Pitch is speed; loudness is size
Twang a ruler pressed against the desk edge and it buzzes. Hold down more, leaving a short stub free, and twang again—the same flick, a clearly higher note. What changed is not the force but how many round trips per second the ruler makes.
That count is the frequency, measured in hertz: one hertz is one round trip each second. High frequency we hear as high pitch, low as low. And note—pitch is about speed, not size.
Plot both notes on a time axis and the difference is immediate: the high note’s crests crowd together, dense; the low note’s crests spread apart, sparse. The wave’s height is a different matter: strike harder and the amplitude grows—a louder sound with the spacing untouched.
“Louder” and “higher” are therefore two independent features of one wave: frequency sets the pitch, amplitude sets the loudness.
Ruler, rubber band, instrument queue
- The ruler. Press a ruler on the desk edge with 20 cm free and twang it; then press until only 5 cm remains. The short stub sounds clearly higher: its free section resists bending more strongly and vibrates faster. Try 3 cm: an almost piercing buzz.
- Three rubber-band changes. Stretch a band and change one thing at a time: tighter, higher; slackened, lower. Shorter vibrating length, higher; longer, lower. A thicker band sounds lower overall. Violinists tune by tightening and play by shortening—the same two rules.
- Line up instruments. The cello sits below the violin, the big drum below the small, the piccolo above the flute. The rule is simple: bigger, heavier vibrating parts make slower round trips. Voices follow suit—longer, thicker vocal cords sound deeper.
Then try: push the loudness slider to maximum. Did the frequency readout move? Did the wave get taller, or denser?
The string formula, the ear’s range, and ultrasound
More: write the observation as an equation
Open: the string frequency formula
For a string fixed at both ends and stretched by tension, the fundamental is:
with L the vibrating length, T the tension and μ the mass per metre (the thickness). For a stretched string: halve L and f doubles; quadruple T and f only doubles (the square root); a thicker string raises μ and lowers f. A ruler is a cantilever, with one end fixed and one free. Its bending frequency must be computed separately. For the same uniform ruler at small deflections, frequency varies inversely with free length squared: halving the length gives about four times the frequency. The animation uses that bending law for the ruler and the string law above for the band. See TU Graz’s cantilever-beam derivation.
Open: the range of human hearing
Ears catch roughly twenty to twenty thousand trips per second:
Below 20 Hz is infrasound; above 20 000 Hz is ultrasound. Bats and sonar navigate and range with ultrasound—time traded for distance at its limits, as the echo lesson showed. Whales and elephants use infrasound to talk across kilometres.
Limits of the model
The experiment draws single-frequency sines; real instrument tones stack a fundamental with overtones, which is where timbre lives, out of scope here. A drum’s “pitch” is approximate—drumheads carry many modes at once. Frequency-to-note names use equal temperament with A4 = 440 Hz. Sources: OpenStax, University Physics, the sound and hearing chapter and NOAA, “What is sound?”.