Set the generator to 220 Hz and cycle through the four waveforms. The pitch never changes. The sound completely does — polite, angry, soft, brassy. Four personalities from one number.
The difference is harmonics: quieter tones stacked on top of the pitch you asked for, at exact multiples of it. Which multiples show up, and how loud, is the entire identity of a waveform. You don't need the math; you need about ninety seconds of listening. Here's your tour guide.
A sine wave is the pitch and nothing else. No harmonics. Zero. It's the flute-like, slightly eerie tone you hear in hearing tests, and there's a reason for that: because nothing else is present, whatever you hear IS the frequency you dialed in.
Use it for anything that needs to be honest — speaker tests, hearing checks, tinnitus matching. All measurement, all the time. It's also the most boring, which in a test signal is a compliment.
Switch to square and the same 220 Hz suddenly sounds like an old video game having a bad day. A square wave carries only the odd harmonics — 3×, 5×, 7× the pitch — and they fade out slowly, so a lot of them are loud enough to notice. That's the buzz.
Hollow and reedy is the classic description; a clarinet is the closest acoustic cousin. Chiptune composers built a whole genre on it because early sound chips could make square waves almost for free. It's also brutal on small speakers at high volume, which makes it a decent stress test. Use kindly.
Triangle has the same odd-harmonics-only recipe as square, but each harmonic drops off much faster. The result keeps a hint of that hollow character with none of the aggression. Soft, roundish, close to a sine with a little texture on top.
Honestly, it doesn't get many test-signal jobs. It gets picked when a sine feels too clinical and everything else is too much — pleasant alarm tones, gentle demos for kids, background tones you have to listen to for a while.
The sawtooth contains every harmonic — odd AND even — which makes it the richest and brashest of the four. Bright, brassy, a bit rude. It's the backbone of classic synthesizer sound precisely because there's so much material there: run a sawtooth through a filter and you can carve almost any tone out of it, the way a sculptor starts with the whole block of marble.
For testing it has one niche: with all harmonics present, it excites resonances a pure sine slides past. Play a low sawtooth and loose objects in the room will announce themselves.
Here's the fastest way to make all of this stick. Set 220 Hz, sine. Now watch the oscilloscope while you click through the four shapes — you'll SEE the corners arrive as you HEAR the buzz arrive. Corners in the waveform are harmonics; the sharper the corner, the more of them. A sine has no corners and no harmonics. A square is almost entirely corners.
That one sentence — corners are harmonics — is most of a synthesis textbook, free of charge.