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Math in music: Do numbers rule the universe?

March 25, 2020

Music is all emotion, right? For some it's a calculated, mathematical puzzle. We've used math to explain music and sound for thousands of years. Pythagoras said numbers ruled the universe. But he was wrong.

Harmony of the World, an illustration of the heliocentric universe from 1806
Harmony of the world Image: Public Domain

Call it human, a natural instinct — our species is desperate to understand and control nature, if not the entire universe. And we do it with numbers.

The ancient Greeks were the first "real mathematicians," says Eli Maor, a retired professor of the history of mathematics and author of Music by the Numbers.

Eli Maor, professor of history of mathematics (retired; Loyola University Chicago) and author of "Music by the Numbers", Foto: Eli Maor
Eli Maor, professor of history of mathematics and life-long music loverImage: Privat

Led in large part by Pythagoras, their motto, as it were, was "numbers rule the universe."

They looked at the cosmos as a single "unity of music, astronomy, geometry and number theory, which they called arithmetic," says Maor. "Music was ranked equal to science and they used it to explain the orbits of the planets and stars."

The cult of math

The Pythagoreans were a cult. They swore to keep their discussions secret. As a result, little or no written records survived. 

But we do know that Pythagoras experimented with vibrating strings. He found that if you divide a string by a ratio of 2:1, 3:2 or 4:3, and pluck the string, as you would on a guitar or violin, the resulting notes have a "harmonious relationship." They are in consonance. 

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Crop of a painting called "The School of Athens," a fresco by the Italian Renaissance artist Raphael. It shows Pythagoras among 21 ancient philosophers.
Pythagoras and his followers believed numbers ruled the universeImage: Public Domain

"And from that he made this huge leap of faith to say that the whole universe ran according to these simple numbers," says Maor.

The idea influenced science "negatively," says Maor, for a few thousand years, right up until the astronomers Johannes Kepler and Galileo Galilei came along.

"Kepler was a Pythagorean. A true believer. I dare say that 30 years of his short life were wasted, searching for the orbit of the planets in musical laws of harmony," Maor says. "Finally, he realized the idea was wrong."

That didn't stop a group of scientists at Yale University in the 1970s, among them Willie Ruff, a jazz musician and musicologist, from turning Kepler's inaudible planetary calculations into sound using computer synthesis.

More math in digital music 

"Everything is math when you get down to it. My talking to you now is being mediated through mathematical operations on ones and zeros," says Matt Black, a musician and creative software pioneer, who has a background in science.

"And people say music is basically mathematics — like harmony, relationship. I was never very good at math, I was into chemistry," says Black, "but I do have that respect for it. Math underlies everything."

In February, Black's music label, Ninja Tune, released an iPad app for digital music production and performance called Jamm Pro. 

Read more: One Thousand Hear Change of Note in World's Longest Concert

As a piece of computer software, it relies heavily on math. But it's also got mathematical principles built right into the user interface. It has as a X/Y pad that lets you influence sounds by moving your finger up and down, and left and right.

Book cover of Eli Maor's "Music by the Numbers," published by Princeton University Press
Image: Princeton University Press

"We talk about hand-eye coordination. This is hand-eye-ear coordination. I might not know I'm controlling the bit crusher [a sound effect] but my ears tell me 'Oh, that sounds good,' and the more I move my finger to the left, the more extreme the sound gets, I can add dynamics to the sound," says Black.

The key is coming home to harmony

Now think back to Pythagoras. His theory of universal harmony may have failed, but his ratios live on.

A ratio of 2:1 gives you an octave — two of the same note, with one pitched at double the frequency of the other. A ratio of 3:2, meanwhile, gives you a perfect fifth — from the root to the "top note" of a basic "major triad" chord.

The human ear yearns for such familiar structures, sounds that resolve, or "frames of reference," as Maor puts it. It's a tonality that gives us harmony — perhaps a similar sense of harmony to that which Pythagoras saw in the stars.

Tonality was a guiding principle in music from around 1600. The idea being that music was confined to a certain key and if it deviated, it had to return to that key.

The 20th century philosopher and music theorist, Theodor Adorno, with a musical manuscript
The 20th century philosopher and music theorist, Theodor Adorno, was an advocate of "Die Neue Musik" (the new music), developed by composers like Arnold SchoenbergImage: picture-alliance/akg-images

By the 1900s, composers such as Arnold Schoenberg started to think tonality had "run its course," says Maor. 

"Schoenberg set about replacing tonality with a form of serial music," says Maor. "It reminds me of Einstein's Theory of Relativity, because the frames of reference are all equal." 

It's an interesting view, given that Schoenberg and Einstein were contemporaries. They met, at least once, at a concert at Carnegie Hall in 1934. It was a meeting of music and math, personified. But that may also be where the connection ends.

Schoenberg experts will tell you there are no musical laws in the composer's music. 

Left to right: Composer Leopold Godowsky, Physicist Albert Einstein, Composer Arnold Schoenberg at Carnegie Hall, April 1, 1934, for a concert in honor of Einstein
Albert Einstein with Arnold Schoenberg (right) and Composer Leopold Godowsky at Carnegie Hall, 1934Image: Clyde Fisher/Arnold Schönberg Center

New music, new rules

Serialism, as it is sometimes known, came and evolved in various ways.

It applied rules to the way the notes of a scale can be placed in a piece, the way they appear in a series, and how they form a sense of unity in the music. Schoenberg disallowed repetition in a series of notes or dictate that all notes must be equal in value — that no single note may be emphasized more than another. 

It may not be math, but you would be forgiven for describing the technique as mathematical. Serialism creates patterns based on rules that can lead to a dissonance that our ears — our emotion — might otherwise reject, but the serial theory allows. Especially as the lack of repetition means there are few frames of reference.   

"I tried listening to Schoenberg's Wind Quintet five times. I looked for a frame of reference but couldn't find it," says Maor, while conceding that that's only to be read as his subjective feeling, not fact. 

Later, the musical theorist and philosopher, Theodor Adorno, suggested that the "Neue Musik" (new music) of the early 20th century was about more than just serialism, or non-harmonic rules. It was about expressing sounds that reflected the increasingly industrial, non-harmonic world around them, describing the universe as they saw it in art. Which is what the Pythagoreans had hoped to do before them.

But, then, Schoenberg's serial music accounts for only a brief period. Some of his best-known work is romantic and harmonious.

Interfacing with the world

Others followed the serialists. Steve Reich, Philip Glass, John Cage, Karlheinz Stockhausen and the musique concrète. 

"Music is an attempt to interface with whatever environment you're in, to make sense of it, and as the environment has become increasingly industrial and techno, that music has come to the fore in our attempt to understand our environment, how we're changing, and humanity is evolving," says Black. "To become a more techno-human-cyborg hybrid. To me, techno is that sound: a conversation between man and machine."

But there's no getting around it — the most timeless music is less extreme, a careful balance between harmony and dissonance, science and art, order and chaos. 

Read more: The world is music: John Cage

John Cage, experimental composer of pieces such as 4'33'' (1952)
John Cage's composition 4'33'', consisted of incidental, environmental sounds. The orchestra were instructed to not play their instruments for 4 minutes and 33 seconds. Image: Getty Images

"I think everything is a balance between order and chaos," says Black. "If you think of a sine wave [a smooth periodic oscillation in sound], a sine wave is an example of perfect order, it's a completely repetitive signal, and boring because you can predict it. White noise, on the other hand, is complete chaos. Chaos is full of information but it's shapeless, and it's not very useable. And I think life and consciousness exist at that phase-transition between order and chaos, at the meeting point."

Too much order is stasis, says Black, boring, nothing interesting ever happens. And when there's too much chaos,  nothing has a chance to coalesce or organize because it gets torn to pieces.

"A groove or a piece of music is something that has structure and some amount of repetition to seize onto as a pattern — humans are hungry for pattern, meaning — but music must also evolve and change, because life is like that. So, I see that music is a model of this relationship between order and chaos."

Is everything under control?

An obsession to control nature 

As systems, math and music are imperfect. But math likes to think of itself as pure — a system we've developed to explain, calculate and control nature. But if we could do that fully, "we'd be God," says Black.

Music, on the other hand, allows some imperfection, dissonance. We don't need to understand or control it fully. We can feel it. And perhaps that's why legions of top scientists play instruments, purely for the love of it.

"I have always considered and experienced music as a counterpart to my scientific interests," says Reinhard Brinkmann, an accelerator physicist at the German Electron Synchrotron (DESY), who plays a mean jazz piano (this writer can vouch for it). "Music has helped me stay healthy, mentally, and that's motivated me in my work."

Albert Einstein played violin, it's said, because it helped him think. But he played a beautiful Mozart, too. Which was, incidentally, Maor's favorite composer growing up.

"We are obsessed with controlling things," he says. "We think any unusual phenomenon has to have some reason, and that reason is based on numbers. But music was created to move our souls, to touch our feelings, our emotions. And that's why most attempts to subjugate music to math have failed."

Eli Maor plays clarinet.

DW Zulfikar Abbany
Zulfikar Abbany Senior editor fascinated by space, AI and the mind, and how science touches people