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Same note, different instrument

So a clarinet, and a piano, both play a middle C (261.63 Hz).
But they both make distinct sounds..
What is it about the waves produced that allows you to differentiate between the instruments?

I've pondered this for a long time.

8 Solutions
Guy Hengel [angelIII / a3]Billing EngineerCommented:
an interesting article about sound/waves in general:

now, why do different instruments have different "sounds", despite the same frequency?
well, because the pure "frequency" is not everything, the sound is not 1 single wave.
instruments will produce a multitude of different waves, with different amplitudes, and also having intermediate frequencies.

also, with the middle C on a piano, you also would have other "C" tones at the same time produced, which are in "harmony" with the actual C, as 1 octave lower means half the frequency ( C, is 130.8Hz; C,, is at 65,4Hz etc... )
Kent OlsenData Warehouse Architect / DBACommented:
Hi InteractiveMind,

middle C is simply a sine wave at 261.6 Hz.  The distinct sound generated by different objects (instruments) is due to the unique shapes of the waves and the secondard waves that accompany it.

When a note is struck on a Piano, 4 feet of 'wire' vibrate.  (More when the wire is wrapped by another wire to generate the lower freqencies.)  The wire may move 1/4" at the center and less toward the fixed ends.  Additionally, the vibration is maintained by the mass at the center of the length seeking equilibrium.  As the mass moves it literally pulls the remaining wire into the vibrating frequency so the sound wave consists of the dominate wave, plus a multitude of smaller waves NEAR the base frequency.

The tone of a clarinet starts with the vibration of a 3" piece of bamboo.  Much differerent characteristics than a 4' length of wire.  The vibration is transmitted throughout the clarinet, which modifies the original wave due to the construction of the clarinet.  The transmitted sound may have the same base frequency as the sound generated by the piano, but the wave has a slightlly different shape, as do the secondary waves.

Good Luck,
Sound is produced by a vibrating string in a piano, a reed in clarinet, a column of air in an organ.

The primary tone is usually the fundamental mode or the lowest resonant frequency.

But you can always excite higher order modes.  And the higher the mode, the faster they tend to decay.

If you pluck a guitar string in the center, you preferentiall excite the fundabmental.  If you pluck it near the end, you also excite higher modes.

You can hear the difference.

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In particular:  

A piano string is fixed at both ends.  Normal modes are integral numbers of half cycles.  So a piano tone will include all harmonics.

The reed in a clarinet is fixed at one end.  Normal modes are odd multiples of quarter cycles.  So a clarinet will include mostly odd harmonics.
For you, the best explanation lies in the application of Fourier analysis to the sound produced by different instruments.
If a Fourier analysis is done on the output of different instruments, all will produce the same fundamental but each will have a different mix of harmonics.
Thibault St john Cholmondeley-ffeatherstonehaugh the 2ndCommented:
The 'attack' of the note is the most noticeable difference between different instruments. The piano string is struck and so has a sharp attack, initial rise in volume. The clarinet has a much smoother increase.
If you take a sample of the notes where the initial few miliseconds has been cut off there will be much less obvious difference between the sounds. The decay of the note also makes a difference, The piano note is decaying from the moment it starts whereas the clarinet note will sustain as long as the player can blow. If you take just the middle section of a note, no attack and no decay then the individual harmonics as mentioned by others will be the main difference, but these are not so obvious a difference as the attack and decay.
Skipping the differing attacks, the diminishing volume of the piano note, the way each note ends, and the fact that a clarinet is in the key of B-flat and a piano is in the key of C...

A clarinet note has a timbre distinct from a piano note because pianos and clarinets have differing harmonics.  

A simple sound that corresponds to y(t) = a * sin(w*t) sounds a great deal like a whistle or flute.

More complex timbres come from sounds that correspond to:
y(t) = Sum for i = 0 to n of (a_i * sin (i*w*t + p_i))

In other words, you take the fundamental tone (frequency w/(2 pi)), and add in a bunch of harmonics (with amplitudes a_i and frequencies wi/(2 pi), with phase shifts p_i) and you get the distinct timbre.  
InteractiveMindAuthor Commented:
Thanks.  Sorry for delay.

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