Symmetries in the Harmony of Nature

by Shane L. Larson

In 1619, Johannes Kepler published “Harmonices Mundi” (Latin for “The Harmony of the World”), in which he described his discovery of the Third Law of Planetary Motion.  It was the first empirical description of the  “musica universalis,” the ‘music of the spheres’ that medieval philosophers had appealed to in order to describe the motions of celestial bodies.  This naming was not by chance; it reflects the rather profound fact that a simple, consistent mathematics underlies many phenomena in Nature.

The mathematics describing sound and music shares a deep heritage with the mathematics that describes the repetitive motion of the planets in their orbits, which in turn shares a deep connection to objects that exhibit repetitive motion like pendula or spinning wheels.  In science we give a name to these phenomena that will be at once familiar and pleasing to music aficionados: harmonic motion.  The “harmony” of the motion derives from two fundamental observations.  First, the repetition of the motion can be thought of like a wave, a motion that repeats itself in time.  Sound itself is a wave phenomena, an idea that was first put forward by Greek philosopher Chrysippus in 240 B.C. and was solidified by Marin Mersenne (the “Father of Acoustics”) in 1636.  Sound, like ripples on the surface of a pond, is little pieces of air moving back and forth in place.  Second, increasing the speed of repetitive motion exhibits pleasing symmetries.  With sound, this is most familiar in the “octave scale” of music.  Two musical notes, separated by eight well defined steps in pitch (a sound one octave above another is double the frequency of the first sound), resonate together, often in a pleasing way.  Orbits can be the same — if you change the size of an orbit, you change how long it takes a planet to repeat each circuit.  Neptune and Pluto show a “harmony of motions” — for every two times Pluto orbits the Sun, Neptune orbits exactly three times.  Similar harmonies are seen in the cold shining moons of Jupiter — for every single orbit made by grey Ganymede, icy Europa makes two orbits and volcanic Io makes four orbits.

The mathematical connections between sound and music and harmonic motion are astonishing, but common.  As physicists often quip, “Everything in Nature is a harmonic oscillator.”  The metaphorical idea that the Cosmos is alive with musical harmonies is appealing, owing to the deeply pleasing nature of music. Humans ascribe great meaning to music, witnessed by the fact that we often label our most successful achievements with musical adjectives — a monumental work is a “symphony,” and peace among the myriad cultures of the world is “harmony.”  Why is music so special?  By and large, humans find symmetry and patterns pleasing in many facets of our sensory interface with the Cosmos.  Our brains are good at discovering and parsing patterns; they always have been.  Our recognition of systematic behaviour in Nature naturally leads to one of the most profound ideas of the collective human consciousness: the Universe is knowable.

There are rules that describe the goings-ons of everything: if you increase the pitch of a note by an octave, it will resonate with the first note.  If you have a planet farther from the Sun, it takes longer to come back to its beginning point.  If you throw a stone up into the air, it comes back down to the ground.  We call these rules the “Laws of Nature.”  We write these rules down in a common language, called mathematics.  We can use these rules to figure out how the world is going to behave, how things are going to change, and what the consequences of certain actions might be.  The Laws of Nature embody all that we know about the regular patterns and symmetries of the world.  Of course, Nature does not have to play fair nor make these rules easy to discover or understand.  But if they can be discovered, if they can be understood, then we can know the world around us.

Our brains are so good at deducing symmetry, that an entire field of science has emerged whose specific goal is to destroy and obfuscate symmetry.  We call this science “cryptography.”  The basic premise of of cryptography is to systematically hide easily recognizable symmetry (like the words on this page) using a set of rules that can easily be undone (if they are known).  One of the simplest encryption algorithms is the “substitution cipher,” which employs a letter by letter substitution to hide a message.  To create a simple substitution cipher, write out the whole alphabet twice, once forward and once in reverse:

ABCDEFGHIJKLMNOPQRSTUVWXYZ   ZYXWVUTSRQPONMLKJIHGFEDCBA

To encrypt a message, you take a letter from the top row and substitute the corresponding letter from the bottom row.  The message “music is the sound of Nature laughing” would encrypt as “nfhrx rh gsv hlfmw lu Mzgfiv ozftsrmt.”  An encrypted message such as this appears as utter gibberish, but can easily be understood if you know the rules (let’s call these rules the “Laws of Cryptography”) that created the message.  If you don’t know the rules, then you must look for patterns and hidden symmetries that reveal the true nature of the message.

There is an easy way to break a substitution cipher like this, with its own musical name: frequency analysis.  Written language is the repetition of a few letters, organized in patterns called “words.”  Some letters in English appear more often than others.  In any given body of English text, the letter E is about 12% of the letters, T is about 9.1% of the letters, A is about 8.1% of the letters, and so on.  With enough text, one can look at the whole encrypted message and evaluate the number of times a given symbol appears, surmising that the encrypted letter appearing the most often might be an E, and the next a T, and so on.  One rewrites the message using educated guesses about the identity of letters to decode enough parts of the pattern that your brain can deduce what the true text really says.  Even in our attempts to hide and destroy the symmetry of Nature, patterns remain that can be used to discover that which is hidden to cursory inspection.

Like Nature, a cryptographer does not have to play fair.  Exceptionally complicated mathematical algorithms are utilized in text encryption, immunizing messages to simple symmetry investigations like frequency analysis.  If the algorithms are known, messages can easily be understood.  If the algorithms are not known, it can be all but impossible to decipher messages, even using today’s modern computers.  Interestingly, one important approach to cryptography is to use prime numbers, a subject for which the father of acoustics, Marin Mersenne, is well known.  The mathematical algorithms used to encrypt messages are often seeded by a large prime number which is kept secret.  The resulting encryption can be alarmingly random in appearance.  But with the correct prime number in hand, the mathematical algorithms can be accurately unfolded, reversing the encryption in a precise mirroring of the original Law of Encryption that was used in the first place.  Mersenne probably would have found this symmetry pleasing, a harmonious merging of math, language, art and subterfuge that shows Nature’s unfailing knowability.

I Can’t, but I Can

by Michelle B. Larson

I can’t sing.

In second grade mom was called into a parent teacher conference. The teacher asked if she could talk to me about mouthing the words when the class sang. I sang out of tune (enthusiastically, but out of tune). Apparently, I was distracting everyone else. Clearly, singing doesn’t come naturally. Enthusiasm, screw it.

In college I held a job at a Rangeland Insect Lab. I spent months in Montana pastures sweeping up grasshoppers with only my headphones and singing to pass the long, dry, hot days. I loved the music so much I sang while at the microscope back in the lab. I sang until a fellow college student pointed out that I can’t sing. Why did I feel so compelled to do so? Could I, please, stop.

I can’t sing.
But, wait. I can learn.

Junior year in college I took singing lessons. It started rocky. But, I could see the professor believed in me. He politely ignored my out-of-tune bellows. He ignored them for weeks. Offering technique pointers now and then, but never questioning my ability to, someday, get it right. Then, someday came. I’ll never forget his face. Yes! Yes! Michelle, Yes! You’ve got it.

I can sing!

I joined the choir. Turns out I’m a high soprano. Who knew. I can sing! What doesn’t come naturally was conquerable with hard work and perseverance. I can learn. Singing still doesn’t come naturally. And, high soprano doesn’t sound great along with the car radio. But, I know, when I work at it, I can sing!

You can’t do math.

Perhaps mom and dad say they can’t do math either. Yet, math is interesting, especially as it applies to the world around you. But, math doesn’t come naturally. Enthusiasm, screw it.

You can’t do math.
But, wait. You can learn.
If you work at it, you can do math.

You can do math!

grant us peace

by Adam Johnston

It was with the Vienna Boys Choir that I first sang Dona Nobis Pacem — Grant Us Peace.

Okay, I was never really a member, but I did sing with the Vienna Boys Choir. We were in the same concert hall and we were directed to sing at the same time. It just so happened that I was shoulder to shoulder with the boys of my own elementary school choir, and the Austrians were on the stage. We sang together, as directed, and it was beautiful. It was a big deal at the time. It didn’t create world peace, obviously, but it didn’t hurt, either.

I’ve been thinking about that canon lately. At Christmas, especially, I find myself poking it out of the keys of the piano, the pedal heavy on sustain. The tune, the progression of the chords, the simplicity of the sequences of notes, it all draws me.

Music is one of those things I try to teach in classes. “The Music of Physics,” I call it, rather than the converse, not simply to be witty but to make that point that what we study can be both about the beauty of nature and the nature of beauty. That’s a heavy load to bear, but one I think we regularly try to carry in physics class.

The physics of “good” music is a tricky thing. We would like to believe and even teach that the beauty of the music comes from the relationships between the frequencies. An octave is defined by a factor of two in frequency between two notes. Thirds, fourths, fifths, and all other intervals you hear in chords and scales are each defined by factors, so that each major key may be made up of distinct notes, but the same proportions from one note to the next. When played together, the difference between any two notes becomes another unique “beat frequency.” So, for example, the difference between two notes separated by an octave is the same as the lower note (delta = y-x where y=2x, so delta = 2x-x=x). We think of an octave’s difference as really being the same note simply shifted up or down — so the baritones can sing in unison with the sopranos. On a piano keyboard this just shows up as a different place on the array of keys, but the same note within the pattern of repeating white and black. This is wonderful consequence of this geometry.

What’s really splendid is that this doesn’t have to be. It’s amazing, come to think of it, that this could work at all. Why would two notes with completely different frequencies, ever have any chance to be the “same,” or in “tune” with one another? I suppose the brain hears each note, and hears the difference between these notes, forms the beats in some physiological and mental way. Hearing two notes an octave apart, there’s a pattern that’s detected that creates that beat frequency that’s exactly equal to the base, bass note. Other differences between notes (as well as differences between harmonics heard on a singular note of a singular instrument) that sound good together are created by similar geometry. The difference in frequencies is itself a frequency that is in phase with the notes themselves.

That’s one thing I think about when I’m playing Dona Nobis Pacem on the piano. This chord progression is almost the least complicated one can imagine. A garage band of 17-year-olds could be cranking away on the three chords and we’d scoff or grin at the simplicity of it. And, because the three lines can be sung in unison or in round, they are each modeled after the same pattern. (Pull out your dusty guitar and strum G-D-G-D-C-G-D-G; repeat two more times and you can accompany my colleagues from Vienna.) It’s all so fulfilling that we go to the trouble to re-create it in contraptions like tubas and cellos and pianos. Considering all the design work put into the inner workings of my piano, I realize that these simple patterns are something we are driven to re-create. Not only are the patterns there in nature, but we’re hard-wired and compelled to construct them over and over.

So, I think understand how the difference between two notes produces a beat that itself is in tune with either of the notes; and this geometry of the physics of the music makes it pleasant, interesting, coherent. And then we play with patterns in this, both in the sequences in time as well as the audial space. In class, during that “music of physics” unit, I teach this with a tuning fork, a keyboard, a microphone, and an oscilloscope. I swing a tube over my head to play a series of harmonics that emulates “Taps” and they all laugh at my cleverness, or silliness — or perhaps out of sympathy for the bizarre instructor. I can get an organ tube to resonate with a piece of wire mesh and a blowtorch; and this phenomenon can be used to calculate the temperature of the room. I know where to stroke a violin bow on a piece of aluminum in order to play that octave’s difference, and we can see the standing wave set up by sprinkling some corn meal on the surface of the metal. I’m delighted with how much we can understand, as well as what we can do with it.

But then I throw my hands up and turn up the music. I’ve been known to tear up when, for example, Brandi Carlile sings Hallelujah. When Clapton has his hands on a guitar I understand why anyone called him God, and I believe. When Bach is played on a church organ I forgive him for all the pieces I had to learn in his name (though I still curse him, the asshole, when my fingers return to those pieces). Or when I find the right notes for peace, for prayer, on a piano. My left hand finds a way to march up a scale and meet my right hand in the middle, and I think, maybe, this is what I was meant to learn from it all. The physics of the harmonics and the physicality of sound is all beautiful, but it’s only as beautiful as we choose to make it. Dona Nobis Pacem. It’s stunning, and maybe my own version and belief in a miracle, that such a sentiment can come out of three chords. Grant us peace.

December’s prompt is … music!

As the title says, December’s prompt is music.