“Intonation is a social construct.” If you’ve hung around our store long enough, you’ve probably heard me before, as it’s one of my favorite aphorisms. What do I mean by that and what does that 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. They just got used to how it sounds and may never have thought that music could sound any different. Have you ever tuned your guitar to a perfect G major chord only to find that your E major chord now sounds terrible? 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 all you have to do is learn how to use it. The long answer my friends is:
Music is math at its most basic level. Math not only dictates rhythm, but pitch too: musical notes are essentially vibrations of air pressure that create a waveform that we measure in Hertz (Hz), which is one wave cycle per second. The simplest waveform is a sine wave, which is the root of every note.
Our brain perceives the frequency of these waves as a pitch. We hear faster vibrations as a higher pitch, slower vibrations lower. (For example, a note measured at 200 Hz is heard lower than a note measured at 400 Hz.) The relationship between these vibrations and how we combine them is called harmony.
How do we tune a guitar or 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 essentially looks for the simplest relationships between one frequency and another. Simple relationships sound pure, as if the two notes fit together perfectly. More complex relationships sound contradictory and tend to cause winking at the more gold-eared crowd.
The most basic ratio is 1: 1, which means that the frequency of one note is exactly the same as the other. Here are two notes, both simple sine waves, at 200 Hz (roughly the same pitch as a G string):
If these tones were out of tune, the two waveforms would become out of alignment and amplify some parts of the wave while reducing other parts that our brain perceives to be out of tune. Here is the same fundamental tone of 200 Hz, but with the second tone increasingly detuned at 202 Hz, 204 Hz, 206 Hz, and 208 Hz:
Do you hear each example pulsing faster than the previous one? Did you have that disgusting feeling in your stomach area? Congratulations, you CAN hear when things are upset! You have a better ear than you thought!
Most musical instruments can play more than one note, so the 1: 1 tuning method is not particularly useful. To tune beyond a single note we need to use a more complex 2: 1 ratio. We call this relationship an octave: one note vibrates at twice the frequency of the other, making it sound twice as high. With a root we can tune one octave to another, and then another, and then another, with the 2: 1 ratio applied to each new note: 200 Hz to 400 Hz, 400 Hz to 800 Hz, 800 Hz to 1600 Hz, etc. Octaves blend together almost perfectly, with each note almost blending into the other. Here are two notes, one at 200 Hz and the other at 400 Hz:
Again, music consists of more than just octaves. If so, our music would be incredibly boring – imagine singing the first two notes of “Somewhere Over The Rainbow” over and over again. Our music demands harmony, which means more complex ratios: The next simpler ratio is 3: 2, which gives us a perfect fifth. The notes in a fifth merge almost seamlessly, but each note can be heard independently if you listen carefully. Here is a perfect 5th played with 200Hz and 300Hz:
With a slightly more complex 3: 2 ratio, we can now tune other notes beyond the root and octave and actually create more complex harmonies. If we stack 3: 2 ratios over and over, we can tune all kinds of notes: 200 Hz to 300 Hz, 300 Hz to 450 Hz, 450 Hz to 675 Hz, etc. A fifth over a fifth over a fifth that we are come to name the circle of fifths. We’ve given these frequencies simple names so that starting from 200 Hz, which is roughly a G, we can now tune G to D, D to A, A to E, E to B and all the way back until we agree a G (albeit in a different octave). Perfect right? Not correct.
Ideally, stacking a perfect fifth twelve times would go back to our starting note, allowing us to tune every note in the scale. Mathematically, however, that just doesn’t work. The twelve-fold stacking of perfect fifths can be represented as (3/2) ^ 12, which gives us roughly 129.75, which is close, but not quite equal to seven octaves: 2 ^ 7 = 128. This means that while every single fifth is perfect, our octaves are not coordinated with each other and so are every other interval! Gah! Madness!
If we go back to our original 200 Hz tone and apply the formula above, our two “identical” tones now sound like this:
While this might not seem too bad, keep in mind that most music is made up of more than two notes, and the more detuned notes mixed together, the worse it sounds. The impure conditions reinforce each other and leave behind an unholy mess of inharmonious sounds.
So what is to be done? It is mathematically impossible for your guitar to be perfectly in tune. Over the past hundred years, musicians and theorists have gone to great lengths to try the impossible: instruments with many additional keys and frets, different tuning systems, and many heated debates in search of the perfect tuning system where all the interval is pure. At some point everyone gave up and decided on 12-tone Equal Temperment. This system essentially robs Peter of paying Paul: the entire scale is squashed, each semitone is flattened to align the octaves. This “cleanup of the scale” is known as the twelfth root of two, or 12√2. Every semitone is now slightly flat and every interval is slightly out of tune – no perfect fifths, no perfect thirds. Bach wouldn’t be pleased.
What does it all mean for your guitar? First of all, you have to accept that your instrument will never be perfectly in tune, and that is fine. If you listen carefully, the Beatles were terribly upset, and that didn’t stop them from being the best band in the world. Our greatest hope for our guitar is to detune it as evenly as possible with itself. The key to this is a good setup – if your guitar isn’t properly tuned, fiddling with intonation won’t help. In fact, we set the intonation of the guitars at the very end of the setup process. A well-voiced guitar has a deep saddle, a string position close to the fingerboard, strings that are not too light and a perfect fretwork. The intonation of a guitar is done by lengthening or shortening each string to accommodate the string size, pitch, and scale length. There is a lot to be said about the mechanics of a guitar’s intonation that we will save for another time.
You can improve your intonation at home. After making sure that your guitar is properly tuned, the first thing to do is get yourself a decent electronic tuner. Don’t just trust your ears – remember, you’ve been listening to out-of-tune music all your life and not even realizing it! If you’re in a bind without a proper tuner, tune the guitar to yourself by grabbing the note on the nearest open string – DO NOT use the harmonic method, which is mathematically wrong. Don’t try to tune your guitar by tuning to a chord: your ears can fool you, and if you master one chord perfectly, the rest of the chords are off. Also, watch your technique: if you pinch your neck too hard or are too aggressive with your attack, you will bend the strings and put everything out of tune. The intonation is in your hands as well as in your head.
Or, you know, you could just assemble a Fretless Strat like I did.
If you want to delve deeper into the intonation and history of our current tuning systems, I highly recommend that you start with that 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, please feel free to contact me at firstname.lastname@example.org.