Guitar Tuning is Imperfect (Part 1 of 2): Tuning Systems

And it's not only guitars. Many instruments — from pianos to flutes to saxophones — are tuned in equal temperament. Contrast this, for example, with just temperament. We'll take a look at the mathematical and perceptual differences between the two systems in this post. To start off, Paul Davids has a great video showing what a justly tuned guitar sounds like. Can you hear the difference from a standard guitar?

The other day, I was practicing guitar and was getting frustrated by the B string — no matter how much I would tweak it, it would still sound slightly out of tune. What I at first chalked up to a possible flaw in the guitar turned out to be a side effect of the tuning system itself. Guitars, like most discretely pitched instruments, are tuned using equal temperament.

Tuning Systems

First thing: Yes, multiple tuning systems exist. There is not only one way to tune an instrument, and each choice comes with its own pros and cons.

There are two kinds of tuning that we'll look at here.

1. True Temperament
2. Equal Temperament

True ("just") temperament is a tuning system in which the frequencies of successive notes in the scale are tuned to harmonic ratios of a root note.

The harmonic series (in the musical sense since "harmonic series" has a different meaning in math) are the tones produced by the ratios

The $n$th musical harmonic number is then $\frac{n + 1}{n}$. These numbers are special, because they represent the simplest relationships between harmonically related frequencies. To see this, consider a string (say, a guitar string) that is vibrating to produce a sound. The sound it produces is related to its mass, length, tension, etc., but we'll take it as a baseline. Now, the sound it produces is composed firstly of a fundamental tone (this is the tone that our brains process and recognize) which also happens to be the lowest discernible pitch. Then, the string also produces other, quieter and higher, overtones. The pitches of these overtones are iteratively related by the harmonic series. The first overtone is 2 times the frequency of the fundamental, the next is 3 times, then next 4 times and so on. If you take the ratio of each successive pair of overtones, you get the ratios $\frac{2}{1}, \frac{3}{2}, \dots$ as above.

Physically, these frequencies correspond to the tones that would be produced if we cut the string into halves (thirds, fourths, etc.) and vibrated the shorter pieces.

The existence of overtones and their simple integer relationships to the fundamental has been known for thousands of years. And as musicians began developing instruments with fixed size scales (like the twelve-tone chromatic scale we have today), they were faced with deciding the frequencies that those notes would take on. This is not as much an issue for, say, a violin, which can continuously vary its pitch when the player slides their fingers along its strings.

One natural tuning choice is to use the harmonic relationships of the overtones that already exist in nature to produce the frequencies of the scale. This results in a Pythagorean tuning system.

Pythagorean tuning only works in the context of a key. If you tune based on A, then those same tunings will sound out of tune when switching to another key, like G.

Pythagorean tuning, when played in its intended key, sounds purer than most music we're used to hearing. Since the notes are tuned using harmonic relationships, they minimize destructive interference and instead maintain consonant relationships. Take another listen to the guitar in the above video where this purity can be heard.

Practically speaking, it's not very useful for an instrument to require retuning when changing keys. Enter equal temperament tuning.

Equal temperament is imperfect, but equally so in every key.

Equal temperament depends on a fixed-size scale. Again, take our usual 12-tone scale. Then, the ratio between successive notes will be $2^{1/12}$. This is designed in such a way that by the time we reach an octave (the 12th note after the root), the ratio to the original will be double as expected: $2^{12/12} = 2$.

This is convenient because, no matter the choice of key, the frequency ratio of a fixed interval size will be identical. This is good, but the downside is that equal temperament ratios deviate from the integer ratios found in the harmonic series, and so certain intervals that deviate sufficiently from just tuning will sound "off" to a trained ear.

Relationship to the Guitar

The standard guitar tuning tunes the six strings (in order) E, A, D, G, B and E. Looking at pairs of adjacent strings, we have the five intervals E-A, A-D, D-G, G-B, and B-E. These are all perfect fourth intervals except for G-B which is a major third. Now, recall that I was frustrated with the pitch of the B string on my guitar. Well, it turns out an equal temperament major third deviates from its true temperament counterpart by enough to be detectable by ear.

Let's do a quick calculation to see the extent to which the equal temperament major third is "out of tune".

Take the open G string as 196Hz (which is the standard tuning) and consider the major third G-B. The just tempered major third ratio is $5/4 = 1.25$, placing the B frequency at $196 * 1.25 = 245$Hz. Whereas an equal tempered major third ratio is $2^{4/12} \approx 1.26$. The B frequency is then $196 * 1.26 = 246.96$Hz. So the tunings differ on the B frequency by 1.96Hz. The just-noticeable difference (the smallest difference which 50% of people can detect) for pitches in this range is roughly 1Hz to 3Hz, and hence 1.96Hz is noticeable by some people.

Here are some sketches that show both visually and auditorily what's going on. This first one plays the justly tuned major third from the calculation (196Hz and 246.96Hz).

And this one plays the equal temperament major third (196Hz and 245Hz).

Can you hear the difference?

On the other hand, let's suppose we instead tune the B string to C to make a perfect fourth G-C. In just temperament, the ratio $4/3 \approx 1.333$ yields a C of $196 * 4 / 3 = 261.3$Hz. In equal temperament, we have $196 * 2^{5/12} = 261.6$Hz. Here, the difference is only 0.3Hz, which is not in the commonly detectable range referenced above.

Hence, we have some quantitive evidence as to why the B string can sound out of tune on an in-tune guitar: Because (1) its relationship to the open G string is an equal temperament major third which is perceptively different from a justly tuned major third, and (2) the other strings maintain equal temperament perfect fourth relationships to their neighbors which is perceptually the same as the justly tuned perfect fourth.

Up Next

We've seen some of the pros and cons of different tuning systems. It's interesting that, even though the equal temperament major third is sufficiently out-of-tune to be detected, we've become accustomed to the resulting dissonance to the point that it sounds natural in the music we hear every day.

In Part 2, we'll take a look at methods of tuning the guitar (and show why some are better than others) and how they relate to the different tuning systems.