Gary, you are a jazz fan. You know what Billie Holiday sounds like recorded.

Some folks call that sound 'warm'. Some of the warmth is due to a lack of high frequency content - just plain signal loss through the whole signal chain and onto tape. But some of it quite possibly is some tube distortion that may have been added into the signal chain un-intentionally, or even tape saturation distortion.

The kind of harmonics added matters and can impart a pleasant sound. Yes it's distortion, no it doesn't have to be necessarily a negative thing. It's even harmonic distortion for the most part. You can think of a square wave as a sine-wave with even harmonics added. Still pretty smooth sounding, even though 'distorted'. A triangle wave, on the other hand, is odd harmonics and has a more brassy/grating sound.

Does that clear things up at all for you?