Why does the same frequency sounds differ?

[sophiecentaur]

For the linear problem, one approach might be to do a Fourier decomposition of the initial "plucked" shape, as MikeW describes, into a sum of string modes. Since you know the amplitude (and phase) of every mode at t=0, and you get the time evolution by advancing the phase of all the independent modes and adding them together again.

Once the modes are established, in their relative amplitudes, then the output (time varying) waveform can be established. My point was that the modes are not simply the result of Fourier analysis of the 'spatial shape' of a system - that is found differently. The modes are strictly Overtones and not necessarily harmonically related - although they are pretty near, numerically for a string. It is probably easier to consider a 'bowed' string, rather than a 'plucked' string because it is a steady state situation.

 

[olivermsun]

My point was that the modes are not simply the result of Fourier analysis of the 'spatial shape' of a system - that is found differently. The modes are strictly Overtones and not necessarily harmonically related - although they are pretty near, numerically for a string.

The "modes" ARE the spatial Fourier modes of the string! For each mode number n with wavelength \lambda_n = L/n, where L is the string length, there will be a corresponding overtone at frequency f_n. This is results directly from the frequency-wavelength relationship f_n = c/\lambda_n, where c is linear wave speed in the string. Since every mode has nodes (zeros) at the ends of the string, it follows that every overtone must occur at exactly a harmonic of the fundamental frequency f_1.

For the ideal undamped string, it means that you can get the frequency spectrum for all time by simply Fourier transforming the initial shape of the string in space and converting the wave numbers to frequency using the wave speed relationship!

It is probably easier to consider a 'bowed' string, rather than a 'plucked' string because it is a steady state situation.

A bowed string is NOT a steady state condition and is much more complicated than a plucked string! (See "Helmholtz motion" for anyone who is interested.)

 

[sophiecentaur]

A string is probably the most ideal of all the instruments and a piano or harp are seen more ideal. You say the end is the same for all modes. That can't really apply to strings resting on a nut, bridge or fret. With an air column (very three dimentional) it's even more extreme, so the modes do not have the same end points. Is this not acknowledged by the use of the term 'end effect'?
This means the modes are not necessarily harmonically related. So we are now dealing with what can only be described as something different: why not Overtones?
Would it not be inconsistent to use the term Harmonic sometimes and Overtone on other occasions? Where would you draw the line?

 

[Borek]

A synthesized pure frequency sounds very dull.

Quite close to recorder if memory serves me well.

 

[sophiecentaur]

Recorder. AArrrrgh!
"Little bird, I have heard, what a pretty song you sing... etc"

 

[someGorilla]

Would it not be inconsistent to use the term Harmonic sometimes and Overtone on other occasions? Where would you draw the line?

Usage, tradition, history. Musicians do call them harmonics, even if formally they are not. Study of harmonics began far before precise instrumentation was available to show the difference between an 880 Hz and an 880.03 Hz overtone. "Harmonics" was used in this sense at least from the Renaissance while a glance at a dictionary gives me "overtone" as a 19th century calque from German (so I suppose from Helmholtz). So let musicians and physicists use whatever term they like!
Of course, strictly speaking vibrating bodies don't generate exact pure harmonics.

To answer the OP, you won't get a naturally-sounding waveform like an instrument's just by adding more frequencies. Check attack-decay-sustain-release (which is however a very poor model if you want very realistical results.) For many instruments the attack phase of the sound is very important to make it recognizable. A piano doesn't sound like a piano if you cut away the attack phase (when the hammer hits the string). Try to listen to a piano recording backwards. The overtones are the same but their evolution is time-reversed, and what counts more, the attack phase is lost. You would never guess it's a piano.

 

[sophiecentaur]

@someGorilla
You make some very good points in your post and the historical note would tie in with the improvement in understanding of vibrations around that time.
I would agree with you, wholeheartedly, about pretty well all you wrote, if this were a music forum discussing harmonics. We all use the term in guitar playing. But when you look at what's said (on PF) about the 'ignorance' of non-physicists in other areas of the subject and the way the 'public' misuse all our other precious terms (force, weight, momentum - you name them), it would seem that to ignore the essential difference between the meaning of a harmonic and an overtone is to miss the point about how oscillations occur in most musical instruments. In an RF context, if you were looking for the harmonic of a known fundamental you would be using a narrow band receiver and you would be tuned to a precise twice that frequency. You would completely miss an, off- harmonic overtone. You would never be looking for it, unless the resonator you were using had possible overtone modes.