# Why is the 5th stronger than the 1st in guitar spectrum?

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.

• Don’t you mean the 5th, not the 7th? If the fundamental is E2 then the octave, E3 would be the first natural harmonic and an octave plus a 5th, B3 would be the second one. Jul 9 '20 at 8:38
• @Tim maybe their logic is if you’re dividing something into equal segments the first division is by one? Jul 9 '20 at 14:25
• @AlessandroCarinelli no problem, interesting question and graph! It would be interesting to see the results played from different locations on the instrument and at different volume levels to see how much difference there is. Jul 9 '20 at 15:40
• Related and recent: physics.stackexchange.com/questions/563765/… Jul 9 '20 at 16:22
• I would say that the total power in the peak at 82 Hz is quite a bit more than the total power in the peak at 247 Hz. The peak is broadened and thus the total power has been spread out among surrounding frequencies and the peak is somewhat lower. If you plotted the integrated power for each peak in a window of plus or minus 10 Hz, the 82Hz peak would clearly be dominant
– Kai
Jul 9 '20 at 16:54

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.

• +1 I think this is the most valid explanation. The total energy (integral of the plotted value) in the neighborhood of each harmonic is descending as the frequency of the harmonic increases. You have suggested a lot of ideas for further research. Jul 9 '20 at 18:15
• Be wary of what the DFT is telling you; wide peaks are not necessarily due to dispersion, and low peaks are not necessarily representative of a lack of power. See my answer, below. Jul 11 '20 at 15:47
• The third harmonic of E is B, not D. The D string is about 147 Hz, which is not particularly close to 247 Hz. (More precisely, if E is 82.5 Hz then D is 146 + 2/3 Hz using just intonation whereas the third harmonic is exactly 247.5 Hz.) There is however a B string, the fifth string, that should be tuned to 247.5 Hz. Jul 12 '20 at 13:05

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.]

1. 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).

2. 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.

3. 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.

• I know that is the 3rd harmonic, in the title i was referring to scale degrees (and was also wrong, i've edited it) Jul 9 '20 at 10:37
• This seems like a very unlikely explanation to me. Much more likely is just that the acoustic and electronic signal chain imposes layer after layer of filtering.
– user9480
Jul 9 '20 at 15:45
• @BenCrowell: If you try plucking a low E string on a well-tuned guitar you'll notice that the b and e strings start vibrating, so I don't agree with this being a very unlikely cause of what the OP sees. Jul 9 '20 at 16:07
• @BenCrowell: that's perfectly okay with me, let's wait for the OP if he/she is willing to repeat the experiment with muted strings. Your assertion "normally pretty weak" is of course also purely speculative. Weak compared to what? Probably not compared to the amplitude of the third harmonic. Jul 9 '20 at 16:54
• @BenCrowell, the problem here is that there is not enough info from the OP to even decide if you or Matt L. is on the right track (pun intended). I disagree with your comment though. If this is an acoustic then a lot of the initial energy could have been redistributed to the open string in sympathetic resonance where it will stay. You can clearly hear it in any acoustic guitar. Also, there is no telling what the time window is for the signal and that will change the spectrum.
– user50691
Jul 9 '20 at 18:14

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?

1. What kind of guitar, electric or acoustic?

2. If electric did you mic the speaker or plug the ax into a data acquisition system?

3. How did you mic the system if a mic was used?

4. 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!

• This is incorrect and actually the opposite of what happens. For instance, if you pluck a string at 1/3 of its length, this is the node point of the second harmonic, the point at which the string doesn’t move if you play this harmonic. Therefore, plucking the string at this point will excite very little of this harmonic, as it is initially moving most at this node point. You can test this very easily: if you touch a node of the second harmonic at fret 7 (1/3 string) and try to pluck it at the other node (fret 19, 2/3 string), you get hardly any sound. Jul 9 '20 at 12:31
• However, you are correct that plucking closer to the bridge excites more of the higher harmonics, and so you can change the tone of a played guitar note (and so it’s spectrum) in this way. If you pluck a guitar string at fret 12 (half way) you are encouraging it to vibrate most over its full length. So this allows the fundamental to come through most clearly, not the octave harmonic. Jul 9 '20 at 12:40
• Finally, it is worth noting, that the acoustics of this situation are the same, no matter what you name the different overtones! I see that in this thread some people refer to the fundamental as the first harmonic, whereas I would refer to the octave harmonic (1/2) as the first harmonic. Jul 9 '20 at 12:42
• Yeah, @Tim. I checked out the Wikipedia page for harmonics and I think I have been using the wrong terminology. To be really clear it seems it’s best to talk about the fundamental and then overtones 1,2,3 etc. Jul 9 '20 at 15:57
• I think you got your terminology mixed up. Plucking near the 12th fret will KILL the octave harmonic. You need a node there. You will get mostly fundamental.
– user50691
Jul 9 '20 at 18:18

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.

• I'm sorry but a viola is really a bad substitution for a guitar for the purpose of this experiment due to its much smaller sound box, so any resonance effects on other open strings are totally different. It's not that I'm sure that my answer correctly explains the observed phenomenon, not at all, but your experiment doesn't prove or disprove anything. Jul 9 '20 at 17:36
• This is like saying oranges are not a citrus fruit by taste testing a banana. Sorry but this is a really bad rebuttal to what has been stated so far.
– user50691
Jul 9 '20 at 18:19
• I would consider deleting it.
– user50691
Jul 9 '20 at 18:19
• I think you've got the most accurate answer so far, Ben; I've tried to buttress it with my own. Jul 11 '20 at 15:45

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:

• The sample rate;
• The actual frequency being played;
• The DFT size (number of samples);
• The amount of audio being analyzed (the length of the recording);
• The choice of "window function" (this item has a particularly outsize effect. Audacity has several to choose from), and:
• The fact that the OP is trying to represent a DFT with over 8,000 data points on an image which is only 1300 pixels wide. Some points are being omitted.

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.

1. Filtering effects from the microphone, as mentioned by @BenCrowell. The importance of this cannot be overstated. Consumer-grade computer microphones tend to have very "peaky" responses; they pass some frequencies much more easily than others. Even if the OP is using something more professional-grade, like an SM57, the frequency response is less than uniform:

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.

1. 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.

2. 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.

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.

• Attack is only half the answer. There is also the set up. If the guitar is mic'ed then lobe patterns are frequency dependent and the OP may have not detected the fundamental at all.
– user50691
Jul 9 '20 at 18:23
• To me, the spectrum looks like the string has been plucked rather close to the bridge: It's overtone series is pretty complete. It has most certainly not been plucked anywhere near the 1/3, 1/4 (normal guitar plucking position), 1/5, or 1/6 point. So, indeed, a rather weak fundamental is to be expected. Jul 9 '20 at 21:53

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.