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.
by Jennifer Ouellette, Nautilus | Read more:
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