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Flutter is caused by tiny imprecisions in the mechanism of the tape recorder, and also by vibration of the tape itself as it slips and sticks against the metal surfaces of the heads. Good analogue tape recorders have roller guides to reduce this kind of vibration, but there will always be some, contributing to the general 'dirtiness' of the sound, or analogue warmth as we now know it.
Technically speaking, the result is to add sidebands to the signal. For a sine wave input, the output will be a cloud of sideband frequencies separated by the frequencies of the cyclical variations produced by mechanical imprecision, together with a general fog caused by the tape vibrating. In some respects it could be considered that this adds noise to the signal, but it is a kind of noise that is very closely related to the input signal and follows it up and down in frequency.
And this is before we consider the noise produced by residual randomness in the orientation of the magnetic domains, and also the once-dreaded modulation noise caused by the Barkhausen effect (you can look that one up!). All in all, analogue tape is a rich source of signal-related impurities and distortions, so much so that hardly anyone is making tape recorders any more. But they don't understand warmth too well, do they?
We know what warmth sounds like, and now we know what is causing it: the addition of signal related frequencies, either harmonically related or of a similar nature to noise but close in frequency to the original input. Imagine this - you are recording a singer with a pleasant but thin voice, very precise in frequency with the result that any out of tune singing is painfully obvious.
Record this through a transistor preamp onto a digital recorder and that is precisely what you will get - thin and out of tune. Now replace the transistor preamp with one that uses valves and you will find that the voice isn't so thin, and although you can hear slight tuning irregularities, they don't seem to matter too much. Alternatively, record onto analogue tape rather than digital media and you will hear something that in all likelihood is entirely satisfactory.
Auto-Tune? Never use it! You will note that I said that the voice was basically pleasant to begin with because even the combination of a valve preamp and analogue tape won't turn a silk purse into sow's ear, or whatever that old saying was. The addition of harmonically related frequencies and sidebands will however turn something that is lacking in technical precision into a recording that allows the artistry to shine through.