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Music under Darwinism


Editor’s Note: The Atlantic features a debate between a pair of psychologists on the evolutionary background of the human ability to appreciate music, asking

Is music a deep biological adaptation in its own right, or is it a cultural invention based mostly on our other capacities for language, learning, and emotion? And if music is an adaptation, did it really evolve to promote mating success as Darwin thought, or other for benefits such as group cooperation or mother-infant bonding?

Good questions! We asked Dr. Robert J. Marks II for a comment. Dr. Marks, a founder of the Evolutionary Informatics Lab, is Distinguished Professor of Engineering in the Department of Engineering at Baylor University.
There is beauty in both a rose and a spider web spotted with morning dew. Roasting coffee beans makes my nose happy and there is nothing more delicious than fresh sweet corn. I have a new grandson and there are few experiences comparable to holding and rocking a baby to sleep. Sight, smell, taste and touch all resonate with nature for our enjoyment.
But what of hearing? The peaceful sound of a running stream or rain on a metal roof come to mind. And there is also music. Western music, like the rose and sweet corn, is a direct manifestation of nature. The beat of music is similar to the pounding of one’s pulse. And harmony is a direct result of physics.
The historical pioneer in western harmony is Pythagoras, who flourished about 500 BC. Pythagoras is best known for his triangle formula: the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. He also played with music. The story, disputed, is that Pythagoras walked by a blacksmith shop and noticed that some of the hammers sounded nice when struck at the same time while others were dissonant. Pythagoras began experimenting by plucking strings hung to the ceiling weighted by known masses. We now know Pythagoras discovered that frequencies related by ratios of small integers sounded good together.
Many years later, in the 19th century, Jean Baptiste Joseph Fourier showed that any tone could be represented by a fundamental frequency and its harmonics. Engineers, physicists and mathematicians know this math as the Fourier series. It comes from a solution of the wave equation for vibrating strings, e.g. guitars, and vibrating air columns, e.g. trombones. (The wave equation is a partial differential equation derived from physics that must be obeyed when vibrating or making waves.) If the fundamental frequency is f, the harmonics are 2f, 3f, 4f, etc. One of the reasons a trumpet and a violin playing the same note sound different is that the mix of harmonics is different. Harmonics, like good-smelling foods, are ubiquitous in nature.
Here’s what Pythagoras discovered. Going from f to 2f means you’ve gone up an octave. The next harmonics are 3f, 4f and 5f. If you play all of these notes together, you get a nice sounding major chord. These are nature’s harmonics. If you divide a frequency by two, you go down an octave. Thus 3f/2 is an octave below 3f, and 5f/4 is two octaves below 5f (divide by two twice). The frequencies f, 5f/4 and 3f/2 make a beautiful major cord. This is an example of what Pythagoras discovered. Notes that sound good together have frequencies whose ratios are small integers.
If you expand the third harmonic and the third subharmonic (f/3) into major chords, you get the major scale. If f were the frequency of the note C, then

C chord: f 5/4 f 3/2 f
G chord: 3/2 f 15/8 f 9/8 f
F chord: 4/3 f 5/3 f f

Arranging these notes in order gives the Pythagorean major scale.

1f 9/8 f 5/4 f 4/3 f 3/2 f 5/3 f 15/8 f 2 f

This is pretty close, but not exactly, to what you hear when you play the white keys on the piano starting at C. We have all the notes in this scale we need to make C, F, and G major chords and D, E and A minor chords. This Pythagorean musical structure flows directly from nature’s wave equation.
The problem with the Pythagorean harmony is that you can’t change scales. If we changed to the key of D, for example, A should have a frequency of 27/16 f instead of 5/3 f.
There is a compromise, though, that allows changing keys. We have to change each note’s frequency just a teeny bit. The result is the tempered scale. The way the math works is an astonishing coincidence of nature. There is no reason it should work. It just does. After adjusting (or tempering), adjacent note frequencies on the piano keyboard have a ratio of 2^(1/12) = the twelfth root of two = 1.05946. As a result, it turns out that if you measure the distance of adjacent frets on a guitar to the bridge, their ratio is 2^(1/12).
Does the tempered scale work? If we go seven half steps up the scale from f, we get the frequency 2^(7/12) f = 1.49830708 f which is pretty close to 3/2 f needed for the note G. Other notes are also close. Tempering the scale allows changing keys.
Bach celebrated the tempered scale by writing The Well-Tempered Clavier. In his composition, each of 24 scales — 12 major and 12 minor — are used. The tempered scale allowed him to play the entire composition on a single instrument. Today, everyone uses the tempered scale and changing keys is no big deal.
Harmony is born of nature. I believe, for that reason, we resonate with harmony in a manner similar to how we enjoy other aspects of nature. I suspect we no more had to evolve this enjoyment than we had to evolve our taste for fresh sweet corn.
For further details, including minor keys from subharmonic expansions and the possibility of using 18 notes per octave, see the first section of Chapter 13 of my book, Handbook of Fourier Analysis and Its Applications (Oxford University Press, 2009).