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Friday, March 27, 2009
Ok, so I've decided to put down my guitar today. I will not play guitar for more than an hour a day on weekdays, at least until my exam is over.
However, just as I was putting it down, I thought to myself, how can I apply what I've learnt in 113 to whatever I'm doing and I was just thinking about guitar harmonics. I thus decided to derive a mathematical expression for which frets on which one is able to find harmonics.
In music, the most common tuning is the equally tempered scale. Say we have a tuning fork, which gives us the note of concert A at 440Hz. At one octave up, the frequency of the note would thus be twice that of 440Hz, which is 880Hz. The same applies for all notes, an octave up would mean that the frequency of the note would have doubled. The frequency of each semitone from our key note (meaning A# if our key note is concert A) must therefore be some multiple, r, of our original frequency.
So for A#/Bb, we have
f' = r*f,
for B, we have
f'' = r*r*f
and so on. This goes on until we have our octave which is 2 f = f * r^12. Hence, we find that the ratio between each semitone,
(stupid blogger scales the pics weirdly....)
For a guitar string, every time we pluck an open string, we receive a fundamental frequency that is related to the length of the string by f = v /2L. If say we press down within the first fret, this fundamental frequency increases by 1 semitone, as such we have,
which is the length of the distorted string. As such, the ratio between frets is also 1/r, where r is the twelfth root of 2.
Harmonics are produced by creating standing waves on the string (refer to diagram below).
(photo from wiki)
For the open string, or the fundamental frequency, we create a standing wave as in the first diagram. However, to create a standing wave as in the second diagram, we need to create a node (or a non-vibrating point of string) where it is at half the length of the string (this corresponds to the twelfth fret). For subsequent harmonics, we have to create nodes at the fret which corresponds to the corresponding fraction of string.
Furthermore, we know that the sum of all the “frets” (including those off the guitar neck) is the length of the string. As such, we have the following expression. Note that k represents the fret number.
as the series is geometric with a common ratio of 1/r.
For the first overtone, the frequency of the standing wave produced is twice that of the fundamental and as such, is one octave higher than the fundamental.
The length of string required for the first node is half the entire length of string. Thus,
Solving for k, we find that k = 12.
For the next overtone the frequency of the standing wave is 1.5 times that of the fundamental, which makes it somewhere around a perfect 5th.
The length of string required for the first node is one quarter the length of string, thus using the same process, we find that,
and k = 7.
Thus, for any overtone in general, the fret number can thus be found by the eqn,
where n is the n-th harmonic :)
