Modes That Stop Being Harmonic
Twelve masses on springs. The first few modes are the harmonics you would expect. The last few are nowhere near.
What it is
Twelve masses, springs between them, both ends pinned. The chain cycles through its twelve normal modes, one every two and a half seconds.
Below, each mode’s frequency marked twice: where a continuous string would put it, and where this chain actually puts it.
Two buttons play the first twelve partials as a plucked tone, once at the chain’s frequencies and once at a string’s. They are not the same instrument.
How it works
Mode m of a chain of n masses has frequency 2√k · sin(mπ / 2(n+1)), and its shape is a
half-sine sampled at the mass positions. Those shapes are eigenvectors, so excite one and
it stays pure forever. The tests check that by projecting onto every other mode after
twenty thousand steps and finding nothing above 10⁻⁶.
The simulation is velocity Verlet with each mass pulled towards the average of its neighbours. Timing sign changes of a mode’s projected amplitude recovers its frequency from the simulation alone, and it agrees with the formula to five decimal places: 0.28463 against 0.28463 for the fundamental, 1.97958 against 1.97964 for the twelfth.
What surprised me
I knew discrete chains were dispersive. I hadn’t appreciated how large the effect is, or how early it arrives.
A ten-mass chain, mode by mode:
| mode | a string says | the chain gives | short by |
|---|---|---|---|
| 2 | 2 | 1.980 | 1.0% |
| 4 | 4 | 3.799 | 5.0% |
| 6 | 6 | 5.310 | 11.5% |
| 8 | 8 | 6.392 | 20.1% |
| 10 | 10 | 6.955 | 30.4% |
The top mode isn’t slightly flat, it’s a musical fourth below where the harmonic series would put it. And it isn’t confined to the top either. By the halfway point it’s already 8%, which is well over a semitone.
The cause is a ceiling. sin is bounded, so mode frequencies are bounded, and no mode of
any chain can exceed 2√k however many masses you add. A string’s harmonics rise without
limit, a chain’s crowd up against a wall. The modes near the top all get squeezed into the
last little bit of frequency space below that wall, which is why they end up nearly on top
of each other.
That’s audible, and the two buttons make the point better than the numbers do. The string version sounds like a plucked string. The chain version, with identical amplitudes and decay, sounds like struck metal, because “sounds like a bell” is largely what “has compressed, inharmonic partials” means to the ear.
So the thing that makes a string sound like a string isn’t the springs and isn’t the masses. It’s having enough masses that every mode you can hear still lives in the part of the sine that looks linear.
What I would do next
Vary the number of masses with the pitch held constant and find how many you need before it stops sounding metallic. My guess is a few dozen, which would say something about how far up the series the ear is actually listening.