I uploaded some abstractions that I used in this piece:

The first set (dphasor1, dphasor1~, dphasor2, dphasor2~) reduce aliasing by constraining phasor~ periods to integer numbers of samples, then correct the resultant pitch error by either slowly dithering between the two nearest frequencies, or applying noise modulation. An ordinary phasor~, by comparison, essentially does periodic, audio rate modulation between the two nearest frequencies, generating sidebands that are audible as aliasing. The advantage of my approach is that all aliasing falls back onto the harmonic series (in the dphasor1~ case) or is spread out into broadband noise (in the dphasor2~ case). With oversampling, non-band limited synthesis algorithms can be implemented with very high quality. And at low sample rates the artifacts generally aren't too horrible.

The second set (pdist~, pdistx2~, pdistx4~) implement a phase distortion algorithm that smoothly interpolates between a sinusoid and an arbitrary waveform, or between multiple arbitrary waveforms.

Also included are a number of other "helper" abstractions.

The (carelessly thrown together) example patches are _dphasor_ex_1.pd, etc.

As far as I know, these are novel algorithms. I wouldn't be surprised to see that they'd been published or patented already, but it was quicker for me to implement them than to dig through a bunch of literature.

I'm not making any promises about these being well-documented or useful. At this point they've only been made for my own personal use, so they mostly just conform to my own arbitrary and nonsensical conventions. I plan to eventually write better descriptions that discuss theory, etc. For now I just want to make it available.

http://www.pdpatchrepo.info/hurleur/pdist_dphasor.zip

Great ! I will be looking in to this

Small update: I added the ability (via creation argument) to constrain fractions of a period to integer numbers of samples. This is useful if you want to play samples with multiple periods, or just reduce the distortion to the harmonic series (i.e. make sure your square wave is always square). Conversely you can improve pitch resolution at the expense of subharmonics by using a fraction.

So it works like this: [dphasor1~ 2] ensures that both halves of the phasor period have the same number of samples. [dphasor1~ 4] ensures 4 equal portions, etc.

[dphasor1~ 0.5] ensures that two consecutive phasor periods will have an integer number of samples. This improves pitch accuracy, but it also means that the output may contain harmonics of f/2. Often this is acceptable, so it's a good trade off.

Really this is trivial stuff, but I'm surprised I haven't heard anyone advocating for this sort of thing. None of it is new, and it works pretty well even in less than optimal cases. It opens up some useful territory that's ordinarily scary (arbitrary nonlinearities, etc.), so why not use it all over the place?

http://www.pdpatchrepo.info/hurleur/dphasor1~.pd

I should also add some quick math/theory stuff so that it's a little less confusing.

Imagine a mod n digital counter that receives a clock pulse at frequency Fin and outputs a pulse whenever it resets to 0. Fout = Fin/n, i.e. integer-n frequency divider. The output pulse has no jitter, but Fout is constrained to integer divisions of Fin.

Compare this to a fractional-n frequency divider, i.e. a phase accumulator with fractional increment that keeps a remainder when it resets. Fout can be set to any frequency, but as the output pulse still only occurs on clock edges, there's some jitter. The time between output pulses varies between the nearest two integer divisions of the clock, n (i.e. the fractional denominator rounded down) and n+1 (the fractional denominator rounded up). The output pulse is essentially a pulse train at the correct frequency that's naively sampled at Fin, resulting in aliasing. You could also think of this jitter as frequency modulation by a pulse wave, where sidebands show up as unwanted frequency components. It's really only counting to n or n+1, but it's alternating between them in a way that produces the correct output frequency. Given this, you can change the way that it alternates between the two frequencies to change the properties of the resultant sidebands.

A phasor with an arbitrary frequency is analogous to the fractional-n divider. A frequency that's not an integer division of the sample rate will have some jitter on its output, and if you attempt to use it as a sawtooth wave, it will have a crappy aliased sound (i.e. an ideal sawtooth naively sampled at Fs). If it's constrained to integer divisions of the sample rate, any aliased harmonics fall back onto the harmonic series. This also holds if you put the phasor through any sort of non-bandlimited nonlinearity, and arbitrary wavetable or whatever.

So if you're generating frequency f from sample rate Fs, the closest integer-n divisors are n = floor(Fs/f) and n+1. The idea is to alternate between frequencies fh = Fs/n and fl = Fs/(n+1) so that the long term average frequency is f. Basically you want to find a sort of duty cycle that describes the correct proportion of fh and fl. For error E = fh - f and step size fstep = fh - fl, duty cycle D = E/fstep. The end result is D = -(n+1)*(n*f/Fs - 1). Then you're picking your output frequency fout = Fs/(n+x) where x = 1 if D > M, and x = 0 otherwise. M is some arbitrary modulating signal (which may be periodic or random at any frequency) that has a uniform distribution between 0 and 1.

So this means that you can arbitrarily control the characteristics of the "frequency dither" by using any suitable signal for M, and this determines the characteristics of the artifacts that result. The audio rate periodic FM that you get from ordinary aliasing is about as awful as you can get. It's useful to use either a slower periodic signal (producing vibrato, i.e. dphasor1~) or a rapid noise signal (which spreads out aliased frequencies into broadband noise, i.e. dphasor2~).

Of course oversampling still helps a great deal (with really great results at 16x or more), but my test piece did everything at 44.1 kHz and doesn't sound objectionable.

Hey, thanks for this, I will be digging through your patches in the next few days. Your music is great by the way.