Sinusoid
This effect is not surprising since different periodic signals have different Fourier series representations and, consequently, different content in terms of harmonic frequencies, as explained in the experiment Fourier Series and Gibbs Phenomenon.
You can listen here to different signals with the same frequency (1KHz) but with different periodic waveform shapes.
You will also notice in this experiment that phase changes in a signal do not affect the way it sounds. This is because the human ear is insensitive to phase offsets.
By combining several harmonic components with different amplitudes, we can obtain different timbres, which explains why the same notes on different musical instruments can sound differently. More specifically, when we play a note on an instrument, we not only excite the fundamental frequency fo of the note (440 Hz for A) but also the harmonics nfo of the fundamental frequency (880Hz, 1320Hz,... for A). This is the reason why a piano key sounds more natural and rich than the basic sinusoidal signal you have just heard. The sound generated by the piano simultaneously contains a lot of harmonics with different amplitudes and the sound is then said to be polytonic. Now even for the same note on a trumpet, the amplitudes (or energy) of the harmonics might be different than those on a piano. Therefore, by varying the amplitudes of the different harmonics that compose a polytonic note, we can give different timbres to the note.
You can listen here to different sound signals composed of the same fundamental frequency (440 Hz) and the same harmonic frequencies (880 Hz, 1320 Hz, 1760 Hz, 2200 Hz) but with different harmonic amplitudes.
Finally, by combining sinusoids at close enough frequencies we can generate beat signals, as explained in the experiment Time and Frequency Representations.
