The app controls are labelled in Italian: below, each label is given as it appears in the app, followed by its English meaning.

Distortion by amplitude clipping

1. Instructions
   • Click here to start the app and in the "Forme d'onda" (waveforms) window select Seno (sine)
   • Activate the "suono" (sound) command
   • Using the right mouse button directly on the waveform, cut off the upper part (and, if you like, the lower part too) of the waveform.
2. Observations
   • the sine function, which (obviously) contained only the fundamental harmonic, is changed into a function rich in higher-frequency harmonic components (we could say that the Fourier series expansion of the modified waveform requires higher-order harmonic contributions);
   • as for the sound, its timbre changes while the perceived pitch stays the same.
   • repeat the experiment selecting the other available waveforms as the initial waveform (Triangolo (triangle), Dente di sega (sawtooth), etc.). You will observe that the change in timbre is less noticeable the more the initial waveform, even before being "clipped", already contains significant higher-order harmonic contributions.
   • there is another way of clipping the amplitude of the sound: try turning up the volume of your computer's speakers: at some point you will hear a distorted sound. This distortion is not due to the app, since the contribution of the various harmonics of the sound has not been changed, but to the fact that every sound amplification and reproduction device has a maximum amplitude that it can "deliver". Our ear too (see physiology of the auditory system) can have "saturation" problems: if the stimulus is too intense, perception in effect clips the amplitude of the sound wave. There is even a protective mechanism, called the stapedius reflex, that prevents our ear from being driven by stimuli that are too intense.

Nonlinear distortion

1. Instructions
   • Click here to start the app and in the "Forme d'onda" (waveforms) window select Seno (sine)
   • Activate the "suono" (sound) command
   • Using the right mouse button directly on the waveform, modify the sinusoidal waveform (try to draw with the mouse a waveform that reproduces the initial sinusoid: however accurate you are, you will deform the initial waveform; this is the effect of an amplifier with a nonlinear response).
2. Observations
   • in this case too, the sine function, which (obviously) contained only the fundamental harmonic, is changed into a function rich in higher-frequency harmonic components, with significant changes in the timbre of the perceived sound.
   • the greater the deformation, the greater the contribution of the higher-order harmonics; the sound becomes harsh and very "piercing" (the contributions of the higher-frequency harmonics produce a sort of very high-pitched "squeal" that is superimposed on the initial sound).
   • the distortion produced by the nonlinear response of an amplifier is not just a technical matter: our ear (see physiology of the auditory system) is a nonlinear amplifier. Perception therefore introduces into the original sound harmonics that were not initially present.

Frequency distortion

1. Instructions
   • Click here to start the app and in the "Forme d'onda" (waveforms) window select Triangolo (triangle)
   • Activate the "suono" (sound) command
   • Start changing the amplitude of the various harmonics using the corresponding sliders
2. Observations
   • you will certainly have observed the change in the waveform and perceived the change in the timbre of the sound. The experiment seems trivial: after all, it is obvious that by changing the amplitude of the various harmonic components a new sound has been "synthesized" (that is, a periodic wave with a different Fourier expansion);
   • the real purpose of the experiment is to show what happens in a musical instrument:
     • there is an oscillating part (string, membrane, air column) which, set in motion by the player, generates a sound wave.
     • if this sound wave were radiated directly into the surroundings, the sound intensity would often be too weak;
     • many instruments have a "sound box" able to "amplify" the amplitude of the various harmonic components. The amplification, and this is the point we want to stress, is frequency selective, that is, the various harmonic components are amplified differently because of the phenomenon of resonance;
     • the end result is that the radiated sound not only has a greater intensity but also a timbre different from that of the sound originally generated by the oscillating part.