Quora user Marc Ettlinger recently sent me a paper by Sherri Novis-Livengood, Richard White, and Patrick CM Wong entitled Fractal complexity (1/f power law) determines the stability of music perception, emotion, and memory in a repeated exposure paradigm. (The paper isn’t on the open web, but here’s a poster-length version.) The authors think that fractals explain our music preferences. Specifically, they find that note durations, pitch intervals, phrase lengths and other quantifiable musical parameters tend to follow a power law distribution. Power-law distributions have the nifty property of scale invariance, meaning that patterns in such entities resemble themselves at different scales. Music is full of fractals, and the more fractal-filled it is, the more we like it.
One of the best discoveries I made while researching my thesis is the mathematician Godfried Toussaint. While the bookshelves groan with mathematical analyses of western harmony, Toussaint is the rare scholar who uses the same tools to understand Afro-Cuban rhythms. He’s especially interested in the rhythm known to Latin musicians as 3-2 son clave, to Ghanaians as the kpanlogo bell pattern, and to rock musicians as the Bo Diddley beat. Toussaint calls it “The Rhythm that Conquered the World” in his paper of the same name. Here it is as programmed by me on a drum machine:
The image behind the SoundCloud player is my preferred circular notation for son clave. Here are eight different more conventional representations as rendered by Toussaint:
My last post discussed how we should be deriving music theory from empirical observation of what people like using ethnomusicology. Another good strategy would be to derive music theory from observation of what’s going on between our ears. Daniel Shawcross Wilkerson has attempted just that in his essay, Harmony Explained: Progress Towards A Scientific Theory of Music. The essay has an endearingly old-timey subtitle:
The Major Scale, The Standard Chord Dictionary, and The Difference of Feeling Between The Major and Minor Triads Explained from the First Principles of Physics and Computation; The Theory of Helmholtz Shown To Be Incomplete and The Theory of Terhardt and Some Others Considered
Wilkerson begins with the observation that music theory books read like medical texts from the middle ages: “they contain unjustified superstition, non-reasoning, and funny symbols glorified by Latin phrases.” We can do better.
Wilkerson proposes that we derive a theory of harmony from first principles drawn from our understanding of how the brain processes audio signals. We evolved to be able to detect sounds with natural harmonics, because those usually come from significant sources, like the throats of other animals. Musical harmony is our way of gratifying our harmonic-series detectors.
I’ve undergone some evolution in my thinking about the intended audience for my thesis app. My original idea was to aim it at the general public. But the general public is maybe not quite so obsessed with breakbeats as I am. Then I started working with Alex Ruthmann, and he got me thinking about the education market. There certainly a lot of kids in the schools with iPads, so that’s an attractive idea. But hip-hop and techno are a tough sell for traditionally-minded music teachers. I realized that I’d find a much more receptive audience in math teachers. I’ve been thinking about the relationship between music and math for a long time, and it would be cool to put some of those ideas into practice.
The design I’ve been using for the Drum Loop UI poses some problems for math usage. Since early on, I’ve had it so that the centers of the cells line up with the cardinal angles. However, if you’re going to measure angles and things, the grid lines really need to be on the cardinal angles instead. Here’s the math-friendly design:
Octaves are notes that you hear as being “the same” in spite of their being higher or lower in actual pitch. (Technically, notes separated by an octave are in the same pitch class.) Play middle C on the piano. Then go up the C major scale (the white keys) and the eighth note you play will be another C an octave higher. The “oct” part of the word refers to this eight step distance up the scale.
From a science perspective, octaves are pitch intervals related by factors of two. When a tuning fork plays standard concert A, it vibrates at 440 Hz. The A an octave higher is 880 Hz, and the A an octave lower is 220 Hz. Any note with the frequency 2^n * 440 will be an A. It’s a central mystery of human cognition why we hear pitches related by powers of two as being “the same” note. The ability to detect octave equivalency is probably built in to our brains, and it isn’t limited to humans. Rhesus monkeys have been shown to be able to detect octaves too, as have some other mammals.
A musical pitch is a blend of many different frequencies beside the fundamental. Here’s a visualization of the different vibrational modes of an ideal string. The string’s movements are the sum of all these different modes simultaneously.
In high school science class, you probably saw a picture of an atom that looked like this:
The picture shows a stylized nucleus with red protons and blue neutrons, surrounded by three grey electrons. It’s an attractive and iconic image. It makes a nice logo. Unfortunately, it’s also totally wrong. There’s an extent to which subatomic particles are like little marbles, but it’s a limited extent. Electrons do move around the nucleus, but they don’t do it in elliptical paths as if they’re little moons orbiting a planet. The true nature of electrons in atoms is way weirder and cooler.
Pictures are a terrible way to understand the nature of quantum particles. Music theory is much better.
Music is richly mathematical, and an understanding of one subject can be a great help in understanding the other.
Geometry and angles
My masters thesis is devoted in part to a method for teaching math concepts using a drum machine organized on a radial grid. Placing rhythms on a circle gives a good multisensory window into ratios and angles.
The brain turns out to be adept at decomposing sinusoids into their component frequencies. We can’t necessarily consciously compare the partials of a sound, but we certainly do it unconsciously — that’s how we’re able to distinguish different timbres, and is probably the basis for our sense of consonance and dissonance. If two pitches share a lot of overtones, we tend to hear them as consonant, at least here in the western world. There’s a strong case to be made that overlapping overtone series is the basis of all of western music theory.
The concept of orbitals in quantum mechanics made zero sense to me until I finally found out that they’re just harmonics of the electron field’s vibrations. I wasn’t at all surprised to learn that Einstein conceptualized wave mechanics in musical terms as well.
Octave equivalency is really just your brain’s ability to detect frequencies related by powers of two. The relationship between absolute pitches and pitch classes is an excellent doorway into logarithms generally. You also need logarithms to understand decibels and loudness perception.
If you had to name the most influential drummers in contemporary music, who would you pick? If you’re a rock fan, you might go with Ringo Starr, John Bonham, or Keith Moon. A jazz fan might talk about Max Roach, Elvin Jones or Tony Williams. You probably wouldn’t think to name Gregory Cylvester Coleman. He was the drummer in a sixties soul band, The Winstons. His claim to fame is a five and a half second break in an obscure song called “Amen, Brother,” the B-side to the minor Winstons hit “Color Him Father.” That doesn’t sound like much of a case for Coleman’s importance. But his short drum break is widely considered to be the most-sampled recording in history, ahead of “The Funky Drummer” and “Apache” and “Cold Sweat” and all the rest of the classic breakbeats.
Here’s “Amen, Brother.” The famous drum break comes at 1:27.
Probability is a deeply weird and disturbing topic. The harder I look at it, the weirder and more disturbing it becomes. I find the many-worlds interpretation of quantum mechanics to be the least weird and disturbing way to think about it.