Why does string strummed with finger sound different from the one strummed with pick?

Question

What is the physical reason for difference in sound when using fingers instead of pick and vice versa?

Answer 1

Per a suggestion, I am converting my comment into an answer.

WARNING: math ahead

(uh-oh, it looks like Music.SE doesn't support MathJAX -- I am going to go ahead and post the TeX code anyway and try to explain it in plain english along the way. I also added a meta request to see if we can't fix the MathJAX problem.)

Discounting the inharmonicity due to the bending stiffness of the string, and discounting the damping behavior, the behavior of a guitar string can be modeled as a one-dimensional initial value problem (partial differential equation in space and time with boundary and initial conditions).

$$
\frac{\partial^2 u }{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \\
u(0,t) = u(L,t) = 0 \\
u(x,0) = g(x) \\
\frac{\partial}{\partial t} u(x,0) = 0
$$

In english:

First line: The acceleration of any particular bit of string is proportional to the local curvature.

Second line: The two ends of the string are fixed.

Third line: At the moment of release, the string is pulled to some particular shape.

Fourth line: The string is simply pulled to a shape and released -- the "plucker" does not impart any velocity to the string (i.e., you are pulling and letting go, not "throwing" or "pushing" the string).

---

The solution to this equation cannot really be written in a closed form for a bounded problem. Instead, the behavior generally consists of a linear combination of eigenfunctions, which are of the form

$$
u(x,t) = \sum_{n} k_n \cos \left(\frac{n \pi x}{2L}\right) \cos \left(\frac{n \pi ct}{2L}\right)
$$

If either you can see through this TeX code or Music.SE enables MathJAX, those who are mathematically inclined should recognize this as simply a weighted sum of the harmonics of the fundamental frequency $c$.

What does all that mean?

The sound you hear is simply a function of the different $k_n$ values (where $n$ is the number of each harmonic). Those values are simply the Fourier coefficients of the initial shape of the guitar string at the exact moment of release.

I'll spare you the math on these, but in general, the rule of thumb is that the "pointier" the initial shape, the higher the $k_n$ values are for large values of $n$ and the more brilliant the sound. The rounder the shape, the smaller those values are and the duller the sound.

Ok, but how does this circle back to picked vs fingered?

Simply put, the pick is stiffer than your finger, and it comes to a sharper point than your finger does. If you look at the initial shape at the exact moment of release, the picked string is going to come to a sharper point than a finger-plucked string. The $k_n$ rules above then apply, and the rest is just math.

What about the assumptions?

Thick guitar strings will have an inharmonicity term proportional to the fourth spatial derivative of the displacement. This severely complicates the appearance of the solution, the end result of which is that higher level harmonics are not exact multiples of the fundamental frequency. For the purposes of answering this question, this is immaterial.

There are also damping terms proportional to the velocity of the string (first time derivative of displacement) for viscous damping, and proportional to the square of the velocity of the string for aerodynamic damping. The general effect of these terms is that each harmonic's $k_n$ value fades to zero over time, with higher harmonics fading faster than lower harmonics. Again, for the purposes of answering this question, this is immaterial.

Answer 2

The reason for the difference in sound is that the release of the string from the pick is faster than with fingers, which means that fewer of the upper harmonics are damped as the string is released. This gives the pick a brighter sound than the fingers.

Answer 3

The difference is caused by the different shape of the plucking implement. One easy way to verify this at home is to take your pick (same material and thickness), and pluck the strings with the back of the pick or the side of the pick. The tone will be different because of the different profile of the pick (pointed versus rounded).

Also, the thickness of the plucking device makes a difference.

The primary link between the shape and thickness of the plucking device and the change in tone is the difference in initial displacement of the string:

Quoting (emphasis mine):

Plucking with a sharp object such as a plectrum accentuates the higher harmonics in contrast with plucking with the finger or a soft object. This is because the initial displacement is highly angular in form. In order to achieve such a displacement curve... many higher order modes must be introduced, which would not have been the case if the curve had been more rounded

^{The Musician's Guide To Acoustics, Campbell & Greated, 1994}

A more rounded and/or softer plucking instrument produces a more rounded initial string displacement which excites fewer high harmonics in the resulting motion of the string.

Regarding the thickness difference between the pick and the finger (or between different picks), we already have a Q&A here: Why do thicker guitar picks result in a darker tone color?

Quoting the accepted answer to the above question:

Thicker picks (tend to) remain in contact with the string longer. The impulse provided to the string is of longer duration. A longer duration pulse imparts more lower frequency and less higher frequency content.

See: http://acoustics.org/pressroom/httpdocs/160th/carral.html

Answer 4

As an addition to the other answers, here's the frequency spectrum of an open low E string of an electric guitar (a Squier strat with a bridge humbucker), strummed with the fleshy part of the finger, plucked with a fingernail, and picked with a plectrum (a Dunlop Delrin-500 .71mm). I played each note a couple of times and then selected one that sounded representative, and was of somewhat equal volume.

For the attack phase I used the first 250 ms of the note, for the sustain phase I used the part from 1 to 1.5 seconds. The graph shows the level in dB of frequencies from 50 Hz to 15 kHz on a logarithmic scale.

^{I don't make any claims as to how scientific this is, but it's better than looking at the black body radiation of an incandescent light bulb :-)}

I think it's fair to say that there is a clear difference in the spectrum of the plectrum-picked note; the harmonics up to around 10 are of comparable amplitude, whereas in the finger-plucked note the first three harmonics clearly dominate the sound. The frequencies between 1 and 2 kHz are also more prominent, and the troughs around 2.5, 6 and 10 kHz are narrower and they all but disappear in the attack phase.

Answer 5

Good answers, but they're either incomplete or very technical.

The simple answer is that the shape and stiffness of your finger is different from the shape and stiffness of the pick. For this description, let's presume that the pick is an infinitely hard single point that releases the string instantaneously. In reality, a pick is an edge, like a tiny violin bow, but that can be ignored if you're comparing it to a finger.

By comparison, your finger is a soft, round shape. When it releases the string, the string slides across the finger's surface. Imagine that you ran a circle of sandpaper across the string. Each of the grains would provide a tiny pluck. Similarly, your finger also sends many microscopic plucks (waves) down the string.

Unlike with a pick, These waves will have origins that are the entire width of your finger, and that will slightly alter the harmonics that they produce.

Your finger is also soft. This means that any wave that runs into it mid-pluck will be absorbed by your finger's surface, decreasing the strength of the wave as it travels down the string. This is a much smaller effect than the one you get when you hold a finger above the string at a fret in order to isolate a harmonic, but it's still noticeable in the resulting sound.

The combination of these factors results in a blurring of the tone that is produced. Although not nearly as extreme, the audio effect can be compared to the visual effect of the differences between the spiky spectrum that a florescent bulb produces

and the smooth spectrum of black body radiation you get from an incandescent bulb.

Answer 6

Another factor not yet mentioned is the direction the string is traveling as it leaves the plucking appendage or implement. Guitar strings support two primary modes of vibration--parallel to the body and perpendicular to it--and the resonant frequency in these modes will be influenced differently by the shape of the contact points on the nut, frets, and saddles, and (for electrics) by the pickups. Some of the "warmth" of a guitar's sound comes from the interplay of these vibrations as they go in and out of phase, and the initial direction of motion will influence that behavior.

Incidentally, on an electric guitar, this effect may be demonstrated to a somewhat extreme degree by raising the neck pickup excessively and then playing notes high on the fretboard. On one of my guitars, doing that can cause a single string to play two pitches simultaneously that are more than a semitone apart. Having the pitches of the modes be that far apart probably wouldn't be useful musically, but it demonstrates their existence and independence.

Answer 7

It's because your fingertips are fleshy pads that cover hard bone. When they strike the string and release it, there's not a strong separation, and your skin mutes and muffles the string somewhat. The skin is elastic and will compress somewhat when striking the string, instead of just pushing it more directly. When it leaves the string, it uncompresses somewhat, so it sort of "stays behind" a brief moment (milliseconds) and muffles the string's vibration

Meanwhile, the pick is usually stiff plastic, not very elastic (though they sell different stiffnesses of picks). In any case, it's not as elastic as skin.

When the pick plucks the string, it's a smaller, stiffer surface that strikes the string. When it releases, the pick doesn't rebound (as much as your fingertip skin does), and the string is free to vibrate.

You can demonstrate these principles to yourself by playing finger-style with thick cotton gloves. The cotton is even more elastic, and will muffle the sound even more.

Then try playing with finger picks (what banjo players often use, to keep the sound bright and unmuffled). Instead of elastic skin striking the string, it's now stiff plastic. It will sound more like a string plucked with a guitar pick.

Answer 8

Sound is caused by vibrations, and the way something vibrates depends on what hits it. Knock on a wall with your knuckle and then with a hammer. They will sound different because the physical properties of a knuckle are much different than a hammer. This will result in a difference in how the force propagates through the other medium (the wall) and thus will result in the atoms of the wall vibrating in a different manner. Likewise, a string will vibrate differently depending on the physical properties of the object that hits it.

The exact reason for why the atoms in the string are vibrating in the manner they are can be release speed, but there are lots of factors that can cause slight differences in the resulting vibration which may or may not be audible to humans.