Welcome! This short tutorial explains step by step the nature of the piano sound and illustrates how physical modelling can recreate such a sound.
To listen to the audio demos, please click on the "speaker" button.
Sound with just one partial
Sound is produced by air displacement (pressure variation) and can be represented as a sum of sine waves with various length and amplitude called partials.
Let's start with just one partial, using a given frequency f. Such a sound is considered "pure" and can be written as follows:
S(t) = a sin(2π f t).
Listen to the sound:
What is missing?
The previous sound has poor sonority. Let us add more harmonics using frequencies 2f, 3f, and so on... (when the partials are whole number multiples of a common fundamental frequency, they are known as harmonics):
S(t) = a sin(2π f t)
+ b sin(2π 2f t)
+ c sin(2π 3f t)
Here is the result: Does it sound like a piano?
In an acoustic piano, each string is bound to a soundboard through a bridge. The soundboard, made of wood, is a transducer between the string and the air which brings the sound to our ears.
In other words, the string energy is transmitted to the air via the soundboard. This way, the string loses its energy and the sound amplitude decreases. But what will be the decay rate?
Decay depends on frequency
C3 of a piano (spectral view depending on time, from 0 to 7 seconds):
Harmonics (horizontal orange lines) start simultaneously but the higher the frequency the shorter the harmonic. By the way, we can observe in this figure the beatings produced by a slight detuning between the 3 strings of the unison. We will explain this shortly.
The strength of each harmonic for each note is determined by the interaction between the hammer and the string, and especially by the strike point. In this figure, each vertical stroke matches an harmonic, and the height defines its strength.
With the exception of the low range, each note of a piano has 3 strings, struck simultaneously by the hammer. The fact that there are several strings makes the attack stronger, and it also increases the sound duration.
No two strings can be exactly in tune. So setting several strings together will create close frequencies. Gradually, two close frequencies will fall alternatively in phase - the sound reaches its maximum amplitude - and in opposite phase - the sound cancels -, that is the beating:
Actually, the sound of a piano is not strictly made of harmonics ƒn = nc / 2L but of partials:
ƒn = (nc / 2L) sqrt(1+Bn2)
The inharmonicity coefficient B depends on the note and the instrument. As a result, the sound is no longer periodic, and the beatings rate between partials is modified, which is taken into account by the piano tuner.
Three examples with different inharmonicity, from high to low: