TwiddleLab / Physics / Waves and Sound
Project: Build an Instrument
What you will learn
- On strings of the same tension, frequency goes as 1 / length: 2/3 of the length plays 3/2 of the frequency.
- Two close notes beat: the wobbles per second equal the gap in hertz. No wobble means in tune.
- Musical intervals are simple ratios: octave 2/1, fifth 3/2, fourth 4/3. Pipes depend on the speed of sound, so they drift with temperature; strings do not.
How the lesson goes
Every instrument is wave physics you can play. Listen to this one.
A 60 cm string plays 440 Hz. Where does the bridge go for 660 Hz?
A plucked string holds one loop, so its wavelength is twice its length. The wave speed on the string is fixed by tension and thickness, so f = v / (2 × length): shorter string, higher note.
Tune your first string to the 440 Hz reference by its peg.
Listen for the wobble, or watch it.
Two notes 3 Hz apart drift in and out of step 3 times a second: they add (loud), then cancel (quiet). That is superposition, heard as a wobble.
Same kind of string: 60 cm plays C. How long must the F string be? (F is 4/3 times the frequency.)
f × length stays the same: 264 × 60 = 352 × 45.
Build your instrument: set all eight strings to the C major scale.
Length = 60 ÷ ratio. Let go of a bridge to pluck it and read its tuner.
Every note is the C string's frequency times a ratio, and length goes the other way: divide 60 cm by the ratio.
Your friend's instrument sounds sour. Find the two bad strings and fix them.
A string that is too long plays flat (too low); too short plays sharp (too high). Each bridge must sit at 60 ÷ ratio.
Compose a melody of 6 to 8 notes for your instrument.
Challenge: a string and a pipe, both in tune at 20 °C. Which drifts in a warm concert hall?
A pipe's note is f = v / λ, with λ set by the pipe's length and v the speed of sound in its air (331.3 + 0.606 × T m/s). Warmer air, bigger v, higher f. A string's wave runs along the string, not through the air.
Math: musical intervals are fractions. Match each one.
When two frequencies make a simple ratio, their waves line up again after a few cycles, and we hear them as blending well.
Game: Ear test. Pick the string that is in tune with the reference.
Three rounds. It gets harder each time.
Draw the wave of the C one octave above your C string.
Frequency counts waves per second. Double it and twice as many waves fit into the same stretch of time.
Your instrument, your melody. Play it.
Previous lesson: The Doppler Effect · Part of the course: Waves and Sound
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