In mathematics, a Hilbert transform can be used to synthesize the imaginary part of a real signal, thereby providing the part needed to make it a complexvalued signal. Complexvalued signals are useful because you can among other things know the amplitude from a single sample. [hilbert~] is a fast approximation using allpass filters but is not quite the same because it also phaseshifts the original real valued signal. This demonstration was inspired by Katja V's discussion and was greatly improved using info from @NicolasDanet. Select each of the three phase comparisons in turn and graph their respective phase shifts to see how the hilbert~ magic is performed.
hilbert~ phase.pd

What [hilbert~] does

Thanks, it makes me want to read the Katjaas's blog!

I was wondering if I could test all the FFT frequencies at once, so I started painstakingly building a 1024 pt wave table with all the harmonics in it, and then I saw it <gasp>: it's just the impulse function. I've read about that equivalence a hundred times, but it never sank in until just now. Two and a half days of thrashing around, worrying about [switch~], [bang~], reblocking, vector discontinuities....ok, it was interesting, but completely unnecessary!
hilbert~ phase2.pd 
Archimedes @jameslo

@whaleav Well, that metaphor applies only if, prior to taking his bath, Archimedes was told a hundred times to try putting the crown in a bucket of water.