Need mathematical explanation for different musical notes sound different on different instruments

I am not expert in music. There are number of musical instruments. One (especially a person who knows about music) can blindly recognize which instrument is being played just by listening to it.

I need mathematical explanation about following

Every instrument (say Guitar, Harmonium , Violin like that) seems to be unique while listening. My basic question is why is it so?

Why different musical notes sound different on different instruments?

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• I know that it has to do with overtones [en.wikipedia.org/wiki/Overtone] (higher frequencies that are heard each time an instrument plays a note) which occur in certain mathematical patterns. Different instruments have different proportions of each frequency. The other important thing is articulation - many instruments actually sound similar holding a long note but the very start of the note sounds different - but I don't know a mathematical model for this. – Seth Jul 11 '15 at 12:40
• "language of music is same all over the world which consists of few number of musical notes." This is not true, in at least two ways. First, a standard piano has 88 keys, which is hardly "a few notes". Second, many different cultures use many different musical scales. It's simply not true that all music uses the standard Western equal-tempered scale. – David Richerby Jul 11 '15 at 15:39
• Also, what does this have to do with mathematics? – David Richerby Jul 11 '15 at 15:39
• @David why do think that my question has nothing to do with mathematics. As I already told here that I am not expert in music but every concept is incomplete without maths behind it. as you can see ,I think both of the questions are concerned with frequency,waveforms harmonics,tones etc. and in mathematics there are many frequency transforms like Laplace, Fourier which can be used for the analysis purpose. so I expected that it could be understood in more detail by studying relevant maths behind the concerned questions. – devraj Jul 11 '15 at 16:40
• This is a question of physics, or maybe biology, not mathematics. – user5174 Jul 11 '15 at 20:00

When you pluck a string or hit a drum or sound a not on a flute, the instrument and the air in and around it vibrate and this vibration propagates as sound waves in the air to your hear drum.

When you hear an instrument being played, what you recognise as the note is the base frequency. 'C' corresponds to $261.6$ Hz and is the same for a piano or a guitar. But a 'C' played from a guitar, played from a piano or simply a $261.6$ Hz sounwave played from a computer speaker sound totally different. This is because of the overtones.

Let's look at the case of a string for a concrete example. If you pluck the 'C' string on the guitar, you will hear the characteristic sound of it makes. This is because the string is vibrating at $261.6$ Hz, but also at a bunch of higher frequencies. These higher frequencies are called "overtones", and they are determined by the shape and build of the body of the guitar as well as the way you set the string in motion.
This is guitars with different shapes sound different. You can also try plucking or strumming the guitar string in various different places, and you will hear different tones of 'C'.

Overtones of the vibrating instrument are what makes each instrument (or voice, for that matter) sound different. The material, shape, and way the instrument is played all contribute to determine which overtones will be present.

The reason instruments sound more similar while holding a long note is that the overtones dissipate energy faster. Higher frequency vibrations generally lose energy quicker. So once the string is plucked or hit, the overtones start losing energy (thus lowering volume) faster than the base note, and after a while you only hear the base note.

TL;DR: Instrument sound different because of the overtones they produce. These are higher frequencies than the note being played, and are determined by factors such as shape, material and way of playing the instrument and give the characteristic flavour to each instrument.

• The change in amplitude and overtones in time is also important. Violins have a slow attack (amplitude rises slowly), while pianos have a quick attack. – Stefan Jul 11 '15 at 16:35
• Timbre can be roughly defined as those qualities of a sound that aren’t just frequency or amplitude. These qualities might include: spectra: the aggregate of simpler waveforms (usually sine waves) that make up what we recognize as a particular sound. This is what Fourier analysis gives us (we’ll discuss that in Chapter 3). envelope: the attack, sustain, and decay portions of a sound (often referred to as transients). (From music.colombia.edu) – Stefan Jul 11 '15 at 16:38
• Adding to your answer, here are two wonderful videos that demonstrate overtones. The first describes some physics and shows a frequency spectrum of the tones on a guitar string: youtube.com/watch?v=q-Z4kndewSw. The second shows off polyphonic singing: youtube.com/watch?v=Mwv-E0Gzg8k . Note the many smaller peaks that accompany the fundamental (large) peak of the note. – zahbaz Jul 12 '15 at 1:04

1) A vibrating string produces not only the basic (fundamental) frequency, i.e. vibrates up-down, but also lots of other (multiple) frequencies, i.e. double frequency when two halves of the string vibrates in the counter-phase etc. See more on the video.

2) Each acoustic instrument has a resonating body where the sound waves with all those frequencies travels around and undergoes multiple reflections, producing a complex resulting wave in which all such reflections are added up with some frequencies having been amplified and other attenuated, depending on how the wave and the reflected wave tops meet in a point.

A human ear distinguishes this particular spectral character of the signal with different amplitudes on different frequencies. Different sounds of different instruments comes mainly from the geometry of their resonators.

P.S. With modern digital sound processing it is easy to synthesize any spectrum of the wave and to mimic whatever instrument.

• With modern digital sound processing there are still a few intruments that are harder to synthesize the sound of, particularly those that have non-linear or chaotic mechanics in major parts of their sound production. – Tally Jul 12 '15 at 6:05

Pitch, in music, is equivalent to frequency. How often the wavefore cycles. This is usually defined by length, i.e. how long the string is, how long the pipe is, etc. It can also be affected by the tension (how tight the string is.)

Timbre, the sound of a specific instrument, is defined by the "shape" of the wavefore, whether spikes, round, square, or whatever other shape that instrument makes. This wavefore shape is defined by the construction of the instrument. A trumpet, for instance, makes a very "spiky" shape that just sounds "brassy" to an experienced ear.