Beating your Guitar into Tune

Posted on November 16, 2007
Filed Under Physics |

If you have any musical friends, this should get their attention (if it doesn’t, just spill petrol over their musical instruments and ignite. THAT will).

Basic trigonometry tells us that the rule for adding two sines is as follows:

\sin(\omega_1 t) + \sin(\omega_2 t) = 2 \cos \left( \frac{(\omega_2 - \omega_1) t}{2} \right) \sin \left( \frac{(\omega_2 + \omega_1) t}{2} \right)

It turns out that if our ear is exposed to two sounds which have similar frequencies, it only hears the term which contains the difference of the frequencies. The fast-oscillating term,

\sin \left( \frac{(\omega_2 + \omega_1) t}{2} \right) ,

is “averaged” and goes unnoticed. This is illustrated in the figure below, where I have plotted (in blue) the sum of sin(4t) + sin(4.5t) for values of t between 0 and 40. The red envelope corresponds to the function

2 \cos \left( \frac{(\omega_2 - \omega_1) t}{2} \right) = 2 \cos(0.25t),

which is what our ear would hear:

Beating

The behavior shown above - that of a fast signal modulated by a slower one - is called beating in physics. This very phenomenon is utilized by guitar players, albeit unknowingly in most cases, to tune their guitars. Let’s take a look at a guitar fretboard, which is the name guitarists give to the long “neck-part” of their guitars:

Guitar Fretboard and String Frequencies

There are 6 strings on a guitar. Every string on the guitar, should we pluck it, resonates at a different frequency, as each string has a different girth. The frequencies are listed above, along with the corresponding notes (E2, A2, D3, G3, B3 and E4). By pressing down on the strings at different points, though, the guitar player can effectively shorten the strings’ lengths and vary their pitch - and hence produce melodies. This is the basic physics of all stringed instruments. As a string is shortened its pitch is raised, so it sounds “higher”.

As it so happens, guitars tend to go out of tune with time, so, for example, the 2nd string from the top (B3) might have a frequency different than 247 Hz (perhaps 240 Hz), due to slackening of the string (caused by rust and mechanical wear). Guitarists have devised a quick test that helps them tune their guitar using the phenomenon of beating described above.

Suppose, for the sake of concreteness, that we suspect the 2nd string has gone slightly out of tune, and suppose that we know somehow by divine intervention that the 1st string, E4, is perfectly in tune. We then press the 2nd string down at the 5th, shortening its length such that - if plucked - its pitch would be 330Hz, the same as that of the uppermost string, E4:

Frequency of second string

We can now make use of the beating phenomenon to see if the 2nd string is out of tune: simply hold it down as in the above figure and pluck it together with the 1st string. If they are in tune, the two strings will vibrate at the same frequency and we will hear, effectively, a single note:

s(t) = \sin(2 \pi \times 330 \times t) + \sin(2 \pi \times 330 \times t) = 2\sin(2 \pi \times 330 \times t)

However, if they are not, and the 2nd string is not perfectly in tune, its frequency will differ slightly from 330 Hertz - say, 325 Hertz - and we will hear the beating produced by the two strings:

s(t) = \sin(2 \pi \times 330 \times t) + \sin(2 \pi \times 325 \times t) \\ = 2 \cos(2 \pi \times 2.5 \times t) \sin(2 \pi \times 655 t)

The guitarist’s ear will only detect the \cos(2 \pi \times 2.5 \times t) term. He will then retune the string - using special tuning pegs at the end of the neck - until the beating goes away. This will indicate that they are both in tune. This beating can even be felt if you hold the guitar’s body against yours and pay careful attention.

Beating can only be used in a comparative way. It can tell us if two strings are out of tune with respect to each other. We will still, however, need some sort of reference to tune the absolute pitch of one of the strings.

Comments

5 Responses to “Beating your Guitar into Tune”

  1. Mark Palmer on November 16th, 2007 11:27 pm

    So how long would it take to get the two strings in perfect pitch where there would be zero beats?

  2. admin on November 17th, 2007 6:47 am

    For an experienced guitarist it would take about 10-20 seconds per string. This holds even in a noisy room, as you can feel the beats against your body. This is approximately the same order of magnitude it takes to do things with an electronic tuner.

    The same trick is also used by piano tuners. That’s why they strike their tuning forks and listen. The fork is tuned to resonate at 440 Hz, ideally the same as the middle A note on the piano.

  3. Mark Palmer on November 17th, 2007 11:55 pm

    Well, Dr., I did have that on a test in the early seventies. To get full credit a qualifying statement that what is a good enough absence of beats at the rate of say 1 per 10 seconds could be bettered if you tuned until the beats were 1 per hour, then further 1 per day, and on and on. We called it a trick question then. And now.

    By the way how long you going to let me hang on your energy conservation puzzle.

  4. admin on November 18th, 2007 4:12 am

    Indeed, to tune a guitar to an accuracy of x Hertz, one needs about 1/x seconds to observe to beats.

    Regarding the puzzle - read the comments, one of the replies (by Himanashu) got it right.

  5. Mark Palmer on November 18th, 2007 12:31 pm

    Thank you. Wandering the cosmos gave an explanation stirring memories of a ball on a spinning turntable viewed from two frames. Been a long time for me.

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