Admittedly, I'm early into moving from analog circuits to digital signal processing, so I could be off the mark on my answer. :) Hope it helps, though.
When you clip in discrete time, the spectra is finite (more technically, it's periodic with a period of the sample rate frequency). That means the energy that would go into harmonics past nyquist gets "wrapped" around.
This is the big difference between analog and digital distortion. In analog, it's really quite difficult to create energy at non-harmonic frequencies of the signal. In digital clippers like you have here, it's trivial, and the design problem is figuring out how to deal with it. Most products will use some kind of anti-aliasing strategy (usually oversampling before clipping) to handle it.
This circuit is not a diode emulator, it's a comparator. It's the worst-sounding of all distortions. It sounds even worse in digital because of the aliasing.
And it will always alias, no matter how much you oversample it, because a vertical edge - aka "Heaviside Step Function" - has an infinite harmonic series. If you oversample it enough it won't alias much because the series terms become smaller. But they never disappear.
A better way to do this kind of clipping is with a tanh (logistic/s-curve) approximation. That can give you a variety of valve-like [1] smooth clipping curves. Unfortunately tanh is pretty expensive computationally, so a more practical alternative is a piecewise curve, perhaps with some interpolation.
Although if you only have 8-bit or 16-bit resolution you may as well just use a lookup table.
OP might want to consider learning a little more about signal theory and practical DSP before posting more how-tos.
[1] Not really because real valves are more complicated. But it will do for a first approximation.
The lower the barrier of entry, the more cool things that people can come up with! I'm hoping that more without a math/EE/audio background like myself can get started and explore some more of these deeper topics :)
The classic soft clipper is something like
tanh(kx) / tanh(k)
If you want something even more valve-like, a bit of DC bias to the signal before tanh will give you even order harmonic distortion, which sounds warmer (it beefs up the signal and puts energy into the octave harmonics, where as hard clippers usually only shove energy into the odd harmonics which sound harsher).
One trick for doing nonlinear waveshaping without introducing too much aliasing is to perform the wave shaping at a higher sample rate than the rest of your system and then downsampling with a low pass filter. Thankfully, the high frequency components introduced by nonlinearity tend to decrease in magnitude reasonably quickly.
In general, the more pointy edges you introduce to a waveform, the more high frequency artifacts you get.
This aspect of pontryagin duality (narrow in one domain means wide in the other) is also what underlies the heisenberg uncertainty principle. If you "hard clip" a photon's position (with a slit) you get a lot of frequency domain (momentum) noise, leading to a spread-out beam.
In the digital (or again more correctly discrete-time, you have a sample rate) realm it totally makes a difference, because many of the harmonics you generate will extend above the Nyquist frequency, half the sample rate, and "reflect" back down.
So for example, one might think drawing a basic rectangle in time would create a pulse waveform... but in the digital domain a square wave has ripples in it. Recall that in the spectral domain, a rectangle is approximated by ever-higher and ever-smaller sinusoids to infinity... in sampled signals, though, only frequencies below Nyquist are available. So the sampled version of the waveform will have ripples that the highest frequency components would have completed. The requirement then is to work with a higher Nyquist/sampling frequency than audible so that audible part of the square is correct.
From 0 to 1, 1 to to, 127 to 128, 254 to 255, nothing surprising happens, the output voltage slowly rises. When the counter rolls over from 255 to 0, something surprising *does* happen though, we get a massive and instant negative-going spike.
Because this jump from 255 to 0 is instant, it has to have a lot of harmonics, up into high frequencies. If you plot the spectrum of a sawtooth wave you'll see that there's a sine wave at every harmonic, reducing in amplitude as 1/f - that is, the first one at full level, the second harmonic, twice the frequency, and at half level, 3rd at 1/3, 4th at 1/4 and so on.
Now here's your problem - some of those harmonics, which extend off to infinity, are going to be beyond the Nyquist frequency. As you increase the frequency of the sawtooth relative to the sample rate, you'll start to creep into an area where those harmonics are actually still quite loud compared to the lower ones, and that gives you a problem.
Aliasing occurs because those harmonics "reflect" off the Nyquist frequency. Think about watching a Western, where the wagon wheels appear to spin backwards at certain speeds - they are aliasing. If they were turning slowly they'd move a little before the next frame, but at some point - where one spoke moves exactly into the space left by the previous spoke - they will appear to stand perfectly still! With a bit of thought, you'll see that the fastest the wheel can go puts the spoke exactly in the middle of the gap between spokes in the previous frame, and any faster will make it look like it's going backwards.
This is exactly what's happening with aliasing. The sinewave that's just above half your sampling rate is coming back down towards you, backwards. What goes up must come down.
In a real-life analogue sawtooth oscillator this is of course a huge problem because that big negative-going spike has infinite energy, and we're lucky that it's also infinitely short because anything that requires infinite energy is going to have terrible battery life. In practice, the maximum frequency of the harmonics in the sawtooth are limited by other parameters in the circuit (mostly how quickly the capacitor discharges through the reset transistor). And herein lies our major clue.
In order to generate a sawtooth that doesn't alias, we must calculate that step so it is no longer infinitely fast. There are a bunch of ways we can do this, but a nice simple way is to calculate an amount to "correct" the sample just before and just after the step so that you've got a rough approximation of passing the signal spectrally through your sinc window. This is crude but effective, and can be done with a couple of adds and multiplies making it ideal for real-time computation on even fairly crappy chips.
You can actually generate a sinc wavelet and "paste" that over the step, and when you get that right it sounds perfect. If you've got a fairly chunky DSP or general-purpose CPU with good math support to play with, this will give best results.
It's important to note that if you take a sinewave and clip it, then sample it, you'll still get aliasing because you'll still have a big infinitely fast step somewhere, which is why audio equipment is so big on having steep lowpass filters before and after digital (or, sampled, at least) bits. A great example is in the resolutely analogue Roland Juno 106, where the chorus board has three 24dB/octave Butterworth filters at around 10kHz, realised as cascaded pairs of Sallen-Key filters. The first rips off any signal above about 12kHz so it cannot be passed to the chorus chip, the second two pairs are "reconstruction filters" to remove fizzy-sounding sampled steps which would be apparent as a swooshing sound as the chorus went through the "long" end of its delay.
The reason they need this is that the analogue "Bucket Brigade Delay" chips are a kind of analogue dynamic RAM, where on every clock pulse a capacitor charges up from the input to its left, and on the next discharges into the output on its right. Stick a few hundred of these capacitors together and it will "pass buckets of signal" down the line, delaying it by whatever the clock rate is divided by the number of buckets. It's all analogue, but it's still sampling!
Ah-ha. Gotcha. So my intuition was more or less correct. It's not that clipping in digital is especially susceptible to aliasing, it's that when people clip in analog they brickwall the signal to get rid of the harmonics.
You can't create a naive sawtooth (for example) and *then* filter it because the damage has already been done. That being said, the "supersaw" oscillator in the Roland JP8000 generated a bunch of naive saws and *highpass* filtered them just below their fundamental to remove the gurgly "beat note" from aliased partials.
I don't see you doing anything in particular to bandlimit your waveshaping, but I might well have missed it.