The Mystical Mathematics of Rock and Roll

By George Johnson | February 20, 2013 11:32 am

“The Siren,” Edward Armitage, 1888. Wikimedia Commons

Anyone who has wondered about the mysterious way numbers seem to be woven through the universe will eventually be led to a famous essay by the physicist Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Why, Wigner asked, are the laws scientists discover so readily expressed in terms of mathematical constants like π and in precise numerical equations? Wigner isn’t really able to answer the question, and pondering it leads down a rabbit hole of other mysteries: Is mathematics invented or discovered? And why is the music that sounds so harmonious to our ears based on simple mathematical proportions?

According to legend it was Pythagoras and his mystical cult of number worshippers who first glimpsed the musical connection while plucking the lengths of strings. Tune a string so it plays the note A above middle C, or 440 cycles per second. A string half that length will play the same note an octave higher — 880 cycles per second, and so forth. Other simple ratios will give you the fundamental notes of a chord. A string 3/4 as long plays a D (called the fourth or subdominant), and 2/3 gives you an E (the dominant fifth). The I-IV-V chord progression is the basis for almost all popular music. Pythagoras is the father of rock and roll.

There are quarter-tone musical scales in Middle Eastern civilization, and dissonances are savored by avant-garde Western composers. But it is the music rooted in the Pythagorean tradition that has come to dominate the world. There seems to be something fundamental connecting its simple mathematics and the architecture of the brain.

Of course McDonald’s restaurants are also sweeping the planet, and a recent post by British science writer Philip Ball on the BBC website suggests the jarring possibility that harmony may be something that is learned, not innate. Maybe Pythagoras discovered a system of notes that comported to exact mathematical proportions and then, over the centuries, it was drilled so deeply into our heads that we learned to love it, whether in the intricate renditions of Mozart or the stripped-down simplicity of the Rolling Stones.

The occasion for the BBC post was a new paper in the Journal of Experimental Psychology: Consonance and Pitch. After describing their experiments at the University of Melbourne, Neil McLachlan and his team conclude that “harmony results from the adaptation of sensory systems to reproducible and recognizable stimuli, regardless of their physical properties.” If harmony is an acquired taste, the authors propose, that may explain “the diversity of music tunings that do not conform to simple mathematical proportions that have emerged in isolated societies all over the world.”

Pianos, in fact, are not tuned precisely by Pythagorean ratios. To be capable of playing harmonious music in all keys, compromises must be made according to a system called equal temperament tuning. If A is 440 cycles per second then E should be 330. Instead it is 329.628. Nothing seems very elegant about that.

I’m finding it hard to shake the feeling that there must be some profound neurological-cosmological connection rooted in mathematics. But now I’m starting to wonder. Could that belief be just so much numerological mysticism? The Pythagoreans believed all kinds of weird things. They were fascinated that the numbers in the musical ratios — the tetraktys, 1, 2, 3, 4 — add up to 10. That was supposed to be the perfect number. It was connected, they believed, with the Oracle at Delphi — and with the seductive song of the Sirens.

  • hilbertthm90

    Odd. I wasn’t aware of this recent research, yet just last week I made almost exactly the same (set of 3) blog posts: A Mind for Madness.

    I’d argue that one can actually derive our current tonal system using the overtone series rather than the overly simplistic ratio method, and that the reason what sounds good to us sounds good is that it has its physical basis in the overtone series (something inherent in nature).

    Even music from other cultures which divides their tonal system up in another way still has a basis in the overtone series. While it is possible that you have to learn to like certain tonalities, I don’t think it is an “anything goes” type of situation.

    • ktygbns

      I dont agree– listen to chinese music sometime.

  • David

    Correction: A string half the length plays a note an octave higher.

    • byGeorgeJohnson

      Thanks for noticing that. I’ve made the fix.

  • http://twitter.com/dlpprogram dlp Music Program

    If you study a little jazz improvisation you eventually accept the Lydian scale (adding an F# to a C scale) and Dorian minor (D to D with no sharps or flats) among other more exotic sounds. But no matter how ya slice it, your basic G, E minor, C, and D chords or a standard blues progression generally sound pretty pleasant. It may be “learned” or it may be a bit of ‘laziness’, but it seems most folks are fairly content with basic chords and simple melodies. And that’s ok too. After-all, when you remove the math, you get down to human emotions, reactions, and communication – and therein lies the beauty.

  • http://www.facebook.com/dwayne.stephenson Dwayne J. Stephenson

    I thought the harmonic relations were logarithmic, and that’s why E comes out all funky when you set A to 440. There is a website here that compares log scales and linear scales (the linear scales sound terrible).

    http://www.phy.davidson.edu/fachome/dmb/JavaSounddemos/scales.htm

    I also read somewhere that Galileo’s father Vincenzo had something to do with modernizing the pythagorean music scales, but I’m having a hard time figuring out exactly what it was he did to modernize the music system. His problem with the practices of the time was that they sounded terrible, which presupposes that wherever he got his ideas about what good music was supposed to sound like, it wasn’t the music being played in the tradition he was raised in (cuts against the theory that harmony is purely cultural). He was especially suspicious of the music theorists that were relying on the Pythagorean influenced math to determine musical scales, which is what drove him to a more experimental approach.

  • http://www.facebook.com/people/Colin-William-Den-Ronden/1375054437 Colin William Den Ronden

    I have often wondered what distinguishes music from random noise (I am not a musician). When the news item about Pi being converted into music was announced I listened to it. It sounded partly musical. If they flow smoothly up and down they are more pleasant than ones that jump all over the place. Separate from that is the timing of and between the notes. So repetition comes into it. But how do you recognise sounds spaced wider and wider apart as being a piece of music? The cosmos is sending off waves that can be translated into sounds, so what about when they are spaced 1 second, 1 minute, 1 hour, 1 day, 1 year or 1 century apart? Maybe it has something to do with how our nerves bounce back from repeated stimuli (called refraction). Just something I’d like to know the relationships between.

    • Steve M

      I teach band instruments to beginners, and I find my students enjoy trying to explain of the difference between music & noise in their 1st lesson. They often say music is played on a musical inst., so I play a rhythm with drum sticks on my office door…followed by me throwing the sticks at the door. Even though the door is not a musical inst., they then realize it has to do with rhythm, not necessarily the source. The Rap genre is so catchy & popular to so many people, even though harmony is so minimal or nonexistent, because they are drawn to the beat & rhythm of the song. I believe there is a tie between musical rhythm and our bio-rhythms. We memorize things much easier when they are set to rhythm; our heart rates will often follow the direction of the tempo (or speed of the beat) of a song: faster song=excitement & faster HR, slower tempo=relaxation and slower HR. Even though there might be slight or wide variations of harmony between the music of different cultures, one thing they have in common is rhythm.

  • Sunny D

    If we focus science on these kinds of things, surely we will have amazing breakthroughs in our understanding of the universe and science in general. But if not…well….

  • http://www.facebook.com/shane.milburn.5 Shane Milburn

    ” If A is 440 cycles per second then E should be 330. Instead it is 329.628.”

    If we never changed keys this would not be needed and most people would say the resulting music would sound “better.” Our musical machines are built to be multi-purpose devices to accomodate different keys so trade-off of slight dissonance is permitted and our mind seems to resolve things well enough.

    Here’s an interesting comparison of equal temperment vs. other slight variations in tuning.

    • Steve M

      Some might find it interesting to realize that professional level musicians all over the world, who play instruments that can “shade” the pitch slightly up or down, often do so in order to make intervals or chords sound more pure, or without dissonance. 8va’s, 4th’s, & 5th’s have a pure tone naturally, but if a brass trio would play a major chord or triad, they would shade or bend the 3rd note slightly flat, which makes the chord have a more pure tone. I would be interested how close this move is to reaching the quarter tone the writer spoke of. Players that do this aren’t thinking of a certain distance, but use their ears to make the adjustment till they accomplish the sound they want to achieve.

  • JazzZyx

    I appreciated the nearly poetical, songlike expression of language in this post as much as the information contained within.

  • Dmitri DB

    I came across this research while trying to find out why dogs ignore music:

    http://www.nature.com/nature/journal/v451/n7175/full/nature06476.html

    I found Gurdjieff’s musical theory to be pretty amusing stuff, especially in its examination of eastern music taking on a different tone. It’s interesting to consider what people used to connect in their own cosmologies of things. http://www.gurdjieff.org/werbock1.htm

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Whether a subtle new pattern shows up in an experiment on the Higgs boson, an epidemiological report about a suspected cancer cluster, or a double-blind trial purporting to demonstrate ESP, it can be maddeningly difficult to distinguish between what we see and what we think we see. "Fire in the Mind" takes a look at the big questions behind today’s science news.

About George Johnson

George Johnson writes about science for the New York Times, National Geographic Magazine, Slate, and other publications. His nine books include The Cancer Chronicles: Unlocking Medicine's Deepest Mystery (August 2013), The Ten Most Beautiful Experiments, A Shortcut Through Time, and Fire in the Mind. He is a winner of the AAAS Science Journalism Award and has twice been a finalist for the Royal Society science book prize. Co-founder and director of the Santa Fe Science Writing Workshop, he can be found on the Web at talaya.net. Twitter @byGeorgeJohnson.

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