Pythagoras and his school did experiments to discover the relations between musical notes.The pitch of a note being played on, say, a guitar depends on:
Pressing the finger onto the string clamps the string onto the metal fret to the left of where the finger is pressed. The effective length of the string become the distance between the fret and the bridge on the left hand side. If we say the length of the string is one in some system of units, we show the effective lengths of the string for each of the notes in the scale. These harmonious "consonant" lengths were discovered by Pythagoras.
The lengths are related to each other by the ratios of whole integers. In fact, the Pythagoreans believed that this was a general principle of the universe: everything is related by the ratios of whole integers. Thus, the cosmos sings in harmony, the "Music of the Spheres."
The Pythagoreans also discovered the existence of irrational numbers. For example, the square root of 2 is:
1.414213562...