that's his real name) stands out for his extensive use of a specific technique of mathematical origin, a "poor relation" of Fourier analysis (more on this below) whose rudimentary effectiveness is an essential ingredient in the success of his peculiar approach to mash-up – the extensive combination of samples from well-known artists to create new musical narratives. In "Yasiin Gaye: The Return (Side Two)", Gazaway combines tracks by Marvin Gaye and Mos Def/Yasiin Bey, tackling the challenge of working without vocal tracks or "clean" instrumentals. To pull it off, he openly relies on phase cancellation to separate vocals from arrangements.

Mathematical undertones

Phase cancellation is based on a simple observation: two waves with an identical profile but in opposite phase (the positive peak of one corresponds to the negative peak of the other, and vice versa) produce the same auditory sensation – but cancel each other out when combined. This principle is widely applied in electronic noise reduction, from high-end earbuds to noisy workplaces. The trick Gazaway exploits relies on the stereo nature of recordings: by inverting the phase of one of the two stereo channels and layering it over the other, centered elements – like the vocal – cancel out, while the instrumental parts, often spread across the channels, remain audible. Once the instrumental tracks are obtained, a second inversion pass can even isolate the vocals.

Connections and examples

Although Gazaway is one of the few explicitly documented cases (the artist has discussed the process in numerous interviews), phase inversion and cancellation are well-known tricks in hip-hop and mash-up circles. In the latter field, vocal track separation is achieved through various techniques: for "The Grey Album" (2004), the crossover of Beatles and Jay-Z and a direct inspiration for Gazaway, Danger Mouse had access to acapella versions, while renowned mash-up artists like 2Many Dj's and Neil Cicierega have likely used a combination of several methods for their own projects, including phase inversion.

Phase cancellation is feared by sound engineers due to the risk of losing certain frequencies, but it's also the basis of effects like the phaser, used to manipulate the timbre and amplitude of an audio signal.

The seismic orchestra of data sonification:

Volcano Listening Project – The Volcano Listening Project (2024)

The self-titled album by Volcano Listening Project turns geological, seismic and chemical data into perceptible soundscapes. The goal of the project's creator, geodynamics scholar Leif Karlstrom of the University of Oregon, isn't just to make music, but to give tangible form to the patterns hidden within the data, rendering audible the micro-earthquakes, tremors and gas emissions of the volcanoes under study. The listening experience is dense and immersive: the flow of data drives the rhythm, pitch and density of the electronic timbres, through which a fourteen-member band builds an expressive dialogue that doesn't hide but instead highlights the patterns emerging from the statistical measurements.

Mathematical undertones

Just as the more familiar data visualization aims to represent and summarize statistical data through maps, diagrams, videos and illustrations, the field of data sonification aims to make complex trends and phenomena perceptible through sound. As Karlstrom explains in a lengthy interview, the project employs various mathematical data-processing techniques, the simplest of which is audification, or direct sonification: the time sequence of measurements is resampled and sped up so that the signal falls within the audio band – the range of frequencies from 20 Hz to 20 kHz that our brain perceives as sound.

The track "Mount Saint Helens Drumbeat Seismicity And Eruption, 2004" is one example of this approach. Elsewhere, as in "650 Years Of global Explosive Eruptions", the project instead explores Parameter Mapping Sonification (PMSon), particularly well-suited to multivariate data: this is a rigorous remapping of the different dimensions of numerical information onto various sound parameters (e.g. the magnitude of a seismic event controls volume, frequency corresponds to position in time, timbre to the type of volcano involved...).

Connections and examples:

Volcano Listening Project sits somewhere between approaches oriented toward maximum transparency, like many of the contributions gathered by the Data Sonification Archive, and more artistic manipulations, like Diegetic's "Above A Burning Earth", where the pursuit of musical pleasantness can override the data, reducing it to a mere noise generator. Karlstrom's project seeks a balance, ensuring that the scientific information comes through clearly and successfully guides the musical outcome.

Another significant project, where sonification meets land art, was "Land Music" by the duo Nessun Dharma, made up of Italians Matteo Bonera and Michele Zuccarelli Gennasi. For Bergamo and Brescia's Italian Capital of Culture 2023, the artists staged a data-driven performance involving thousands of bell towers across a 7,500 km² area spanning the two provinces. At 8 a.m. on 18 March 2023, on the National Day of Remembrance for all victims of the Coronavirus pandemic, each tower rang a bell toll for every victim in its own parish. Unfortunately, no recordings of this event exist – nor could they have, since it was a hyperobject in the sense meant by philosopher Timothy Morton: an entity so vast and complex in space and time that it cannot be directly perceived as a whole.

For further reading on the intricate field of data sonification, academic-level sources include the volumes "The Sonification Handbook" (Hermann, Hunt, Neuhoff, 2011) and "Hidden Music: The Composer's Guide To Sonification" (Mermikides, 2025), which cover the theory and practice of the discipline in great detail.

Beyond the twelve notes – xenharmony:

The Mercury Tree – Self-Similar (2023)

Bach convinced us. He composed "The Well-Tempered Clavier" to demonstrate the potential of dividing the octave into twelve equally spaced semitones (equal temperament), and for a few centuries now, the West has followed his path. Getting past it is simple enough: just detune your instrument. Doing so with musically satisfying results… well, that's another matter entirely. Mercury Tree have devoted themselves fully to the task: album after album, they've built a cohesive, fully realized alternative/prog songwriting style while moving through alien harmonic landscapes, alternating dense, layered passages with more suspended, rarefied moments – all built on scales with 19 or 24 divisions. "Self-Similar" moves between math-rock density and moments of acoustic clarity, developing a new language that seems to open up a parallel dimension of musical perception.

Mathematical undertones

Like equal temperament, xenharmony (from the Greek xenos: foreign, strange) can also be built on a regular structure of frequency relationships. While our usual twelve notes have frequencies whose ratio is a power of 2^1/12 (the twelfth root of two), in "Self-Similar" the interval between two consecutive C notes is divided according to powers of the nineteenth or twenty-fourth root of two. The resulting harmonic world has enough similarities to make certain chords feel familiar, but also enough differences to allow for entirely unusual emotional trajectories. Alternative – and equally mathematical – approaches include just intonation, based on ratios between small prime numbers (with 3 and 2 you get the Pythagorean scale used by the ancient Greeks, though other choices open up further expressive possibilities), or scales not based on the octave – such as the Bohlen-Pierce scale, where the fundamental ratio is 3/1 rather than 2/1.

Connections and examples

Ever since 2017's “Flying Microtonal Banana”, psych-rockers psych-rocker King Gizzard And The Lizard Wizard have repeatedly experimented with quarter tones. Along a path closer to minimalism, Horse Lords combine polyrhythms, odd time signatures and guitars with modified keyboards to explore the sonic spaces of just intonation. An important historical precedent for the rich xeno-rock landscape of our times, explored on OndaRock in a dedicated article, is synth pioneer Wendy Carlos: on “Beauty In The Beast” (1986) she had used the Indonesian slendro and pelog tunings along with two original scales, alpha and beta, pushing electronic music toward new intonations as far back as the 1980s. 

Spectralism and Jazz-Hop Ghosts:

Sélébéyone – Sélébéyone (2016)

Influenced by French composers Gérard Grisey and Tristan Murail, but also by the concept of “Afrological improvisation” put forward by theorist George E. Lewis, saxophonist Steve Lehman has built a wholly personal fusion of avant-garde jazz, electronics and Senegalese hip-hop. Among the distinctive traits of this marriage is a conscious reliance on the principles of Grisey and Murail's spectralism: by integrating the timbral and harmonic aspects of music, the frequencies produced by individual instruments are deliberately layered to construct unprecedented sound profiles – timbral chimeras that can replicate the character of known instruments (though absent from the lineup) or sound like entirely new, alien-seeming instruments.

In Lehman's case, this translates above all into the use of a synthesizer to create sets of frequencies that are inherently dissonant on their own, but which, once combined with the other instruments, produce a timbral enrichment that – even when the synthesis isn't perfect – casts an arcane aura over the tracks, at once tactile and deeply evocative.

Mathematical undertones

Spectralism is grounded in the discoveries of the branch of mathematics known as harmonic analysis, largely based on the Fourier transform. This computational tool, developed by Joseph Fourier in the third decade of the nineteenth century to study the propagation of heat, makes it possible to break down any periodic signal – sound, for instance – into a sum of sine waves. Large swaths of signal engineering, digital image processing and crystallography wouldn't be what they are without Fourier analysis, and its use also underpins the electronic synthesis of sound.

Specifically, spectralism is a sort of DIY version of what's known as additive synthesis, which combines multiple sine waves – each with a specific frequency – to generate new timbres at will, or ones resembling those of known instruments. While digital additive synthesizers perform this operation starting from digitally generated sine waves, in spectralist works the starting sine waves are those inherent in the sounds of pre-existing instruments. Lehman's specific approach is something of a mix of the two paths: by adding acoustic timbres to synthesized frequencies, the tracks on “Sélébéyone” are built around cyborg instruments. It's no coincidence that the album's title means “intersection” in Wolof, the language in which MC Gaston Bandimic raps on many of the record's tracks.

Connections and examples

Among the many approaches to electronic synthesis, the additive one has seen comparatively limited use. The first, revolutionary digital sampler used in pop music, the Fairlight CMI, nevertheless had an additive synthesizer at its core. Among the records that have made the most of the tool are Peter Gabriel's fourth solo album, the Cars' “Heartbeat City”, Trevor Horn's work with Yes, Abc, Art Of Noise, Frankie Goes To Hollywood, but also “Future Shock” by Herbie Hancock, Coil's “Horse Rotorvator” and “L'arca di Noè” by Franco Battiato.

On the Ep “Windowlicker” (1999), British producer Aphex Twin used similar principles to hide actual images within the tracks. To view them you need a spectrogram: a Cartesian diagram in which each point is colored according to the intensity with which the two frequencies tied to that point's coordinates are reproduced on the right and left audio channels. Visualized this way, the ending of the title track depicts spirals and concentric circles, while the close of the track fans have nicknamed “[equation]” or “[formula]” (whose actual title is a rather intricate mathematical expression) reveals a leering portrait of the track's creator!

Jesuits Euclidean Rhythms:

Fractal Sextet – Fire and Grace (2022)

Polyrhythms and odd time signatures are among the musical elements most often linked by rock critics to the adjective “mathematical.” In the case of Fractal Sextet, an international ensemble based in Switzerland, the term takes on special relevance: to build the tracks' rhythmic patterns, percussionist Andi Pupato uses the Euclidean algorithm, developed by Euclid over two thousand years ago to find the greatest common divisor between two numbers. The musical result closely echoes the minimal/jazz/rock layering of fellow Swiss musician Nik Bärtsch's Ronin (with whom Pupato played until 2012) and of the Zurich group Sonar (from which the band's guitarist, Stephan Thelen, hails). The progressive cut of the sound – distinctly Crimson-esque – is certainly heightened by the presence of Colin Edwin, longtime bassist of Porcupine Tree.

Mathematical undertones

Euclidean rhythms are based on the principle of maximal evenness: computational geometry expert Godfried Toussaint discovered in 2004 that, given a number of beats k to be arranged over a time span of n beats, the Euclidean algorithm can be adapted to calculate the arrangement that makes the distance between each beat as uniform as possible. For instance, if you need to distribute 3 hits over 5 beats, you might arrive at the sequence 2 2 1 (or 1 2 2, or 2 1 2); for 5 hits over 16 beats, on the other hand, the optimal arrangement would be, say, 4 3 3 3 3. Here (in simplified terms) is how this sequence relates to integer division: 16÷5=3, with a remainder of 1. This means each hit should be spaced 3 beats from the next, but that one gap (since the remainder is 1) must be one beat longer. And how do you get from 3 hits and 5 beats to 2 2 1? Exercise left to the reader!

Connections and examples

In the article “Kid Algebra: Radiohead's Euclidean and Maximally Even Rhythms” (Perspectives of New Music, 2014), musicologist Brad Osborn notes that numerous Radiohead songs (“Pyramid Song”, “Morning Bell”, “Reckoner”, “The National Anthem”, “Codex”…) feature structures based on Euclidean rhythms. Are Yorke, Greenwood and co. mathematical geniuses, then? Fans would love to claim so, but then they'd have to grant the same credit to the creators of reggaeton: the tresillo rhythm underpinning the genre is nothing but a Euclidean pattern E(3, 8), with 3 hits over 8 beats! The ability to balance regularity and asymmetry means Euclidean rhythms are just as ubiquitous in rock'n'roll as in traditional music – largely on an intuitive, unconscious basis.

Consciously deployed, by contrast, is the effort of indietronica artist Imogen Heap, who included a tool called Euclidean Beats within her Box Of Tricks plugin, bringing real-time generation of Euclidean patterns into contemporary art-pop production.

Algorave, the dance of algorithms:

Various Artists – Algorithmic Art Assembly v1.0 (2020)

Born in the early 2010s with Alex McLean and Nick Collins, the algorave movement places the real-time generation of electronic music through algorithms at its core, often tweaked on the spot via live coding — that is, programming performed live in front of an audience. Artists write and modify code that produces rhythm, harmonies, timbres and formal structures, frequently projecting the code onto a screen as an integral part of the performance. The listener thus witnesses the birth of music as a transparent process, tangibly observing the relationship between computational logic and sound.

“Algorithmic Art Assembly v1.0” is an anthology gathering tracks by some of the movement's most significant figures, including Renick Bell, Kindohm, and William Fields. Its thirteen tracks, which will certainly appeal to fans of Autechre and glitch textures, offer a broad view of the possibilities of this recent frontier of Intelligent Dance Music, where metamorphosis and unpredictability form an unbreakable bond with rhythmic experimentation and the evocative qualities of synthetic sound.

Mathematical undertones

The mathematical heart of algorave lies in the use of TidalCycles, a software environment created by Alex McLean designed to manipulate musical patterns in real time. A pattern, in this context, is an ordered sequence of events (kick hits, notes, samples) treated as a mathematical object to which transformations are applied. So the function slow 2 doubles the duration of a pattern, while rev reverses it. Every 3 applies a transformation only every three cycles, while sometimesBy 0.3 (rev) introduces a 30% probability that a pattern will be played backwards. In this way the music is never entirely deterministic, but thrives on algorithmic choices that intertwine periodicity and chance. Further stochastic elements (rand, shuffle) introduce unpredictable deviations, turning concepts from discrete mathematics, computational logic and probability into real-time compositional tools.

Connections and examples

Different in outcome but close in premise, the “generative rock” of artists like the American band Doctor Nerve could be seen as the “flesh and blood” counterpart to generative electronic music. In their “NerveWare” project, Doctor Nerve — led by guitarist and composer Nick Didkovsky — experimented with the Hierarchical Music Specification Language on the Commodore Amiga: developed by researchers at the Center for Contemporary Music at Mills College of Northeastern University, it was an object-oriented language for generating scores for avant-prog pieces lasting one to three minutes, which could then be performed via MIDI or by an ensemble of musicians, as was done on the album “Beta 14 ok” (1991).

A natural field of application for the algorithmic strand is videogame music: as early as 2008, the life simulator “Spore”, developed by Maxis, featured a generative soundtrack created with the contribution of Brian Eno using the open source software Pure Data. In the following decade, the hugely anticipated galactic survival game “No Man's Sky” was accompanied by procedurally generated music courtesy of the British electro-post-rockers 65daysofstatic, which evolved according to the player's actions, combining pre-recorded fragments from the band in novel ways.

Numbers, bells and braids – change ringing:

Meurig Elis Huws – Bellectronic (recordings 1980–1984, released 2021)

“Bellectronic” gathers pioneering recordings in which Welsh engineer Meurig Elis Huws transposes the methods of combinatorial bell-ringing (change ringing) onto synthesizers and electronic percussion. Traditional change ringing is a British art form in which a set of bells is rung, typically by a close-knit team of ringers, avoiding immediate repetitions and generating all possible permutations according to predetermined patterns. Huws, inspired in the early 1980s by the sounds of Human League and Gary Numan, brought this discipline into electronic music, creating a mesmerizing proto-techno derived from combinatorics. A treasure that lay buried for decades, until his son discovered three tapes labelled “Bells” among his recently deceased father's belongings. The recent reissue by the Spanish label Strategic Tape Reserve has allowed the world to encounter this universe where spirituality, play, electronic experimentation and mathematical rigor merge creatively.

Mathematical undertones

In their mathematical essence, change ringing performances are systematic mappings of the configuration space generated by an initial sequence of sounds and a set of possible transformations. In the method known as plain hunt (track 8), each sound moves forward or backward by one position from one sequence to the next: plotted graphically, it corresponds to an interweaving of oblique lines all sharing the same slope. In plain bob (tracks 1 and 13) the pattern is modified to cover a greater number of configurations, introducing inversions and extra position changes to reshuffle the deck in an orderly fashion. The resulting diagrams follow the same geometry as multi-strand braids: transposed musically, the classic three-strand braid is nothing other than a plain hunt on three sounds.

Connections and examples

Change ringing recordings are rare. On “Change Ringing On Handbells” (1981), alongside the chiming of the handbells, one can also hear the calls with which the performance's conductor signals to the other ringers which transformations to execute. The thickening and thinning of these calls — incomprehensible to the uninitiated — as the piece unfolds adds a further layer of esoteric fascination to the hypnotic flow of the chimes.

When “mass-produced” isn't an insult:

Arrigo Barnabé – Clara Crocodilo (1980)

Emerging in São Paulo in the final years of the Brazilian military dictatorship, the vanguarda paulista movement took advantage of the political and cultural thaw promoted by General Figueiredo to produce music dense with experimental ambition, irony and social critique. One of the most celebrated albums of the movement is Arrigo Barnabé's debut, “Clara Crocodilo”, which stands out for its fusion of Mpb (musica popular brasileira), zigzagging avant-rock somewhere between new wave and Zolo, and forays into atonal and twelve-tone territory. It's precisely this last feature that makes this debut a natural candidate among the most “mathematical” rock records of all time. Picking up the approach pursued in the first half of the twentieth century by Arnold Schönberg, Alban Berg and Anton Webern, the eight tracks use twelve-tone series manipulated through retrogression, inversion and transposition to eliminate a stable tonal center and create continuous harmonic tension. Compared to its inspirations, and thanks also to the rhythmic drive and often playful nature of the music, Barnabé's compositions stand out for their balance, managing to combine complex, disorienting structures without sacrificing immediacy of listening.

Mathematical undertones

Introduced as an attempt to move beyond the tonal harmony that had dominated European art music for centuries, twelve-tone technique is founded on rigorous permutations of a series of twelve notes: each note appears only once before the series begins again. Operations such as retrograde (the series read from end to beginning) and inversion (reflecting the intervals from descending to ascending and vice versa) are systematically employed to generate variants that retain a formal connection to the original series.

As algebraists were quick to point out, the twelve notes can be interpreted as a finite set acted upon by formal symmetries: concepts from group theory and modular arithmetic are therefore - even when left implicit - at the very heart of every twelve-tone and serial construction. Flirting even more deliberately with mathematical thought, the serialism developed in the aftermath of World War II extended the use of these techniques beyond the purely twelve-tone field, allowing for series of varying length and applying the same transformations to timbre, duration and dynamics as well.

As documented by musicologist André Cavazotti in the article “Serialism and free atonalism land on Brazilian Popular Music” (Per Musi, 2000), Barnabé's record employs twelve-tone techniques in as many as five of its eight tracks, and features serial elements in the remaining three!

Connections and examples

Blotted Science, the project of Watchtower guitarist Ron Jarzombek, is another example of twelve-tone experimentation in rock - in this case, tech-metal. The tracks on the debut album, “The Machinations Of Dementia” (2007), were described by Greg Burk of Metaljazz.com as “the most radical metal instrumentals in history”, and according to Jarzombek, “around 75%” of the album's music was composed using serial techniques applied to sequences of twelve notes.

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