If you were to break music down into pure mathematics, you would end up dealing with acoustics and harmonics. An instrument's acoustics and harmonics define that instrument's unique sound; they're also the reason you rarely hear songs that use a tuba or a bassoon for the lead instrument. The following paragraphs give you a little of the music theory behind acoustics and harmonics.
Harmonics
Any sound, not matter what the source, is caused by something vibrating. Without vibration, there can be no sound. These vibrations cause the air particles next to the source to vibrate, and those air particles, in turn, cause the particles next to them to vibrate, and so on and so on, creating a sound wave. Just like a wave in water, the farther out the sound wave moves, the weaker it gets, until it completely dissipates.
If the original vibration creates a strong enough wave, though, it eventually reaches your ears and registers as a sound. You hear a sound because air vibrates against your eardrums, causing them to vibrate also. These vibrations are analyzed by your brain and registered as music, traffic, birds singing — whatever.
Each complete vibration of a sound wave is called a cycle. The number of cycles completed in one second is called the frequency of the vibration. One of the most noticeable differences between two sounds is the difference in pitch; it's the frequency of a sound that mostly determines its pitch. Frequency is measured in hertz, with one hertz (Hz) being one cycle per second. One thousand hertz is called a kilohertz and is written as 1 kHz. A high-frequency vibration produces a high-pitched note; a low-frequency vibration gives a low-pitched note. The human hearing range (audible range) is about 16Hz to 16kHz. The frequencies of notes that can be played on a piano range from 27.5 Hz to just over 4kHz.
The musical note produced by a tuning fork is called a pure tone because it consists of one tone sounding at just one frequency. Instruments get their specific sounds — their timbre — because their sound comes from many different tones all sounding together at different frequencies. A single note played on a piano, for example, actually consists of several tones all sounding together at slightly different frequencies.
Acoustics
Next time you go out to see an orchestra or a big band play, or even when you watch one of those late-night show bands perform on TV, take a look at where the performers are sitting in relation to each other. Especially pay attention to which instrument is the "lead" instrument.
You should notice two things. First of all, especially in an orchestral setting, all the performers playing the same instruments sit together. This isn't because they all have to share the same piece of sheet music; it's because when you stick two violins or flutes or clarinets together, they sound louder and fuller. Stick ten of them together, and you have a wall of sound coming at you from that area of the orchestra.
Secondly, notice that the lead instruments are in front of all the other instruments, especially in acoustic performances. This is because of volume and perception: the sound waves from the instruments in the front of the orchestra pit will be heard a microsecond before the rest of the orchestra and will therefore be perceived as being louder than the other instruments.
This principle applies to a regular four-piece band setting, too. If you want your singer to be heard above the guitars, make sure the amplifier carrying his or her voice is placed closer to the audience than the guitar and bass amp.
