Seamless audio loops via Fourier transforms? I am trying to create short seamless loops of continuous sound.

I recorded myself making an "Aaaaah" sound at 44.1 KHz, and cut this recording to a section containing 4096 samples (giving me slightly under 1/10th of a second of audio).

When I play this in a loop, the sound is very much recognizable but there is of  course an audible ticking noise at the point where it is cut.

I applied an FFT transform to this data. When I apply an inverse FFT to the Fourier tables to recreate the signal, the defect is of course still audible.

Is there anything I can do to the Fourier tables to eliminate the defect without changing the other audible properties of the recording? I'm very much uneducated in signal processing, but intuitively I feel that since the ticking sound repeats once per loop, it might be possible to work in the frequency domain to achieve my goal of cleaning it up to produce a seamless continuous sound.

Note: In case there are Javascript programmers here, I created a CodePen here with the "Aaah" loop and its conversion to and from Fourier tables, to facilitate experimentation in code.
Try using some sort of pitch estimator to determine the pitch of your sample.  Or hum in tune to a note of a known musical pitch (from a pitch pipe for instance).  Then overlap and cross-fade frames, not 4k apart, but a shorter distance apart that is an integer multiple of the period corresponding to that pitch.