Did you know that much of what we understand as music rests on a big mathematical coincidence? Namely, that two to the power is approximately equal to 1.5.
If you’ll stick with me for a minute I’ll show you what I mean.
Consider these two basic facts of music:
- The circle of fifths tells us that if we start on a note and move up or down a fifth 12 times, we get back to the same note we started on (though in a different octave).
- The two most fundamental intervals in music are the octave and the fifth. The frequencies of two notes an octave apart have a ratio of 2:1. For a perfect fifth, the ratio is 3:2.
The problem is… (1) and (2) can’t both be correct. We have to choose!
Let’s see why that is.
The A string on a bass has a frequency of 55 Hz, or 55 vibrations per second. (The unit of frequency, hertz, abbreviated Hz, means vibrations per second.) When we multiply 55 by 1.5, what we get is the frequency of the note (E) a fifth higher—namely, 82.5 Hz. We can get the frequency of B by going up another fifth from E: it has a frequence of 123.75 Hz because 123.75 = 82.5 × 1.5 = 55 × 1.5 × 1.5. If we continue to generate all the notes by going up in fifths twelve times, we arrive again at A:
A → E → B → F# → Db → Ab → Eb → Bb → F → C → G → D → A
The frequency of this high A, 7 octaves above the original one, is 55 Hz multiplied by 1.5 twelve times, which is to say or 7136 Hz.
Since this A is seven octaves above the low A (55 Hz) we started with, we could have found its frequency by just doubling 55 seven times. After all, moving up an octave just doubles a note’s frequency, right?
By that reasoning the frequency of that high A is , which is to say 7040 Hz.
That’s close to 7136 Hz but not the same. The two pitches differ by 23.45 cents, or approximately a quarter of a half-step.
Listen to what I’m talking about. Here’s an example of a starting note (middle C), followed by G, D, A… going around the circle of fifths and finally arriving back at C. In each case the next note is a fifth higher, 1.5 times the frequency of the previous one. (Sometimes we lower it an octave so it stays in the same general range.) We keep the original C playing underneath. Listen to the sequence and the final C that it arrives at. And notice how out of tune it sounds with the original C:
What you’re hearing is the 23.45 cent difference that’s accumulated over the course of the journey around the circle of fifths.
What’s up with that?!?
The inescapable bad news is that our notion of an octave—the most basic interval, found throughout nature and representing a 2:1 ratio of frequencies—and our notion of a perfect fifth—the second most basic interval, also found throughout nature and representing a 3:2 ratio of frequencies—are incompatible with the very notion of the circle of fifths—the notion that moving up a fifth twelve times should move us up seven octaves from the original note.
But the good news is that they’re close enough that we can “fudge it” and never even worry about the difference. (Almost never, anyhow. It does cause subtile problems in situations where musicians do tune their fifths to a 3:2 ratio, such as a choir in a church.)
The Solution: Equal Temperament
The solution is to redefine the perfect fifth so that it is compatible with the octave. Instead of using the 3:2 ratio, we divide up an octave into twelve equal half steps and define a fifth to be equal to seven of those half steps. In other words, every half step interval represents a pitch ratio of , i.e., — we design it that way so that twelve half steps together yield a ratio of multiplied by itself 12 times, or , which is equal to or, simply, 2. Thus twelve half steps make up an octave. We call this solution the Equal Temperament tuning system. In it, the perfect fifth, which is made up of seven half steps, has a ratio of multiplied by itself seven times, which is to say or . Which equals 1.4983. That’s not 1.5 but it’s pretty darn close.
Listen to the difference. Here’s a “just” perfect fifth interval—two notes with a frequency ratio of 1.5:
And now listen to the same interval played with equal temperament, so that the frequency ratio is 1.4983:
Can you tell the difference? Most people don’t notice it. The telltale sign is the “beats” present in the equal-tempered version that are absent in the other one. Those are the pulsations in the sound—in this case, a bit more than one per second—caused by the misalignment of the lower and the higher pitched waves. A 3:2 ratio means that the higher pitch vibrates three times in the same amount of time that the lower pitch vibrates twice… so every other time the lower pitch vibrates, the two waves align. But with equal temperament, the alignment is not perfect. Like windshield wipers that are almost, but not quite in sync with the beat of the music that’s playing on the radio—or two metronomes keeping slightly different time—they are in and out of sync, periodically meeting up.
The effect is more pronounced when you listen to the equal tempered fifth sounded against the just tempered fifth:
For all practical purposes, the adoption of Equal Temperament has put an end to the issue of systems of tuning in western music. Alternate tuning systems are still in use, and are discussed at great length in academia, but rarely, if ever, appear in the music listened to by the vast majority of people.
But if you’re interested…
Still, tuning systems are a fascinating subject. For instance, it turns out that the octave and the perfect fifth are not the only intervals to correspond to simple ratios of whole numbers. Rather, all of our western intervals can be expressed that way. Some more simply than others. (For example, the major third, as it appears in nature, represents a 5:4 ratio, whereas the tritone is 45:32.) Not only are these intervals incompatible with equal temperament, they are also incompatible with each other and the circle of fifths.
For instance, let’s look at the major third. The simplest formulation of the major third, called the just major third is taken from the fifth member of the overtone series, and represents a frequency ratio. But conventional harmony tells us that we ought to be able to derive the ratio of a major third by going up a fifth four times and down an octave twice. If the ratio of a fifth is then that means the ratio of a major third ought to be or . This is called the ditone or Pythagorean major third, and it’s different from the just major third because is equal to , not . Listen to a just major third:
And now a Pythagorean major third:
Can you hear the difference? The pitch discrepancy is especially noticeable when you hear the two major thirds together:
Both these major thirds are different from the equal-tempered major third which we are used to hearing from our instruments, though the Pythagorean third is closer to equal temperament: It is only 8 cents above the equal tempered major third, whereas the just third is 14 cents below. Here is the equal tempered major third:
The difference between the just and Pythagorean major thirds—a ratio of 81/80 or about 21.5 cents—has a name: the syntonic comma. (The gap in pitch between two variants of a given note is called a comma.)
Here’s one last example of incompatibility, this time between the just major third (5:4) and the octave (2:1): Three stacked major thirds should add up to an octave, right? That’s what our music theory tells us. But in fact three just major thirds combined yield or , which is 1.953, not 2 (which equals ). This comma has a name too: the diesis. It amounts to about 41 cents, which is super easy to hear:
Epilogue: What is a coincidence, anyhow?
There are thousands upon thousands of books and articles written about tuning systems, in much, much more detail than I’ve touched on here. What surprises me so much is that every one I’ve seen “buries the lead.” For me, the most gripping part of the story is the obscure mathematical fact that lies at the heart of all of the music I listen to. There’s no shortages of “coincidences” in math. For instance, there’s a sequence of six 9’s in a row within the first 800 digits of π. What makes so different is that it has consequences. HUGE consequences. We think of music as not a random construct of society but something much bigger, many would say an essential part of the universe. How can it rely on a two things being not-quite-equal-but-really-close? Doesn’t that make all of music a coincidence?
Maybe the answer is that music is the same kind of coincidence as carbon based life on Earth. Life as we know it depends heavily on various properties of chemistry in conjunction with specific conditions on Earth. As far as I know, those properties and conditions have no special significance apart from the fact that they give rise to life… much the same way is of no particular importance except as far as music is concerned. In which case, music may be a coincidence, but it is no more of a coincidence than life itself.
Me, I suppose I’ll settle for that, for now.
As always, I welcome your thoughts!Almost all of the #music we listen to relies on a crazy #coincidence of #math. Click To Tweet
UPDATED March 3, 2018: added audio examples of major thirds.