# Physics, guitars and pitch harmonics

One of the fun parts about being a scientist is running across “day-to-day” phenomena which can be understood much better using a little bit of one’s knowledge.  One of these I’ve had bouncing around in my head for a while: the idea of ‘pitch harmonics’ on guitar.

I started playing the guitar about a year and a half ago on a whim.  Deciding that I spent too many evenings in front of the television doing nothing, I marched into the local music store and declared that I wanted to learn guitar, and needed the works: guitar, case, and lessons!  I have a really excellent instructor named Toby, and my knowledge of guitar has increased rapidly, if not my technique.

Anyway, what is a pitch harmonic?  Here’s a short audio clip of me playing the first few notes from Pearl Jam’s Jeremy (an .mp3 file):

If you can get past the quality of the guitar playing, you’ll notice that the last two notes of the riff have a completely distinct sound from the first few.  Those are the pitch harmonics, which are played differently on the guitar from other notes.  We discuss the physics of them below the fold…

Let’s start with the guitar basics!  A guitar, of course, makes noise through the vibration of strings, though the means by which those string vibrations are converted to audible sound depends on the type of instrument (electric or acoustic).   A picture of my “axe”* is below:

One  normally plays notes of different pitch by pressing down upon the ‘frets’ (raised pieces of metal) on the neck of the guitar, which effectively shortens the length of the string and increases the pitch:

But one can also play harmonics, by touching a finger lightly upon the string right above certain frets, without pressing; this produces the pitch harmonic.  This can only be done effectively at certain frets: namely, at the fifth, seventh, twelfth, nineteenth, and twenty-fourth frets.  In sequence, starting with the fifth fret, the pitch harmonics sound as follows:

They range from high pitch at the fifth, to the lowest pitch at the twelfth fret, and then climb in pitch again.  Curiously, the pitch of the regular tone at the twelfth fret and the pitch of the harmonic are the same:

We can understand all of these properties by understanding the basic physics of a vibrating string, which we cover next.  We then wrap up by testing the principles with measurements on my own guitar.

(Note: I’ll skip the detailed math, which involves the solution of a partial differential equation, and focus on the important results.) Waves on a string travel at a speed determined by the tension on the string and the mass density (mass per unit length) of the string , according to the following equation,

.

Strings on a guitar are of different thicknesses to take advantage of the effect of the wave velocity on the frequency of sound emitted; we’ll come back to that momentarily.  For waves on a string of length fixed at both ends, it follows from the mathematics that any wave excitation on the string can be written in terms of a series of elementary standing waves on the string, referred to as ‘modes’; the first few modes are illustrated below:

The modes are labelled by an integer index n, and spatially take the form of sinusoids, of the form

,

where is the height of the wave as a function of the position on the string.  The quantity is the amplitude of the wave.  The lowest mode, with n = 1, is known as the ‘fundamental mode’.  Looking at the picture, we see that there are ‘nodes’ (points of zero amplitude) of each standing wave spaced at intervals of along the length of the string; for instance, the n = 3 mode has zeros at and at .  This will be important in a moment.

Also from the picture, we can see that the wavelength of a mode (the distance after which the wave repeats itself) is given by

.

Recalling (see my optics basics post here if you don’t recall) that the wavelength and frequency of a wave are related by

,

The frequency of the modes satisfy the equation,

.

From a guitar design and tuning point of view, there are a few things to note in this equation.  The harmonics () have higher frequencies than the fundamental mode (); in particular, the first harmonic has twice the frequency of the fundamental mode.  The frequency of all modes on a string can be reduced by decreasing the tension of the string (which is how one tunes the guitar) or by increasing the mass density.  In fact, the ‘low’ strings on a guitar are thicker than the ‘high’ strings, which allows them to play lower notes than the ‘high’ strings without a significant difference in tension.**

So what mode is played when a guitar string is plucked?  In fact, some combination of all of them!  The mathematical equation which describes the behavior of a string which has been plucked (pulled and released) may be written as

,

where ‘‘ is the mathematical representation of a sum over all modes of the string.  The relative amplitudes depend on the exact manner in which the string has been plucked; a mathematical calculation assuming a pluck of amplitude at the 20th cm of a 25 cm string gives values, for the first seven modes:

{0.74, -0.30, 0.133, -0.05, 0.0, 0.02, -0.02}.

Part of the distinctive sound of any stringed instrument is the relative weights of the fundamental mode and the higher harmonics.  For an ordinary pluck of a guitar string, the fundamental mode will dominate.  The next couple of modes have some appreciable amplitude, but beyond the third mode the amplitude is negligible.  How do we isolate one of the higher harmonics and suppress the fundamental mode?  We’ve already described how guitarists do it: by lightly blocking the string with a finger at a location which doesn’t obstruct a harmonic.

This is illustrated roughly below:

If we wish to have the first harmonic dominate, we place our finger about halfway along the string (the twelfth fret).  The fundamental mode has a maximum amplitude at that point and will be blocked by our finger, while the first harmonic has a node at that point and will ring freely.  To allow the second harmonic to play, we place our finger about a third of the way along the string (the 7th or 19th frets).  This will block both the fundamental and the first harmonic, and allow the second harmonic to play through.  Similarly, blocking the string at a point a quarter of the way along the string (the 5th or 24th frets) will allow the third harmonic to play, while blocking the others.

In principle, one should be able to play harmonics higher than the third but, as we can see from the relative amplitudes listed above, there’s typically just not enough energy put into these higher modes from a single pluck of the string.

Just to be an experimentalist again, I went to my (acoustic) guitar and measured the string lengths and the positions of the various frets.  The results are listed below:

String length: 25.2”

5th fret: 6.3” from head: ~1/4 of string length: 3rd harmonic

7th fret: 8.3” from head: ~1/3 of string length: 2nd harmonic

12th fret: 12.5” from head: ~1/2 of string length: 1st harmonic

19th fret: 16.6” from head: ~ 2/3 of string length: 2nd harmonic

One other question can be readily answered by the equations above.  We noted that the pitch of the regular note at the twelfth fret and the harmonic at that fret are the same.  Playing a regular note at the twelfth fret halves the length of the string, doubling the frequency.  Likewise, playing the first harmonic at the twelfth fret doubles the frequency; the pitches are therefore the same.  The tonal quality of the notes are different, however; because they were played by different techniques, the notes have different weights of ‘s, i.e. they contain different weights of the higher harmonic tones.

There are a few comments one can make about harmonics as applied to other fields of interest.  First, in Central Asia, notably the country of Tuva, there exists a type of ‘throat singing‘ in which a singer can sing an ‘overtone’ or harmonic pitch in addition to the fundamental.  It is not entirely clear, to me at least, exactly how such overtones are produced, but it is quite remarkable to hear.  There is a website, Khoomei.com, dedicated to the art, and a very wonderful documentary, Genghis Blues, about an American blues singer traveling to Tuva and competing in a competition there (fun fact: Tuva became known to American audiences thanks in large part to physicist Richard Feynman, who endeavored mightily before his death to travel there).

The technique of ‘blocking’ the fundamental mode to produce a harmonic, or vice versa, is also applied in laser physics.  A laser can in principle operate in numerous transverse modes.  Some of these modes (referred to mathematically as Hermite-Gaussian modes) are depicted below as they would be seen ‘head-on’*** (via Wikipedia):

These modes are labeled by two indices, m and n, just as the modes on a string are labeled by a single index n.  In most applications, one wishes to have the laser operate only in the lowest mode (the 0,0 mode); since this mode is the smallest in spatial extent, one can introduce a small aperture into the laser cavity.  The higher harmonics are partially blocked by the aperture, and that loss causes them to lose out to the fundamental mode.  Alternatively, one can block the fundamental mode by putting a very small obstruction (such as a strand of hair or wire) across the center of the laser cavity, which blocks the fundamental mode more than the higher harmonics.  For instance, a thin hair arranged vertically in the cavity would cause the laser to favor the 10, 11, and 12 modes.

As I’ve said, this is one of the things I like about physics: when one looks around long enough, one can make connections between day-to-day casual activities (such as guitar playing) and the physics of such activities, and then make further connections to more advanced physics, such as the behavior of laser resonator modes!  I may come back to other guitar/physics connections later, hopefully with some more skillfully played examples!

***************************************

* For best nerdy effect, imagine me making quotation marks with my fingers as I speak.

** In fact, for a steel-stringed guitar, a solid wire for the lowest-pitched strings would be too rigid for practical use.  To get around this, the low strings on such a guitar are actually one wire with another wire coiled around it.  This coiling allows the mass to be increased without increasing rigidity.

*** Though I always feel compelled to point out that one should never look into a laser beam head-on with your eye!