The tape on a compact cassette imposes all kinds of noise and distortion on a recorded signal. Much of it is so quiet that it's practically inaudible, but with some signal processing we can zoom into some fascinating details.
(There's a high-res version of the enclosed infographic at Dropbox.)
I generated a test tone that sweeps from 50 Hz up to 21 kHz in 5 seconds, then recorded it onto a cassette. I played the tone back and plotted it above, in turquoise. I also monitored the "dry" signal passing through the circuitry and plotted it in red, to see what was actually caused by the tape.
There's a prominent banding that seems to follow the rising frequency of the tone. The pattern, with its mirrored sidebands, looks very similar to a frequency-modulated carrier, and that's effectively what it is. The tone frequency is being modulated by rapid fluctuations in tape speed during recording and playback, also known as flutter.
Another persistent signal is visible in the leftmost part of the picture, or the low part of the spectrum. It's the humming sound of the power grid, 50 Hz in this part of the world, leaking to the tape because of an unwanted ground loop somewhere in the circuit. Ultimately, it's the sound of the steam turbine rotor turning at the nearest power station. The hum has several harmonics at integer multiple frequencies.
Some harmonic multiples of the test tone are also visible. These are caused by gross distortion of the original waveform, most probably by magnetic saturation (clipping) of the tape media, or they may be crossover distortion from an amplifier stage. The harmonics are repeated in the end of the sample as a zig-zag pattern, which is probably a digital aliasing artifact.
The noise floor on the cassette is set by what is commonly known as tape hiss. It is caused by the finite size of the magnetic particles that the audio is recorded on. This is similar to the grainy pattern on an enlarged film photograph, or pixels in a digital one. A faster-running tape will have more hiss, but at the same time can record information at a greater bandwidth.