### ‘String’ Theory- Solving the Guitar

A few months ago, I gave a talk about some findings related to Guitars. Here are the details:

Basic Facts:

• Vibrations of the strings in a guitar, are transmitted to the ear through disturbances in air pressure (sound waves). In an acoustic guitar, the string vibrations cause the hollow body to vibrate along, thus amplifying and adding pleasant overtones to the disturbances, which propagate as sound waves. In an electric guitar,  the string vibrations are picked up by the pickup coil inductors, which transmit the electric signals to an amplifier.
• The group and phase velocities of sound waves are the same as they come through a non dispersive medium here. So, we don’t need to worry about different frequency waves coming  at different speeds.
• Small variations in frequencies go unnoticed by the ear, atleast in the frequency range of a guitar ( around 82 Hz- 1319 Hz), with more resolution in the lower frequencies than the higher frequencies. This behavior holds for the whole spectrum, as obviously you can distinguish between 1 Hz and 1.5 Hz, but it is amazingly difficult to distinguish between 10000 Hz and 10020 Hz.
• When more than one pure tone (i.e. sinusoidal sound wave) is played at the same time, the waves interfere to form a wave with a time period equal to the LCM of all the constituent waves.

Experiments and Observations:

I noticed that Fret widths change in a very non trivial way, as one goes down the neck of the guitar. So the basic question leading to these findings was:

Why and how do fret widths change the way they do ?

So, I

• measured all the fret widths along the neck,
• calculated the ratios of consecutive fret widths, and
• measured the distance of each fret from the bottom of the Guitar.

Observations (in cm):

Conclusions:

• From the distance from fret bottom to string end in the table, we get that if we start with any fret i, the fret 12+i  is half the distance from the bottom, compared to fret i.
• Motivated by this, and the Distance from bottom graph, we see that $e^{0.058*12} = 2$ and $e^{0.058x} = 2^{x/12}$.  Therefore, we get the brilliant result that each successive fret’s distance for the bottom is $2^{1/12}$ of the previous fret’s distance.
• This conclusion also explains the constant ratio of successive fret widths as  $2^{-1/12} = 0.94$.

So how does a guitar work?

• The Guitar is tuned by turning the knobs (tuning pegs) on the head. The sole purpose of this is to increase the tension in the corresponding string. Once tuned, the tension in the string is effectively constant.
• The frets are there to change the effective length of the string, i.e. the part of the string that vibrates, i.e. the part of the string between the pressed down fret and the end of the string.
• When we pluck a string, we are in effect, activating the fundamental mode corresponding to the effective string length. Though, other overtones are present, the amplitude of the fundamental mode is the greatest. In fact, if we pluck a string at half the effective string length, we hear a purer tune than when plucked above the pickup coil.
• The 5th Fret of the 6th string produces sound at 110 Hz. Armed with the knowledge of the previous slide and how to tune different strings relative to each other, every position on the guitar can be mapped to a specific frequency.

The Logic behind tuning:

For resonance on a string with fixed ends, the condition is λ= 2L/n where L is the effective string length, n a natural number, and λ the wavelength.

Therefore, $f=v*n/2L$, where v is the velocity of waves on the string, and f the frequency.

Now,  $v= (T/\mu)^{1/2}$.

As, each string’s tension can be varied from the tuning pegs and the mass/length($\mu$) is different for each, according to whether the string is coiled or not, we can set the frequency of any position on the guitar. Now the word tuning is commonly used by guitarists to mean tuning the different strings of a guitar relative to each other. There are many ways of doing this (the most common being the EADGBE tuning), but that is preference and convenience dependent. But the more physical tuning, is the tuning of frets relative to each other on a single string. This is the same for all strings and cannot be changed, which provides a clue to it being a more fundamental tuning. The most important observation is that the sounds transmitted from frets separated by 12 frets are perceived by us as similar. The note from the $12+i_{th}$ fret is said to be an octave (for reasons which will be apparent later) higher than the note from the $i_{th}$ fret. The octave(also refers to the interval between a note and it’s octave) is divided into 12 notes, each note corresponding to one of the 12 frets between the $i_{th}$ and the $12+i_{th}$ fret. The qualifier ‘Harmonic’ comes from music. As can be seen, the condition for resonance is λ= 2L*(1/n), where 1/n is the nth term of the Harmonic Series. In terms of frequencies, the Harmonic series is an arithmetic progression, as f=(v/2L)*n.

Now, as mentioned before, we always produce only fundamental modes in a guitar. This is why we change the effective length of the string instead of n, to vary frequency.

Therefore, from the previous slide, we get the astonishing result that the fundamental mode of a string and the fundamental mode of half the string sound similar. Therefore an octave is twice the frequency of the root note. Note that this also corresponds to the first overtone of the original unhalved string.

So, for any frequency f, $2^n$ times f sounds similar(n here is any integer). The use of this property is called transposition.

Therefore, the word pitch is better suited to music. Although the actual frequency of a sound may be doubled or  halved, the pitch is said to remain the same.