“Intonation is a social construct.” If you’ve hung around our shop long enough, you’ve probably heard me say this, as it’s one of my favorite aphorisms. What do I mean by this, and what does this have to do with your guitar? Well, your guitar is out of tune. In fact, EVERY fixed pitch instrument is out of tune, and probably every song you’ve ever heard is out of tune. You’ve just gotten used to how it sounds, and maybe have never considered that music could sound any other way. Have you ever tuned your guitar to a perfect G major chord, only to discover that now your E major chord sounds horrible? Why do you have to sacrifice a perfect G major to make your E major palatable? The short answer is that your guitar is out of tune, and always will be, and you just need to learn to deal with it. The long answer, my friends, is this:
Music, at it’s most fundamental level, is math. Math doesn’t just dictate rhythm, but it also dictates pitch: musical notes are essentially oscillations of air pressure, creating a waveform that we measure in Hertz (Hz), which is one wave cycle per second. The most basic waveform is a sine wave, which is the fundamental tone of any note.
Our brains perceive the frequency of these waves as pitch. Faster vibrations we hear as a higher pitch, slower vibrations lower. (For example, a note measured at 200Hz will be heard as lower than a note measured at 400Hz.) The relationship between these vibrations and how we combine them is what we call harmony.
How do we tune a guitar, or any other musical instrument? We measure the difference between one vibrating string and another, and shift one note to perfectly match the sound of the other. This process is essentially seeking out the simplest ratios between one frequency and another. Simple ratios sound pure, like the two notes blend in perfectly with another. More complex ratios sound discordant, and are prone to cause some eye-twitching with the more golden-eared crowd.
The most basic of ratios is a 1:1, meaning one note’s freqency is exactly the same as another. Here’s two notes, both simple sine waves, at 200Hz (about the same pitch as a G string):
If those notes were out of tune, the two waveforms would be out of alignment, amplifying some parts of the wave while reducing other parts, which our brains perceive as being out of tune. Here’s the same root note of 200Hz, but with the second tone getting progressively more out of tune, at 202Hz, 204Hz, 206Hz, and 208Hz:
Do you hear how each example pulsates faster than the one before it? Did you get that sickening feeling in the pit of your stomach? Congratulations, you CAN hear when things are out of tune! You’ve got a better ear than you thought!
Most musical instruments can play more than one note, so the 1:1 tuning method isn’t exactly useful. To tune beyond a single note to itself, we have to use a more complex ratio, 2:1. This relationship we call an octave: one note is vibrating at twice the frequency as the other, making it sound twice as high. With one root note, we can tune one octave to another, and then another, and then another, using the 2:1 ratio applied to each new note: 200Hz to 400Hz, 400Hz to 800Hz, 800Hz to 1600Hz, etc. Octaves blend together almost perfectly, with each note almost vanishing into the other. Here’s two notes, one at 200Hz, the other at 400Hz:
Again, music is made up of more than just octaves. If it were, our music would be incredibly boring – imagine singing the first two notes of “Somewhere Over The Rainbow” over and over and over again. Our music demands harmony, which means more complex ratios: the next simplest ratio is 3:2, which gives us what we call a perfect fifth. The notes in a fifth blend together almost seamlessly, yet each note can be heard independently if you listen closely. Here’s a perfect 5th, with 200Hz and 300Hz being played:
With a somewhat more complex ratio of 3:2, we can now start tuning other notes beyond the root and octave, and actually create more complex harmonies. If we stack 3:2 ratios, over and over again, we can tune all kinds of notes: 200Hz to 300Hz, 300Hz to 450Hz, 450Hz to 675Hz, etc. A fifth on top of a fifth on top of a fifth, which we have come to call the circle of fifths. We’ve applied simple names to these frequencies, so starting from 200Hz, which is approximately a G, we can now tune G to D, D to A, A to E, E to B, and all the way back around until we get to a G again (albeit in a different octave). Perfect, right? Wrong.
Ideally, stacking a perfect fifth twelve times would circle back around again to our beginning note, allowing us to tune every note in the scale. Mathematically, however, it just doesn’t work out that way. Stacking perfect fifths twelve times can be represented as (3/2)^12, which give us roughly 129.75, which is close, but not quite, equal to seven octaves: 2^7=128. This means that while each individual fifth is perfect, our octaves are out of tune with each other, and so is every other interval! Gah! Madness!
Going back to our original 200Hz tone, applying the above formula, our two “identical” notes now sound like this:
While this may not seem too bad, remember that most music comprises of more that just two notes, and the more out of tune notes are mixed in the worse it sounds. The impure ratios compound upon each other, leaving us with an unholy mess of discordant sounds.
So what’s to be done? It is mathematically impossible for your guitar to be perfectly in tune. Over the last several hundred years, musicians and theorists have gone to great lengths to attempt the impossible: developed instruments with lots of extra keys and frets, different tuning systems, and got into a lot of heated debates, seeking the perfect tuning system where every interval is pure. Eventually everybody just kind of gave up and settled on 12-tone Equal Temperment. This system essentially robs Peter to pay Paul: the entire scale is squished, flattening every half step so the octaves line up. This “tempering of the scale” is referred to as the twelfth root of two, or 12√2. Every half-step is now slightly flat, and every interval is slightly out of tune – no pure fifths, no perfect thirds. Bach would not be pleased.
What does all this mean for your guitar? Well, first of all, you need to accept that your instrument will never be perfectly in tune, and that’s ok. If you listen closely, The Beatles were horrendously out of tune, and it didn’t stop them from being the best band in the world. Our best hope for our guitar is to make them as evenly out of tune with themselves as possible. They key to this is a good setup – if your guitar isn’t set up properly, no amount of fiddling with the intonation is going to help. In fact, we set the intonation on guitars at the very end of the setup process. A well intonated guitar will have it’s nut cut low, have string action close to the fretboard, use strings that aren’t overly light, and perfect fretwork. Intonating a guitar is done by lengthening or shortening each string to compensate for string size, pitch, and scale length. There’s quite a bit to be said about the mechanics of intonating a guitar, which we will save for another time.
You can make improvements to your intonation at home. First of all, after making sure your guitar is properly set up, get a decent electronic tuner. Don’t simply trust your ears – remember, you’ve been listening to out of tune music your whole life and you didn’t even notice! If you find yourself in a pinch without a proper tuner, tune the guitar to itself by fretting the note of the next open string – do NOT use the harmonic method, which is mathematically incorrect. Don’t try to tune your guitar by tuning to a chord: your ears can fool you, and if you get one chord perfect, the rest of your other chords will be off. Also, be mindful of your technique: if you squeeze the neck too hard, or are too aggressive with your attack, you will bend the strings and push everything out of tune. Intonation is in your hands as much as it is in your head.
Or you could, you know, just put together a fretless Strat like I did.
If you’d like delve further into intonation and the history of our current tuning systems, I highly recommend to start with Temperament: How Music Became a Battleground for the Great Minds of Western Civilization Kindle Edition, by Stuart Isacoff. I also recommend The Fifth Hammer: Pythagoras and the Disharmony of the World, by Daniel Heller-Roazen. If you have any questions, feel free to contact me at firstname.lastname@example.org.