One of the last things you’d expect to see at a physics conference is a physicist on stage, in a dapper hat, pounding out a few riffs of the blues on a keyboard. But that’s exactly what University of Illinois professor J. Murray Gibson did at the recent March meeting of the American Physical Society in Baltimore. Gibson has been doing these wildly popular demonstrations for years to illustrate the intimate connection between music, math, and physics.
While there is a long tradition of research on the science of acoustics and a noted connection between music and math in the brain, science and math have also influenced the evolution of musical styles themselves. For thousands of years, Western music was dominated by the diatonic Pythagorean scale, which is built on an interval (the difference in pitch between two different notes) known as a perfect fifth: where the higher note vibrates at exactly 50 percent higher frequency than the lower note. Anyone who’s seen The Sound of Music probably gets the idea of the perfect fifth, and can likely sing along with Julie Andrews: “Do, a deer, a female deer….” If you start on one note and keep going up by perfect fifths from one note to the next, you trace out a musical scale, the alphabet for the language of music. While a musical scale built like that includes a lot of ratios of whole numbers (like 3:2, the perfect fifth itself), it has a fatal flaw: It can’t duplicate another keystone of music, the octave, where one note is exactly double the frequency of the lower note. Contrary to Andrews’ lyrics, the scale doesn’t really bring us back to “Do.”
To bring the fifth and the octave together in the diatonic Pythagrean scale, various versions of the same interval were forced to be different lengths in different parts of the scale—one was so badly out of tune it was called the “wolf fifth” and composers avoided using it entirely. This meant that a piece of music composed in the key of E sounded fine on a harpsichord tuned to the key of E but dreadful on one in D. It also made it difficult to change keys within a single composition; you can’t really re-tune a piano mid-performance. Johann Sebastian Bach, among others, chafed at such constraints.
Thus was born the “well-tempered” scales, in which each appearance of an interval was tweaked so that it was not far off from the ideal length or from other versions of the same interval, so composers and performers could easily switch between keys. Bach used this scale to compose some of the most beautiful fugues and cantatas in Western music. This approach eventually led to the equal temperament scale, the one widely used today, in which every interval but the octave is slightly off from a perfect ratio of whole numbers, but intervals are entirely consistent and each step in the scale is exactly the same size.
In the 20th century, musicians like Jelly Roll Morton and ragtime composer Scott Joplin wanted to incorporate certain African influences into their music—namely, the so-called “blue notes.” But no such keys existed on the piano; when in the key of C, one major blue note falls somewhere between E-flat and E. So blues pianists started crushing the two notes together at the same time. It’s an example of “art building on artifacts,” according to Gibson. That distinctive bluesy sound is the result of trying to “recreate missing notes on the modern equal temperament scale”: In more traditional scales, the interval called a third represents a frequency ratio of 5/4; and indeed in the key of C, a true third lies between E-flat and E.
Gibson always had a musical bent, and first heard about the “blue note” as a child, while reading a book by Leonard Bernstein from the 1950s. But his serious interest in physics and the blues began when he joined the faculty of the University of Illinois after eleven years at Bell Labs. He used various diffraction techniques to explore the structure and properties of materials in his research, which he sees as the equivalent of harmonic analysis in music. And he realized this would be a good way to bring the concepts of math and physics to life for non-physicists.
For instance, music provides an excellent analogy of the uncertainty principle in physics, which states that you can’t precisely measure both the momentum and position of a particle at the same time. The same is true of a musical tone: It exhibits a kind of particle/wave duality, related to the duration of the sound. “The momentum is equivalent to the wave nature and the position is equivalent to the particle nature,” Gibson explains. “It’s exactly the same physics and mathematics that controls this phenomenon with sound.” Particles are very short in duration, so they sound like percussion—“like banging on the table”—but they don’t have a well-defined pitch; a sound wave of long duration means you can measure the frequency very precisely. “The musical particle is the point in time where you can’t tell what the tone is, and the musical wave is the perfect tone whose frequency can be measured very precisely,” he says.
“We don’t really know how the brain works, but you have to believe that somehow we’re recognizing these fundamental rules.”
Gibson has a nifty way to demonstrate this using a synthesizer to create tailored wave packets of sound. (A wave packet is just the number of oscillations in a sound wave, determined by the decay rate of that wave.) He can program it to play a simple melody, like “Mary Had a Little Lamb,” and then gradually shorten the number of oscillations for each note until it is little more than a very short burst of noise. At that point, people can no longer identify a note as being C, D, or E, and hence don’t recognize the melody. He can do the same exercise in reverse, starting with short bursts and gradually increasing the duration until his audience can identify the tune.
The reason for this relates to how we perceive sound. The tone has to be long enough in duration so there are enough oscillations for your ear to measure the frequency well enough to distinguish between a C and a D. There is only about a 5% different in the frequency between those two notes, which means the ear needs around 20 oscillations to tell them apart. If it’s just a short blast with, say, a mere five oscillations, you can’t really hear the difference.
When Gibson talks to biologists about his ideas, they inevitably raise the question of why human beings evolved this kind of harmonic structure capability in the first place. It may be related to the vocal chords, which vibrate like strings. The ear, too, resonates on different parts of the membrane depending on the frequency of the sound it detects. “We don’t really know how the brain works, but when you see how much the structure of music is influencing the way we hear it, you have to believe that somehow we’re recognizing these fundamental rules,” he says.
Humans’ understanding of pitches may be connected with our rare ability to imitate and learn through sounds around us. Research suggests that a only few other animals, like whales, dolphins, and some birds, can engage in this type of “vocal learning”; no other primates seem to do it. Possibly as a result, we are also one of the few animals that can keep a beat well enough to dance in rhythm.
For Gibson, the rules and axioms that control the composition of music serve as a palette of colors for the composer, much like the laws of physics are nature’s palette for creating things in the world, and those constraints can lead to innovations like the blue note. “We take these kinds of artifacts, these mistakes, and we make them beautiful,” he says. “Constraints help human innovation thrive.”