One problem that can happen with sampled sounds is a discontinuity at the end of the sample. If the sample is long enough, this sounds like a click, but a short sample (e.g. a single oscillation of a waveform) can turn it into a sawtooth buzz, one that can overwhelm the rest of the signal.
One tool available in old MOD trackers was time-reversal of the sample, so instead of playing the wavetable from beginning to end repeatedly, it would play boustrophedonically, first from beginning to end, then backwards from end to beginning, and then repeat. This approach eliminates the discontinuity. It also cuts the size of your wavetable in half.
Audacity has a “zero-crossing finding” tool on the Z key which adjusts your selection boundaries to the nearest positive-going zero crossing. This also helps with the same problem.
The boustrophedon approach has an interesting effect on the signal, though. Since the resulting waveform is reflection-symmetric, it can only contain cosine waves; the phase of all the signal components is forced to zero, or rather all imaginary components are canceled, because the time-reversed part of the signal has conjugated phase. (All harmonics of the period of the doubled-length wavetable are possible, though.) So the extent to which a partial makes it through this procedure is dependent on its phase — if it happens to have perfectly real phase, it’s untouched, but if it happens to have perfectly imaginary phase, it’s entirely canceled. This means that filtering a signal with an allpass filter before this procedure can radically change the frequency content on the output, even though the signal going in may sound identical.
I propose a probably known extension to the procedure: rather than repeat(wavetable[1:] || reverse(wavetable)[1:]), use repeat(wavetable[1:] || reverse(wavetable)[1:] || 2·wavetable[0] - (wavetable[1:] || reverse(wavetable)[1:])), reversing the sample both in time and in sign, eliminating the discontinuity in the derivative at the beginning of the wavetable (though not its end) and extending the generated waveform to four times the length of the wavetable. This allows the generated waveform to contain all of its period’s harmonics with any phase, rather than just real phases, and it cuts in half again the number of samples needed to represent a waveform of a given fundamental frequency.
For example, a 440-Hz A note at a sampling rate of 16kHz (a common rate in MODs) would require a wavetable of 36.3636 samples for a full oscillation; a 36-sample version of this wave would be out of tune by 1%, which sounds inconsequential but is 17 cents, easily audible. So you’d probably use three oscillations, 109 samples, which gives you 440.37 Hz, only off by 1.4 cents. If you try to use this approach with ¾ of an oscillation, it will work, but you’re constrained to 27 or 28 samples, corresponding to 108 or 112 samples in the full three-period time, which would be 36 or 37⅓ samples per oscillation.
I was thinking that you could choose to duplicate the initial/final sample of a segment, or not, to correct this detuning, but I think that probably a better way to inexpensively tune up or down by fractions of a semitone is to occasionally duplicate or omit samples at points where the sample value and, especially, derivative are zero or nearly so. I don’t know how this will sound, but maybe if it works well enough, you could represent an arbitrary 440Hz harmonic sound with a 12-sample 16ksps wavetable. This is very appealing not just because it takes 24 bytes but because optimizing over a 12-dimensional space, whether in the frequency domain or the time domain, whether by hand or using an algorithm, seems much more tractable than optimizing over a 109-dimensional space.
For images, you can use this kind of approach to make a smoothly tiling texture out of any image by reflecting it at the border. A kaleidoscope works this way; the tiling need not be square but it does need to have only vertices of even degree, which probably limits you to variants of square and triangular tilings.
You can also use this approach to make smoothly stretching display window frames from a single example: by inserting reflected sections into the middle of stretched dimensions as necessary, you can make the window whatever size you want, down to the minimum of the original drawing. You won't introduce discontinuities in colors, but you may introduce discontinuities in line angles; if you can choose preferred direction-reversal points where the lines are all either perpendicular or parallel to the edge, you can keep that to a minimum, but you may not be able to eliminate it entirely.