A sawtooth wave contains every harmonic at once — the fundamental, then each overtone above it, fading gently as they climb. Synthesis of this kind starts with that abundance and carves away. The filter is the sculptor's chisel: everything above the cutoff falls to the floor, and what remains is the tone you meant.
This is why the instrument above rewards slow hands. Hold one saw note and drag the cutoff down — you can hear the harmonics leave in order, top first, until only the fundamental hums. Push the resonance and the filter's own voice rises at the edge of the cut, ringing like a wet finger on glass. Two knobs, one note, an entire vocabulary.
Resonance exists to make that vocabulary visible. The scope shows you the shape your ears already know; the spectrum shows the arithmetic underneath it. When the picture and the sound click together — when you see a square wave hollow out as its even harmonics vanish — the theory stops being theory.
The oscilloscope has no opinion. It draws exactly what you played, which is why it is the most honest teacher in the room.