The String Sings Many Notes
When you pluck a string, it doesn't just vibrate as a whole — it vibrates in halves, thirds, quarters, and every integer division simultaneously. Each division produces a distinct pitch called an overtone or harmonic. If the fundamental frequency is f, the overtones are 2f, 3f, 4f, 5f, 6f, and so on — a pure ladder of integers.
The Octave and the Fifth
The first overtone, 2f, is the octave — the note we perceive as "the same note, higher." The second overtone, 3f, is a fifth. Drop it down an octave (divide by 2) and you get 3f/2 — the perfect fifth, the most consonant interval after the octave. This is why the fifth appears in every musical culture on Earth: it is baked into the physics of vibrating strings and columns of air.
Harmonic Ladder
Stack the overtone series. Each harmonic is an exact integer multiple of the fundamental — the basis of every tuning system.
Octave Equivalence
Notes an octave apart are heard as "the same note" — a perceptual fact so deep that even people who sing an octave off by mistake don't notice. Birds, however, often fail to recognize a song played in a different octave. The octave is where physics and perception meet: 2f has the same harmonic relationships as f, just shifted up.
Reflect
What do you notice when you stack several harmonics at once? Do you hear the fundamental change, or does something else shift?