Equal vs. Just

Here’s a fun fact: Nearly every single song you’ve ever listened to has been out of tune. Ever-so-slightly out of tune, but out of tune nonetheless.

That’s because all modern instruments are tuned using a method called Equal Temperament. In a nutshell, Equal Temperament makes all music slightly out of tune so that music in any key sounds equally good.

Think of any musical note (A, D, F#, etc.) as corresponding to a specific frequency. In all modern music, the ratio between any two successive notes is fixed at the unusual ratio of ¹²√2, or 1.05946309436…

But that’s not actually how music should sound, at least not in theory. As recently as a few hundred years ago (during the Renaissance), instruments were tuned in a way that captured their natural harmonics, a method called Just Intonation.

To avoid going too into the mathematical details (although they are fascinating, in case you’re interested), here are a few examples of the difference in these tunings.

Say you were playing in the key of A. You’d set the frequency of your middle A to the standard 440 Hz. Then in Just Intonation (the old way of doing things), the note E would be tuned to 660 Hz. A nice round number. In Equal Temperament, by contrast, E would be tuned to ~659.25511 Hz. Almost 660, but not quite.

Similarly, a C# in Just Intonation would be set at 550 Hz. In Equal Temperament? ~554.36526 Hz.

Between the Renaissance and the Baroque periods (so roughly 1600 AD), musicians decided that they no longer favored the beautiful rounded ratios (5:4, 3:2) between notes. What they preferred, the fixed ¹²√2 ratio, was not only less rounded; it was irrational. From a harmonics standpoint, that means that the sound waves we make with our instruments today can never produce perfect harmonic patterns.

This is what the sound wave of an A and C# looks like in Just Intonation. Notice the perfect regularity of the pattern.

And this is what it looks like in Equal Temperament, the way you hear it in the music of today:

Each bump looks different; there is no regularity. The irrationality of the ratio guarantees that the pattern will never repeat, even if you continued to play these two notes together for the rest of time.

So why did this transition from the rational to the irrational occur?

Changing Keys

Back in the days of Just Intonation, it was common to only play music in one key and never deviate from that key mid-song. And that worked quite well, because those nice Just Intonation ratios are really only nice for notes in the key you’ve tuned to. If I was composing a song in A Major, notes in the key of A Major would be beautifully set to round ratios. But notes that are not used in A Major (Bb, Eb, G, etc.) actually sounded out of tune. So if you changed keys mid-song, you were in trouble.

Equal Temperament did away with that. By having the ratio of each incremental note be identical (the ratio of A to A# is the same as the ratio from D to D#), everything is exactly slightly out of tune by the same amount. And that allows us to do something that is very common in modern music (meaning any music written after the 1700s). We can alter keys mid-song, and it’ll sound just as good.

That epic key change in “I Will Always Love You” (Whitney Houston version)? The chorus in “Hey Jude”? The bridge in “God Only Knows”? Those only sound good because we’ve fixed our ratios to the irrational twelfth root of two.

The music of today sounds equally good in all keys at the expense of it sounding perfect in any key.

To give you a sense of just how much we’ve been trained to hear Equal Temperament as the right way to tune, I recommend you spend a few seconds listening to any part of this album by composer Michael Harrison. He plays piano in Just Intonation. And you’ll be able to hear that everything he plays sounds slightly off in a way that’s hard to describe and that sounds unlike any other music you’ve ever heard.

A Puzzle That Doesn’t Add Up

There’s something about Just Intonation vs. Equal Temperament that fascinates me. Why is it that music, something that is so mathematical in nature, something so deeply rooted in the laws of physics, can be so imperfect? The divide between these two types of tuning sheds light on a stark fact about our universe: It is impossible to have any musical key tuned perfectly without having that cause other keys to get out of tune.

Isn’t that strange? Wouldn’t you expect the math of music to fit together like a perfect puzzle? It’s almost as though, no matter how hard you struggle to complete that puzzle, there is always one piece left over that will never fit.

This flies in the face of our typical expectations of math, physics, and the universe at large. When it comes to science, puzzles should add up to a complete image.

Other Puzzles That Don’t Add Up

And as it turns out, musical tuning is far from the only impossible puzzle with which our universe presents us.

Take the Heisenberg Uncertainly Principle, a cornerstone of modern physics, which states that even with the best measurement devices imaginable, we’ll never be able to perfectly measure both the position and the speed of a particle (say, an electron). This isn’t a limitation of our technology. It’s a proven physical fact of our universe.

Or take non-computable numbers in math. There are real numbers that can never be generated, by any human or machine. In fact, there are more of these types of numbers (which we will never be able to see, feel, interact with) than there are numbers we can generate.

Or take a related concept in computer science called the Halting Problem. It’s impossible to write a piece of software that can accurately predict whether a given block of code will ever stop running. Again, this isn’t a limitation of technology. It’s been proven.

Or take Arrow's Impossibility Theorem, which shows that it’s impossible to design a democratic system that is universally fair.

Or take Gödel's Incompleteness Theorems, which shows that there will always be mathematical statements that can’t be proven or disproven using math.

The list goes on. And the takeaway is startling: We live in an imperfect universe. For all its beauty, symmetry, and complexity, there are corners of science, math, and logic where the rules seem to break down. Or the numbers don’t quite add up. Or the puzzle doesn’t fit perfectly together.

I wouldn’t say that’s a bad thing necessarily. Each of the above facts is actually quite beautiful in its own way—as long as you’re willing to embrace the beauty in the imperfect (or redefine what perfection means in the first place).

Which brings me back to music: If all of the incredible music that exists in the world is built on a tuning system that’s a bit askew, clearly we as a species can get quite far despite all of these universal limitations.

P.S. If you enjoyed this article and know someone who might like it, please consider sharing it. Forwarding these emails is the primary way Z-Axis is able to grow.

P.P.S. If you aren’t yet a subscriber yourself, consider subscribing to this free weekly newsletter by clicking here.