I was experimenting with spectrum analysis of guitar plucks and this is what i got on an open low E recording:
As you can see the strongest peak is at about 247 Hz, which is a B3 while the peak at about 82 Hz (E2), which should be the fundamental frequency of the string is only the second one. Whay is going on here? Is there something I missed?
Please excuse me if I posted in the wrong place.
This answer can hopefully be useful as an additional perspective on top of the existing ones.
Note that the spectrograph is a plot of (the log of) power output vs frequency (power being energy per unit time). It appears that the fundamental harmonic has been "washed out" into the broad background at low frequency, and perhaps broadened (it's hard to tell because the frequencies are on a logarithmic scale).
There are many reasons why this may be, which could depend on, as others have said, things like where and how you pluck the string, how the other strings and the body of the guitar resonate, and the actual composition and behavior of the string itself. The key point, however, is that the "fundamental frequency" (82 Hz) does, in fact, contain the majority of the power output, but that strings do not actually act like a perfect, ideal string treated in an elementary physics course. There are non-linear corrections to the perfect string behavior which depend on the actual tensile properties of the string (e.g. an ideal string can be modeled as a bunch of springs attached end-to-end with a fixed spring stiffness, but in fact the spring stiffness is a function of frequency and amplitude in a nontrivial way), as well as the resonant effects of the other strings and the body of the instrument, all of which contribute to the "timbre" and rich sound of the instrument.
While for a perfect string one would expect perfect peaks at multiples of the fundamental frequency (delta functions for you math nerds), all of these complicated, non-linear corrections contribute to the broadening of the peaks, and how much each peak broadens depends on the frequency (which we physicists call dispersion). In this case, it appears that a lot of the power in the low-frequency end of the spectrum has been significantly spread out over a broad range, so the low-frequency modes are much more dispersive than the high-frequency ones (this is why you have the broad continuum background at low frequencies which tapers off at higher frequencies). Perhaps others can speculate as to what causes larger dispersion at lower frequencies, but in general, the speed of waves traveling along the string is frequency-dependent, as demonstrated beautifully in this short NPR clip about electromagnetic waves traveling through the atmosphere caused by lightning strikes and detected at the South pole.
I quickly played around with this, and we see something interesting,
As shown in this Physics.SE answer, we would expect that the amplitudes of the harmonics go as the inverse of frequency squared. The power delivered is proportional to the square of the amplitude, and so would go as 1/frequency to the fourth power. I have indicated this by the red line on the spectrograph. Following that red line to lower frequencies would give a rough estimate of the expected height of the fundamental frequency if the string were perfect and there were no resonance effects. However, it appears that the fundamental and to a lesser extent the second harmonic (164 Hz) have lost amplitude and their "spectral weight" has been spread out across a broad range of frequencies at the low end. It's curious that this effect seems to decay as the inverse of frequency squared, indicated by the black line. I don't have an intuitive explanation for this behavior, but I am sure someone could take a stab at that.
One more observation is that the third harmonic of the low E string occurs at about 247 Hz, which is about the same frequency as the D string, this is the third peak. It's possible that the D string is resonating and adding additional intensity to the third peak, but I can't estimate how much we would expect it to add. Similarly the fourth harmonic should resonate with the high E string.
Some more info can be found here.
It's probably impossible to give a definite answer by just looking at the spectrum, but here are a few thoughts:
[EDIT: the first point below has become irrelevant since the OP edited the title of the question.]
first of all, the strongest harmonic is the third one, not the seventh, if you start counting from the fundamental, i.e., the strongest harmonic has 3 times the frequency of the fundamental, and hence it is the fifth plus one octave (some people call this the second harmonic).
Note that the strongest frequency is the frequency of the open B string. What could have happened is that the B string started resonating when you plucked the low E string.
The same can be true for the high E string with a frequency of 328 Hz (4 times the fundamental), which also has quite a strong peak in the spectrum.
So you could try to repeat the same experiment while making sure that all other strings are muted and can't resonate.
There are way too many unknowns to completely answer this. Also, I am not used to looking at frequency on a log scale but I suppose it's all good.
The strength of the harmonics depends on the attack. It is very common when plucking a steel string guitar near the bridge to have strong upper harmonics. I would venture to guess that, just based on geometry, your strongest harmonics would be the ones with an anti-node close to the location of the pick.
If you are mathematically inclined you can solve for the ideal spectrum of a plucked guitar by doing the Fourier integrals over the shape of the string, taken to be a triangle with the pick location at one vertex and the nut and bridge at the others. This is grossly simplified but would serve to illustrate conditions under which you can excite upper harmonics.
If you are playing an acoustic guitar here then there are even more factors that would contribute to enhanced harmonics. For one. all the other strings are subjected to a driving force (the vibrating bridge) and will respond with their harmonics. As an example the Low E string will have cause the B and high e string to vibrate as those match the n = 3 and n = 4 harmonic of the E. But also the A string will vibrate at its n = 3 harmonic which matches the n = 4 of the E, the open e string 2 octaves higher. Etc. To really predict which harmonics will dominate you need to count up all the possible sympathetic resonances. Energy will be conserved so you cannot expect more total energy than you gave the system, but this will get redistributed over time to different parts of the guitar. If your spectrum is over a long time period then what counts is which harmonics are excited for longer periods. With damping, the higher frequencies will die faster but once the fundamental of the higher strings gets excited and steals that energy away it usually stays there. Over long periods of time one can hear the higher harmonics of the guitar emminateing from the body and this may be direct resonance of the plates in the body as well as the higher string fundamentals.
Also, if you mic'ed the guitar or speaker then you have lobe effects in the spatial distribution of the acoustic energy and these are frequency dependent. You could have set yourself up to geometrically kill the reception of the fundamental and enhance the harmonics. So, it isn't even fair to try and describe the phenomenon solely based on knowledge of string behavior.
So, as you can see there is a lot to consider. What exactly are you doing?
What kind of guitar, electric or acoustic?
If electric did you mic the speaker or plug the ax into a data acquisition system?
How did you mic the system if a mic was used?
Is this the entire window of time that the sound was captured or just a millisecond?
How, and where you pluck the string also affect the overtones dramatically.
If you pluck the string on the 12th band (half string), you get a lot of the first harmonic (fundamental). If you pluck it 1/3 from the saddle, you'll get a lot of the second harmonic (octave). Now, if you pluck it 1/8 from the saddle, you should get a lot of the 7th harmonic, the minor seventh.
Edit: I just tried it out on a guitar with copper wound nylgut strings, and the effect is there, but it is not pronounced, arguably negligible, in particular in the higher harmonics. However, my original statement holds: The position where you pluck the string also affects the harmonics drastically!
Generally, the closer to the end of the string you pluck it, the stronger the higher harmonics will sound.
Heck, this is why guitar is such a wonderful instrument, you can shape the sound of each note with your plucking position!
Before drawing conclusions from a spectrogram of this type, it's important to understand the limitations of the algorithm being used -- in particular, the errors that are present in the DFT.
Errors in Apparent Power
Using Audacity, this is what a perfect, computer-generated 82.09Hz sine wave looks like.
(Feel free to try this yourself, using Audacity's Generate->Tone function):
Notice how wide the peak is. We can be sure that this is not due to dispersive effects, because we know beforehand that the wave is perfect (neglecting quantization error and so forth); it was generated by a computer.
If I change the frequency just a bit, to 80.75Hz, the resulting peak is considerably more narrow, and the noise floor drops away:
And, just for comparison, a wave of the same amplitude, but at 6000Hz:
Note that all waves were generated at the exact same amplitude (0dB, or "1.0" in Audacity). As a factual matter, they all contain essentially the same power. It is the DFT representation that is so misleading.
I chose those frequencies because they have a particular relationship to the frequency "bins" present in the DFT.
I will not go into a detailed explanation of DFT principles here, but basically, these differences arise from a complex interaction between:
For all the plots above, I used a 16384-sample DFT, 44100Hz sample rate, and a "Hamming" window function. This is based on an educated guess about what the OP is using.
TL;DR:
The bottom line is that it's difficult to compare the total power in two different waves by estimating the area under the graph, in a log/log DFT spectrum like this. Just because the fundamental has a wide peak, does not mean it has more power than taller, narrower peaks. It is entirely possible the OP actually does have less power in his fundamental tone.
Analysis
The actual reason for the louder 2nd overtone is likely a combination of several factors already mentioned in other answers.
Note that the nominal attenuation is > 5dB at 82Hz. That's under ideal conditions; low-frequency attenuation is also a function of distance to the speaker cone (and mic orientation, among other things). The manual recommends a 2.5cm placement (I believe the above chart was created under such conditions). Depending on their setup, the OP could be getting a dramatically different frequency response.
The location and manner in which the string is plucked. As mentioned by @fraxinus and others, this will emphasize certain harmonics. For an electric guitar, pickup location will also play a role.
The fact that the fundamental simply does not have to be the most powerful tone. Even though the fundamental is likely to be the most powerful (all things being equal), there is nothing that says it has to be, and your ear tends to perceive the tone favorably even if the fundamental is highly attenuated.
There is no reason to expect that the fundamental frequency will be the strongest one in the spectrum of a musical note. A common model of hearing, which works pretty well in most cases, is that the ear-brain system picks out the period of the sound. The period is the same regardless of whether the fundamental is present. This is the reason that you can hear low bass in earbuds. The speakers are much too small to be able to reproduce the fundamental at all, but your ear hears the higher harmonics and perceives the pitch as being the pitch of the fundamental.
Physically, the spectrum is the product of a bunch of factors. First, when you pluck the string you establish a certain spectrum based on the initial shape of the string when it comes free from the pick and starts vibrating. For example, if you pick the string exactly at its center, then the first harmonic (twice the fundamental) is completely absent, because that wave has a node (point of zero vibration) at the center.
Next the power spectrum of the string gets filtered through the response of the sounding board, which has a bunch of different resonances. See this answer: https://music.stackexchange.com/a/77480/9480 (This is assuming it's an acoustic guitar.)
Then the sound spectrum gets filtered through the response of the mic. If the mic you're using is a cheap computer mic or the mic in a webcam, then it's probably intentionally designed so that it has little response to frequencies as low as 82 Hz. These mics are designed to pick up human speech. Intelligibility of human speech only requires the frequency range of about 300-3000 Hz. Letting through lower frequencies will just make it harder to understand you when you're doing a video call, since it lets through random sounds like vibrations from passing trucks.
Matt L's answer proposes that this is due to a sympathetic vibration of the other strings. That seems extremely unlikely to me, both because it ignores all the effects above, which are guaranteed to be present and likely to be huge, and because the sympathetic vibration effect is likely to be weak, whereas we're trying to explain a huge effect. Anyway, this hypothesis is easy to test experimentally, so I did that. I don't own a guitar, so I tested this with my viola, plucking the low C string. The viola has a G string that is tuned a fifth above the C string, so that sympathetic vibration of the first harmonic on the G string can be excited by the vibration of the C string. Matt L's explanation makes several predictions: (a) the sound spectrum should change dramatically when the open strings are muted; (b) when the open strings are muted, the fundamental should be the strongest frequency; and (c) playing a fingered note like Eb should not produce any sympathetic vibrations and therefore should also have a spectrum in which the fundamental is the strongest frequency. Below are three spectra that I measured:
Spectrum 1 is from plucking the C string (130 Hz) while leaving the other strings open. Spectrum 2 is C with the other strings muted. Spectrum 3 is playing Eb. These observations disagree with all three of the predictions made by Matt L's model. In all cases, the fundamental is much weaker than the harmonics. The most prominent frequency in all cases is the first harmonic (twice the fundamental).
If anyone wants to try this with an acoustic guitar, that would be interesting. You can do this by recording a note using the open-source, multiplatform app Audacity. After you record the sound, select the loudest part of the note, go to the Analyze menu, and do Plot spectrum.
It depends on where you pluck.
If some harmonic has maximum amplitude at the point where you hit the string, it gets a great deal of energy and if it has minimum (node) there, it gets no or almost no energy.
If you pluck the string exactly in the middle, you'l get pretty specific sound that has no even harmonics.
In order to get maximal 3th harmonic (2nd overtone) you have to pluck at 1/6 of the string length. At 1/4, you will get 2nd harmonic/1st overtone max.
On a piano, where strings have a fixed length and no fretting, the hammer position is precisely engineered to excite a specific combination of harmonics (I think somewhere at 1/8 length for most strings in order to suppress 7th and 9th that are less pleasant at the cost of almost completely losing 8th)
p.s. Obertones and harmonics of a string probably need some clarification: Harmonics are counted at multiplies of the base frequency, 1st is the base, 2nd is an octave higher, etc... Obertones (well, oVertones) are counted next to the base tone and 1st overtone is ~= 2nd harmonic. That's for strings, air instruments have other, different overtone series.
I want to make some super-nerd comments. When a string is first plucked it briefly supports all possible frequencies. After a short time, it settles down and supports only the harmonics. Coming cold to this, the slow background continuum looks like that.
The strengths of the harmonics (correctly described as the area under the peaks) are not certain to follow any pattern. Think about plucking a harmonic tone by touching the center of the string. This has zero power in the fundamental. The spectrum really depends on how the string is plucked.
The width of the harmonic peaks is ultimately limited by the time that the tone is sustained. In this case that’s probably the averaging time of the instrument. The longer the tone, the narrower the width. In fact, the width of the peak multiplied by the length of the sound equals one. Said differently, the better you know the time of the tone, the less you know the tone frequency. This “uncertainty principle” is built into the math. Think of a really short tone on a synthesizer. It’s almost a click, and has a huge range of frequencies. Only a very long note has a single precise frequency.
In quantum mechanics, particles are waves, and this exact effect means that you can’t know, for example, both the exact position and exact velocity at the same time. The Heisenberg uncertainty principle. It isn’t really from quantum mechanics: it’s from music.
Another point to consider is that even if the string were stimulated in such a way that it vibrated perfectly with nothing but the fundamental, the sound of the guitar could (and likely would) end up with other harmonic content. Consider, for example, how the tension on the string varies as it vibrates. It will be at minimum when the string is crossing the centerline, and at maximum when the string reaches the end of travel. Since the string crosses the centerline twice per oscillation, the tension will go between minimum and maximum twice per oscillation. Since the force the strings exert will vary with tension, this force will likewise vary twice per oscillation, thus producing a second harmonic component even if the string were vibrating in ideal fundamental-only fashion.
