In summary, soft tube saturation is a phenomenon that occurs when a tube amplifier is driven beyond its linear operating range, resulting in a gradual compression of the output signal and the generation of harmonics. While it can have undesirable effects, soft tube saturation is often intentionally used in audio processing to add warmth and character to an audio signal.
When a tube is operated within its linear range, the output signal is directly proportional to the input signal. However, as the input signal increases, the tube's gain begins to decrease, and the output signal starts to compress. This compression is known as soft tube saturation.