Math meets music

March 25, 2011 By Libby Fairhurst

Geometry is the force that shapes both the sound of music and the novel research of Florida State University composer-theorist Clifton Callender, whose work explores and maps the mathematics of musical harmony.

Now, Callender’s latest article on that mind-bending research has earned him the inaugural David Kraehenbuehl Prize from the Journal of Theory (JMT), the oldest and most distinguished music-theory journal now published in the United States.

In its citation of his work, the selection committee sings the praises of its first honoree, noting “Callender develops novel ideas in imaginative ways, harnesses a sizable mathematical apparatus with technical aplomb, and presents his work with exemplary elegance and clarity.”

Named for the founding editor of the JMT, the Kraehenbuehl Prize will be awarded biennially to an article judged the journal’s best submission from a not-yet-tenured scholar.

The theory described in Callender’s Kraehenbuehl Prize-winning paper (“Continuous Harmonic Spaces”) complements the “geometric music theory” he formulated at Florida State three years ago in collaboration with researchers from Yale and Princeton universities. The 2008 paper they coauthored, “Generalized Voice-Leading Spaces,” was featured in the journal Science.

But while Callender’s present work is related to that earlier milestone, it also breaks new ground and draws upon different branches of .

“For ‘Continuous Harmonic Spaces’ I used a mathematical technique called the continuous Fourier transform to investigate the aural quality of individual chords and the way in which those qualities differ from one chord to another,” said Callender, an associate professor of composition in the Florida State University College of Music. “It is after all a chord’s innate sound or ‘feel’ that makes it perfectly suited for depicting, say, a murderous turning point in a Hitchcock thriller, but not at all appropriate for an uplifting song.”

While the language of music theory may sound a little strange to the uninitiated, for Callender those continuous voice-leading and harmonic spaces are inextricably linked to the old, familiar tunes.

“As we know, some chords sound more similar than others,” he said. “For instance, while there are several different types of chords in the Beatles’ ‘All You Need is Love,’ all of these chords have a lot in common. Certainly they are more similar to each other than to the bebop-inspired chords of Jimi Hendrix’ ‘Purple Haze,’ or the even more dissimilar dissonant stabs in Bernard Herrmann’s score for the 1960 film ‘Psycho.’

“Imagine these and other chords as existing in a multi-dimensional harmonic space in which similar-sounding chords are located close together and dissimilar chords are far apart,” Callender said. “Music theorists, including my FSU colleague Michael Buchler, have developed ways to map this space and measure the similarity of chords built on a limited number of ‘note types,’ the twelve notes within a single octave.

“But in ‘Continuous Harmonic Spaces’ I map all possible chords –– including those that do not belong to standard Western tuning because they contain notes that lie in between adjacent keys on the piano. I felt it was essential to do this for two reasons.

“First, because the music of many contemporary composers and of non-Western cultures is not limited to the standard Western tuning,” he said.

“And, second, because by looking at the most general case of all possible chords, we can better understand the nature of harmonic spaces and shed light on the relationships and similarities between more common Western .”

Callender is a composer who practices what he theorizes. The notion of ‘continuous spaces’ is important in several of his compositions, including “Metamorphoses,” in which the rhythms and tempos change in a continuous, gradual manner. And he currently is working on “Spira mirabilis,” a set of music canons (or rounds) in which a given melody can be played against itself in an infinite number of ways.

“It is gratifying,” he said, “not only to be the first Kraehenbuehl Prize recipient but also to be a part of a community of musician-scholars who are all pursuing such interesting, cutting-edge research at the intersection of music and mathematics.”

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not rated yet Mar 25, 2011
I used a mathematical technique called the continuous Fourier transform.

This person is literally sitting in front of a physical geometric interpretation of Fouriers' transformations.

Imagining just the strings (of, in this case, the piano), you literally see the geometrical progression.

Now, imagining just the stereocilium (of, in this case, the human ear), you literally see the geometrical progression.

(No imagination necessary in either examples. Both are depicted with actual photographical images).

Yes, dear skeptical science readers, it is merely all coincidental. Really? Oversimplification?

Well, if quantum information theory and energy distribution is an oversimplification of Nature, then Nature will be the first to let us know - if we botch the incision of Nature with cutting-edge research.
1 / 5 (1) Mar 28, 2011
Roll over Johann Sebastian Bach... esoteric, scientistic dilletantism with a strong reductionist flavour...

The formative grasp of a thematic idea in composition is a deeply intuitive approach that lies at the cutting edges of the potentials of consciousness, and informed by technique and technical, compositional expertise. The modernistic c***p that is turned out by scientistic synthesis is marked by the aphorism "von Fachkentniss nicht betrübt"...

There is a natural evolution in the compositional domain, but the influence of sensationalistic dilletantantism, and "effect'- oriented artifice is sorely regrettable.

not rated yet Mar 28, 2011

I do not understand your words, except for the German words. The German words make sense to me. The German adage makes sense, even in light of the article. The rest of your words does not.

Unless your words are a parody on the typical verbal plumage, almost all classical musicians and modern critics can be accused of displaying.

If you are saying 'modern day music and composition' with 'modern day means', lacks classical compositional understanding...,
then you are correct.

I harbor only one regret against those who abuse music as an instrument of torture. Inexcusable. Unforgivable.

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