When we play a certain note on a piano, for example, the note C, we are able to distinguish it from the same exact note played by a guitar, a violin or any other instrument. The difference in the sound we hear is called “timbre” or “tone color”, and each instrument has its unique tone color. For the same reason, we are able to distinguish between the voices of different people, even when they utter the exact same words.
In order to understand the difference between the tone colors of different musical instruments, we must first understand what waves are. A wave is a general name for a phenomenon in which a disturbance in a given medium propagates in time or space. We are familiar with many examples of waves in everyday life, for example, ocean waves, which are alterations in the level of the water that advance towards the coast, or for example, during a soccer game, we can sometimes see a wave of hands moving across the stadium as fans rise from their seats.
The sounds we hear are also waves - sound waves, meaning changes in air pressure and density that our ears are able to perceive. When we play a musical instrument we create waves that cause the air around us to vibrate, thus creating the sounds that we hear. Two important characteristics of waves in this context are frequency - the number of times that the pressure alters and the air vibrates per unit of time, and the amplitude - an indication of the extent of alterations in air pressure made by the wave. In the case of sound waves, the frequency determines the pitch we hear and the amplitude dictates the intensity or volume of the sound.
Fundamental Frequency and Harmonics
Each wave created by a musical instrument can be treated as if it were composed of a variety of smaller waves at different frequencies and amplitudes. The lowest frequency is called a fundamental frequency of an instrument, and it is determined by the length of the string in a stringed instrument or by the length of the air tube in a wind instrument. When we pluck a string, the fundamental frequency is the frequency with the largest amplitude, and it is the frequency that determines which note we hear - C, D, E etc... The other frequencies are integer multiples of the fundamental frequency, termed harmonics or overtones. The amplitude with which we create these sounds depends on the wave that the instrument creates, and they determine the tone color of the sound we hear.
How is all of this manifested in a musical instrument? When we play the note C on a piano, the piano’s mechanism strikes the appropriate string with a small hammer. The strike creates a certain wave in the string, which we can think of as oscillations with a certain composition of frequencies - mainly the fundamental frequency, but also other harmonics at higher frequencies. In contrast, plucking a guitar string creates a different composition of harmonics, some stronger and some weaker. The difference in frequencies produces one wave on a piano and a different wave on a guitar, but as the fundamental frequency is identical, we hear the note C in both cases.
The significant difference here is that each instrument vibrates its strings differently, and thus the composition of harmonics that it produces is different. In fact, each manner in which we vibrate our string will produce a slightly different composition of harmonics and will change the tone color of the sound. This is why different instruments can play the same note and still sound very different from one another.
The ratio between the amplitude of the different harmonics can also depend on time, and this can also affect the tone color of the sound. In fact, the main factor that allows us to identify different musical instruments is the ratio between the amplitude of different frequencies during a short period at the first moment of hearing them. For this reason, if we play a note on a certain instrument, and then we play its recording in reverse order, it will sound as if it were played by a different instrument altogether. The harmonics that compose the sound will remain the same, but their amplitude will change in a different pattern over time and thus will sound different to our ear.
Non-Harmonic Components
And finally, instruments are also distinguished by their non-harmonic components, which are the sounds that are not directly related to the fundamental frequency or the instrument’s overtones. For example, the musician’s breathing sound or the sound that the musician's finger makes while sliding on a guitar string carry great importance. Taken together, these factors give an instrument its signature sound that we learn to associate with it.
Today we know how to use these principles to digitally create sounds of different instruments, with no need for strings and tubes of mechanical instruments to vibrate the air for us. To produce a sound that resembles a guitar, we simply add harmonics at the appropriate amplitudes and obtain a sound that is identical to the plucking of a guitar string. If we want a sound resembling a trumpet, we will compile harmonics in a composition similar to that of a trumpet, and so on.