Tape saturation
The term “saturation” comes from things like tape, where all the magnetic particles are pointing the same way and can’t possibly take anymore signal, and likewise for a transformer.1
Just turning on a tape machine emulator adds brightness via the the second and third order harmonics generated by the tape machine’s amplifier.2
The distortion point of tape is based on frequency as well as amplitude. As you push it, it will first start clipping the high frequencies. This makes tape useful for Controlling harsh high end. Slower tape speeds have less high frequency.2
Within the linear range of tape, you don’t get a clean low-frequency response. A bias signal is a super-sonic signal that puts energy into the tape so there’s something that is adding dynamic range. Then adding in your recorded signal, you get more of a linear response. By turning the bias up, you get a larger dynamic range, but the high frequencies start clipping even earlier (Satin has a bias knob, as does Chow Tape Model).2
By boosting the highs before going to tape, you can excite them even more.2 Satin’s pre-emphasis does just this.