Interactive Tone Generator
Play Intervals:
The Legend
According to myth, Pythagoras (c. 570-495 BCE) walked past a blacksmith's shop and noticed that hammers of different weights produced harmonious sounds when struck. Intrigued, he experimented and discovered that consonant musical intervals correspond to simple integer ratios: an octave is 2:1, a perfect fifth is 3:2, and a perfect fourth is 4:3. This observation led him to conclude that mathematics governs harmony—and by extension, the cosmos itself ("the music of the spheres").
The Mathematics
Frequency Ratios: Musical intervals arise from frequency ratios between tones.
• Octave (2:1): Double the frequency produces the same note, one octave higher
• Perfect Fifth (3:2): Very consonant; in C major, C to G
• Perfect Fourth (4:3): Also consonant; in C major, C to F
• Major Third (5:4): Consonant but slightly less "pure"; C to E
• Major Second (9:8): A whole step; C to D
Why Simple Ratios? When two frequencies relate by a simple ratio, their waveforms align regularly, creating consonance. Complex ratios (e.g., 17:13) produce beats and dissonance.
Historical Note: The blacksmith story is apocryphal—hammer weights don't work that way! But the mathematical principle is real and forms the basis of musical tuning systems.