CURVED PHRASES from Indian ragas melt into languorous blues microtones. The strains of music born on the American South’s slave plantations mutate into the more angular notes and layered melodies of the Western classical genre. The different sounds weave in and out of one another, creating a shifting and arresting aural landscape that feels at once both achingly familiar and dazzlingly novel.

This is the radical new music on a CD, titled Metaraga, to be released worldwide on 17 January by Origin Records, a prestigious US label. The music has grown out of a path-breaking, unifying musical framework developed by Purnaprajna Bangere, an Indian-American mathematician and violinist, using concepts from algebraic geometry, the discipline in which he specialises. Bangere also founded the Purna Loka Ensemble, which performs six of the CD’s eight tracks.

“What I find appealing is that Purna’s concept is highly virtuosic and is built on an incredibly vast traditional musical system, but folds in references to jazz, blues and mountain musics that slowly unfold as he moves through a piece,” said John Bishop, a drummer and founder of Origin Records. “There are many points of reference, rhythmic and melodic, that our ears connect with, but it does feel like something new and significant.”

This framework, which Bangere calls the ‘metaraga system’, combines the full range of Indian microtones and oscillations, such as meend and gamak in Hindustani music, and kampitam and sphuritam in Carnatic music, with Western-style key changes, or modulation, and polyphony, which is music containing two or more simultaneous lines of independent melody. Moreover, in the framework, blues-style melodies evocatively emerge as the best technical and aesthetic bridges between the Indian and Western genres.

In addition to its musical importance, therefore, this system has a profound cultural significance: Afro-American music that grew from pain and protest becomes the link between East and West.

Bangere evolved the system’s musical principles from algebraic geometry, whose concepts form the system’s invisible subterranean plumbing. Bangere has presented the mathematical ideas underpinning the system — which is a work-in-progress that he continues to expand and refine — at several prestigious venues in India and the US, such as the Asian Art Institute in San Francisco, Brown University, the Mathematical Sciences Research Institute-Berkeley, Tufts University, the Chennai Mathematical Institute and the Indian Statistical Institute, Bengaluru. His ensemble has also performed the music at some of these venues.

“I was searching for a musical idiom that transcended civilisational barriers,” said Bangere, a professor at the University of Kansas in Lawrence, US, who grew up in Mysore, during the course of about a dozen face-to-face, telephone and Skype interviews over the past year. “Mathematics showed me the way. The metaraga system integrates different genres, which exist separately, but their key elements dynamically interact with each other to produce new sounds, with no sense of hierarchy.”


Above image: Purna Loka Ensemble. (L-R) Jeff Harshbarger (bass), Purnaprajna Bangere (lead violin), Amit Kavthekar (tabla), David Balakrishnan (second violin). Photo courtesy Purnaprajna Bangere

Six of the CD’s eight tracks reflect this richly textured, pan-civilisational sound. Four of these, ‘Syzygy’, ‘Triality I, Triality II’ and ‘Triality III’, have been composed by Bangere, in collaboration with one of the finest contemporary jazz violinists and composers, David Balakrishnan, founder and director of the three-decade-old Turtle Island Quartet, which has twice won the Grammy Award. Balakrishnan is a key member of the Purna Loka Ensemble, whose other members are bassist Jeff Harshbarger, and tabla player Amit Kavthekar, with guest clarinet player Robert Walzel heard in one track.

A fifth track, ‘Fibration’, is Bangere’s solo composition, and a sixth, ‘Alabama’, is his re-interpretation of jazz saxophonist and composer John Coltrane’s musical take on a speech by Martin Luther King Jr condemning the bombing of a church in Birmingham, Alabama, in 1963, by the white supremacist Ku Klux Klan. The remaining two pieces are Bangere’s renditions of the Indian ragas Abhogi and Hamsadhwani.

“Purna’s experiments with tonal transition, which includes temporal suspension, make him a true pioneer,” said Jan Radzynski, a distinguished composer who is a professor at Ohio State University in Columbus. “Purna is an incredibly talented Indian-American composer and violin virtuoso whose time has come.”

Bangere is now composing new pieces with fellow professor Dan Gailey, a saxophonist, composer, and director of jazz studies at the University of Kansas, who will bring his own aesthetics and vision to the system.

“Purna’s music feels especially innovative in that it seems to defy a pronounced leaning to one world or the other, such as Western jazz and classical vs Eastern and Indian music,” Gailey said. “I was absolutely blown away when I heard it performed live, and my reactions were confirmed upon hearing the mastered recordings that comprise Purna’s CD. By integrating core aspects of both Western and Indian music, he truly has created music that is neither East nor West, and I find that incredibly exciting.”

To fully appreciate the metaraga system’s power, one must become familiar with its nuts and bolts. But just as one ideally needs a few years of training to understand the nuts and bolts of Hindustani or Carnatic music, one also needs systematic initiation into the metaraga system. Towards the end, this article offers interested readers a glimpse into this system’s conceptual landscape.

But for an average listener, the proof of the pudding lies in the listening. Two tracks included here, ‘Syzygy’ and ‘Triality II’, vividly illustrate the seamless synthesis of genres that can be achieved within this framework. Ultimately, only by listening to these tracks will music lovers begin to fathom its potential. Like all good art music, the compositions yield their secrets slowly on repeated hearing, and perhaps never fully. The CD can be ordered through several online retailers, such as Barnes & Noble and Target.

Even more difficult than delineating the metaraga system’s musical concepts is describing the underlying mathematics, which is a highly technical and abstract field. In any case, the mathematics is not relevant even to a musician interested in using the musical framework, let alone to a lay reader. But again, towards the end of this article, readers will find a couple of examples of the correspondence between the mathematics and the musical concepts.

Universally compelling, however, is the human story of one man’s artistic and intellectual quest to unite disparate music systems, and how and why mathematics showed him the way.

Into the unknown

In 1991, Bangere arrived at Brandeis University, in Waltham, near Boston, from India, to do his PhD in mathematics. He arrived full of optimism, carrying a rosy image of the US. In a few months, he saw that despite its many obvious virtues, the US was also home to economic inequality, racism and imperialism. As a more realistic picture of his adopted country began to take shape and the biting winter set in, a deep homesickness began to take hold of him.

Although trained as a Carnatic violinist, Bangere found refuge in the pathos of blues and jazz. He began listening to musicians from a wide variety of genres: Eric Clapton, Aretha Franklin, BB King, Leo Kottke, Thelonius Monk, Charlie Parker, Stevie Ray Vaughn, and above all, John Coltrane, who had tried to incorporate Indian raga music into jazz. All the while, Bangere continued to hone his Carnatic repertoire.

Bangere follows what is called the Parur style, named after the native village in Tamil Nadu of the violinist Sundaram Iyer, the father and guru of MS Gopalakrishnan (1931-2013), often known just as MSG. Bangere, who grew up in Mysore, learned from HK Narasimha Murthy, a wonderful musician and a guru of great repute. Narasimha Murthy learned from MSG, imbibing his teacher’s deep and liberal outlook towards music. In Mysore, from a very young age, Bangere was also exposed to vast amounts of Western classical violin through his father’s friends.

MSG was a pioneer. He became a rare maestro of Carnatic music who also mastered the Hindustani idiom. Like his father, MSG believed that both systems harked back to a deeper, abstract musical base. He developed revolutionary new bowing and fingering techniques that enabled him to achieve startling clarity at dizzying speeds. He also created a completely new sound that exploited the violin’s particular potential, diverging from a longstanding dominant trend among instrumentalists of largely trying to reproduce the nuances of vocal music.

Bangere inherited not only the Parur technique but also its underlying iconoclasm and universalistic philosophy. After getting his PhD in 1996, he did two post-doctoral stints: at Oklahoma State University in Stillwater and at the University of Missouri in Columbia. All the while, he continued to widen and deepen his interest in the blues and jazz, listening to many musicians live. He finally landed a job at the University of Kansas, Lawrence.

In 2008, David Balakrishnan and his Turtle Island Quartet performed at the university. Bangere invited Balakrishnan for dinner and played for him, rendering a couple of ragas and showing him some Parur drills. “This is fantastic,” David told him, heralding the start of an intense musical collaboration that continues to this day.

In 2013, Bangere also began meeting jazz musicians in the Boston area, his old hunting ground. He played with jazz saxophonist Phil Scarff, who also trained in Hindustani music and founded a world-jazz ensemble called Nataraj, and befriended Marc Rossi, a composer and pianist at the Berklee College of Music.

At the same time, Bangere was neck-deep reading Récoltes et Semailles, a radical perspective of geometry and space, by Alexandre Grothendieck (1928 to 2014), and Séminaires du Géométrie Algébique du Bois Marie, the transcript of a nine-year-long seminar that the French-German mathematician ran. Bangere had been introduced to Grothendieck’s ideas at the Tata Institute of Fundamental Research in Mumbai, where he spent a few years as a graduate student before transferring to Brandeis University.

Considered one of the 20th century’s most influential mathematicians, Grothendieck revolutionised the subfield of algebraic geometry. He unified a slew of existing concepts by seeking their underlying abstractions. His theory of schemes and topological theory, to cite just two examples, dramatically expanded the field. Of the various dimensions of mathematics, he said that he was fascinated less by “size” and “number” and more by “form”, particularly “the structure hidden in mathematical things.”

Eureka moment

Propelled by the ideas of two pioneering intellects, namely MSG and Grothendieck, Bangere had his eureka moment some time in 2015, when he began to see that he could use Grothendieck’s ideas to integrate various musical forms under one umbrella theory. The metaraga system gradually began to take shape. But the ideas still had to be translated into concrete music, with compositions that could be performed and recorded.

Bangere excitedly called Balakrishnan, and the two began composing pieces, meeting on and off to perform them, roping in Harshberger and Kavthekar. The skeleton that was the metaraga system gradually acquired flesh and blood. The importance of Balakrishnan’s contribution to the process cannot be overstated. With decades of experience as a director of one of the most reputable jazz ensembles, he has been pivotal in pulling the project together. Musically, in the various compositions, his harmonies epitomise that elusive trait called taste. Lush yet delicately wrought, they beautifully support Bangere’s captivating and at times, haunting, main melodies. Harshberger and Kavthekar are superlative, their accompaniment discernibly elevating the compositions.

“David has been like a brother and mentor to me,” said Bangere. “His contributions are invaluable. His harmonies for ‘Triality II’ are brilliant. He has been centrally responsible for putting me out into the world. Jeff (Harshbarger) is immensely gifted, and Amit’s rhythmic groove is outstanding.”

To be clear, Grothendieck’s work does not provide formulae that, when fed some numbers, will output music: the relationship between the math and the music is not mechanical. Rather, Grothendieck’s ideas offered structural or geometric principles, which Bangere began connecting to music.

Bangere’s mathematical knowledge therefore formed only one part of the equation. In order to translate the mathematical structures into music, Bangere had necessarily to tap his musical knowledge, namely his long years of training in Carnatic music, deep understanding of Hindustani music, and sustained exposure to other genres, such as blues, jazz and Western classical music. In other words, mathematics gave him the structure, music the aesthetics.

The link between mathematics and music is perhaps not surprising. Neither refers to an external reality; they are internally consistent languages of pure abstraction. This might be one reason why an unusually high number of mathematicians are accomplished amateur musicians, and why, conversely, Western music composers, such as the American Milton Babbit (1916-2011), have also used mathematical ideas in their work, said David Eisenbud, a distinguished mathematician who was Bangere’s doctoral advisor and the first ‘David’ in his life — Balakrishnan being the other.

Another reason so many mathematicians take to music may lie in the personality of the practitioners. Smiling, Eisenbud said that mathematics tended to attract introverts, and like mathematics, music allowed its practitioners to access emotions without interacting with people!

Eisenbud himself learnt to play the flute before switching to singing. He became seriously interested in Indian classical music through colleagues and students from this country, hosting many concerts at his home.

“Purna is a fine mathematician, and has published in many very good journals,” Eisenbud said in an interview in mid-December, when he was visiting Mumbai on work. “His metaraga system is compelling.”

Not fusion

Where does the metaraga system fit within the larger world of multi-genre collaborations? After all, such cross-pollination and fusion are hardly new. For decades, musicians from different streams have worked together to find common ground. Among Indo-Jazz unions, the band Shakti was perhaps the most successful in consistently generating a new fused sound. But the vast majority of efforts were the result of trial and error — some pieces of music came together well, others, as is inevitable in any experimentation, remained disjointed. It is unlikely that any was underpinned by a coherent theory.

Indian musicians have also experimented with Western-style key changes through processes they call moorchana and grahabhedam. But these do not incorporate polyphony in a systematic way. Normally, in the Indian system, musicians do not change keys, sticking to one base ‘sa’ for a full concert, if not their whole lives. By contrast, a single Western composition can have several key changes.


Above image: Metaraga will be released worldwide on 17 January. Photo courtesy Origin Records.

Bangere’s music departs from fusion and goes beyond moorchana experiments because it emerges from a systematic framework, one with fundamental building blocks and grammatical rules. It emphatically does not fuse genres or try to find a middle ground; rather, it encompasses key elements of the Indian and Western classical systems — hence the term ‘meta’ in its name. This is why the transitions between the various genres sound so seamless in the various compositions on the CD.

The metaraga system expands the concept of the raga. The building blocks for a metaraga are a collection of several families of infinitely many, continuously varying raga structures. Only some raga structures will be actual Indian ragas. In fact, in some pieces, Bangere has used raga structures that are not in the Indian system.

A typical member of a collection has two components; a family of raga structures that are structurally similar and a fixed base frequency upon which this entire family is viewed. This fixed base frequency differs for each family in a collection. A metaraga is a cohesive patching of the members of this collection: a specific patching gives rise to a metaraga. So one collection can yield several metaragas.

Underlying this system are complex mathematical ideas. These are relevant to lay readers only for the purpose of providing a glimpse into Bangere’s creative process. A couple of examples will suffice: raga structures correspond to mathematical entities called fibres, while a family of raga structures is the musical equivalent of a set of fibres called a fibration, which is also the name of one of the tracks on the CD.

In a metaraga, one can freely change keys by moving from one raga structure to a neighbouring one. In order to maintain an Indian sound in the aalaap sections, or slow passages, however, one that is continuous and not abrupt, one needs to use microtones and oscillations. But often, these links are not to be found in the Indian system. Mathematical concepts in deformation theory spurred Bangere to try out blues-like microtones to bridge different raga structures, which he did using his training in the blues. It turned out to be technically and aesthetically the best solution.

Another dimension

Crucially, the metaraga system allows a musician to move not only vertically along one or more raga structures, the way an Indian musician does, but also horizontally, the way Western musicians do, across several raga structures — either travelling linearly across one point each in several raga scales, or zigzagging through fragments of different raga structures.

Indeed, a part of the system’s power comes from the way it opens up a whole new dimension, one that allows an Indian musician to enter the world of Western polyphony.

“When Purna starts to leave the ‘sa’ in his melody, it is not yet polyphony, but he is implying the possibility of polyphony,” explained Balakrishnan, who has been working with Bangere since 2008. “That’s where I come in. I take his melodic line and I voice them in my system and make it architecturally like a Western piece.”

“The most important thing that he is doing is finding a way to enter polyphony through the mathematics-based system and to change keys in a great, subtle way, without himself learning the Western system,” Balakrishnan said. “Western music is very patterned and mathematical, so his system allows him to approach the mathematical bedrock of polyphony in his own sophisticated way. He’s adding gamakas (oscillations) to known Western chord progressions, and nobody has done that to that level that I have heard.”

Take ‘Syzygy’, included here. The main melody, played by Bangere, begins in raga Dhani. It then moves away from the raga structure, along horizontal components. After a brief violin solo, it undergoes Western-style counterpoint, with a key change to raga Durga. After a Durga solo, it again undergoes a modulatory transformation, this time to raga Malkauns. The composition ends with a rhythmic coda containing two key changes, first to raga Durga and then to a non-raga structure. The piece ends with the blues version of Malkauns.

There is serious Western-style polyphony throughout, especially in the coda, in which Balakrishnan and Harshbarger are the second and third voices. Bangere, for his part, plays the lead voice using the full microtonal facility and oscillations of Indian music. One of the reasons the metaraga framework is so powerful is that it enables the seamless integration of Indian microtones with Western polyphony.

An even more powerful example of the system’s capabilities is ‘Triality II’, because Bangere is playing completely away from ‘sa’ and has left the world of raga far behind.

“He is playing all these chromatic, modulatory lines that he wrote using his system, which allowed me explore all these different Western styles,” explained Balakrishnan. “I am writing equal and active melodies below his melody, ones that are not static but really moving, because that’s my skill, and it’s fun, and they follow the implications of what he’s written.”

These are the technical possibilities. The aesthetics, without which sounds will not be pleasing as music, will evolve over time and with more collaborations. It will also depend upon how many students Bangere can train. Therefore, whether the metaraga system becomes just one man’s personal vision or finds wider acceptance and practitioners in the music community remains to be seen.

Balakrishnan believes that, at the very least, the metaraga system has given Bangere a unique voice.

“The most important thing for me, as his friend, is that he has found a way to be complete inside himself,” Balakrishnan said. “He has found a way to express the experiences in his life — growing up in Mysore and then travelling to America and falling in love with all these different styles, as well his love of mathematics. He has found his voice. You recognise that in another musician. I had a similar thing happen to me when I was in my 30s, and I was on fire, just like he’s on fire now.”

“But he is going to work with other musicians, and each of them will choose a way in which the Western elements occur,” he continued. “Mathematicians will also be testing him. If you have a beautiful theory, people will test you, and it could grow and change, and it could become bigger than you. History will decide.”