If, like me, you’ve heard that notes on a piano can’t be perfectly in tune with each other, but haven’t paused to actually work out why, then this post is for you. Well, strictly speaking, it’s for me to flesh out my own understanding of the same question in the vain hope that writing it down will make me understand it better, but you get the point.
I started by reading the Wikipedia article on Just Intonation but found it utterly impenetrable. So this is my attempt at working out what’s going on and explaining it in a more approachable form.
What are notes, anyway?
Vibration in the air. More specifically, vibration at a given rate. That rate is the frequency of the vibration, i.e. how many times it moves per second. This frequency is measured in Hertz (Hz). So when people talk about tuning to A=440 they mean adjusting their instrument so that when they play an ‘A’, it makes the sound produced by the air vibrating at 440 times per second.
How notes relate to Hz
For reasons we don’t need to go in to (i.e. I don’t yet understand why not), our ears don’t hear differences in frequencies in a strictly linear way. So a 10Hz difference between 40Hz and 50Hz sounds a lot bigger than the difference between 400Hz and 410Hz. This kind of makes sense when we look at the most harmonious‐sounding intervals—octaves. One note is an octave higher than another when its frequency is double that of the lower note. So from our standard tuning note at 440Hz, known as A4, the next highest A, A5, is found at 880Hz.
So far so good. So what’s the problem?
This all makes sense so far, right? I mean, doubling frequencies to make an octave sounds believable, and having fractions of that doubling making up the smaller intervals should be fine, too? Right? Let’s take a look.
Let’s start at A1 which is 55Hz. A2 is therefore 110Hz and the midpoint between those two frequencies is the perfect 5th away, E2, which works out to be at . Which is fine.
If you squint you can see the frequency, or the number of times the amplitude reaches zero, of A2 is double that of A1.
So let’s now see what happens when we expand that further. A perfect 5th above E2 is B2, which should be . And if we keep adding 5th on top of 5th we should eventually work our way around the cycle of 5ths entirely until we come back to A again, at which point our frequency should be a multiple of our original 55Hz. There are 12 notes, so applying the cycle of 5ths 12 times will get us back to an A, which means we need to multiply our frequency by 1.5 12 times, or . That cycle spans 7 octaves, so to go up by the same amount in octave intervals our starting frequency should be doubled 7 times, or .
Let’s plot frequencies against notes for both multiplying by 1.5 and by 2. Each series starts at A1 (55Hz) and ends up at the same end note at A8.
But. Here’s the snag. The frequency of A8 in the Octaves series is different to the A8 in the 5ths. How come? but . We’re off by 96.049Hz!
So we can choose to either have all of our octaves perfectly in tune with each other, i.e. be exact doubles of our starting note, or have all of our fifths perfectly in tune with each other, i.e. be exact one-and-a-halfles1 of our starting note, but not both.
So what can we do about it?
Not a lot, as it turns out. Instruments with fixed pitches, like pianos, have to have their note frequencies fixed at the time they’re tuned. So a pianist can’t adjust the frequency of the note they’re about to play to make it perfectly in tune for the specific intervals they’re playing, nor can players of instruments with frets (guitars), valves (trumpets) or, to a slightly lesser extent, keys (clarinets and saxophones).
Instead they have to make a choice as to which notes are going to be tuned “incorrectly”. Mostly, that choice is to use a different tuning system called Equal Temperament which, instead of aiming for perfect intervals within the octave, splits the octave itself into 12 equal intervals, one for each semitone.
Equal Temperament vs Just Intonation
Equal temperament splits an octave into 12 equal intervals. But because we’re dealing with frequencies that double every octave, i.e. the frequency of octaves increases exponentially, finding the frequency of all twelve notes in an octave is not as simple as dividing the difference between 55Hz (A1) and 110HZ (A2) by 12. That would mean each note was an equal number of Hz away from the next one, but this falls in to the trap mentioned earlier that an increment of 10Hz sounds a lot bigger between 40Hz and 50Hz than it does between 400Hz and 410Hz. This is because it’s the ratio we’re hearing, rather than the absolute difference. So, let’s do some more secondary school maths.
Given we know that octave frequencies double, we can say that the frequency of any number of octaves, , from a starting frequency of is . Therefore to find the frequency multiplier of each note in an octave, i.e. of , we need to take the 12th root of 2. so therefore our E2 (which is 7 semitones above A1) in equal temperament is
So our “perfect” 5th is not so perfect anymore, but we’ve traded that perfect ratio from A1 to gain better ratios with all the other notes. And whilst we’ve shown a piano can’t be perfectly in tune with itself (if you also want it to be able to play in any key), Equal Temperament is a good–enough fudge to make all of the intervals close enough to sound OK regardless of which key you’re in.
To close off, I’ll leave you with this gem. Some sadist out on the internet worked out a just intonation version of Giant Steps' chord sequence. Let’s just say it doesn’t sound like it normally does.
This is Giant Steps, but the whole damn progression is Just Intonated with the ratios 4/3, 5/4, 3/2.. and so on. The notes are tuned such that, at the end of the whole progression there is a decrease in pitch by roughly 22 cents, which is the syntonic comma.
Of course that’s a word. ↩︎