If you want to attempt the analog route: you can start with a fast frequency and the divide down, and you could also filter/add nonlinearity to the base signal to get higher harmonic content in the tone your system is putting out (though without selective amplification, the harmonics will only get so loud), but the biggest problem may be in the nature of the tuning systems.
It's relatively easy to generate harmonic multiples of the fundamental - 100, 200, 300, 400, 500 etc., but these frequencies don't line up equal temperament or just tuning systems exactly, and this is just the nature of the harmonic series vs. the twelve tone systems we've come to adopt. Against equal temperament, the second harmonic will be exactly an octave, the third harmonic will be a little sharp, the fifth harmonic will be a little flat, and there are little deviations for every tone that's not an even octave. So if you generate harmonic multiples of the VCO frequency, you can harmonize, but you'll have a fairly limited range of intervals that will actually sound in tune, and since the second harmonic is at the octave, you won't be able to get a major third, for example, closer than three octaves above the starting tone, and the minor third is another octave beyond that.
So what may actually be a better solution would be to do some analog math on your input voltage and feed it into subsequent VCOs. So you have your base VCO that produces the fundamental, then you have a second VCO that produces a fifth by using opamps to convert the input voltage into what the second amp needs to sound a fifth in the same octave - this will limit the input range in terms of what will be properly harmonized with the whole system because there will be some error and nonlinearity introduced by the voltage conversion circuits to feed the VCOs, but with sufficient tweaking and listening, you could probably make a fully analog harmonizer.
For octaves and close harmonic stuff, you could filter the signal to add in higher harmonic content or use a frequency multiplier or something to just get a tone an octave up.
It's probably going to be finicky if you try to go fully analog, but it sounds interesting. It simplifies things a lot to use a micro to do the voltage trimming for the VCOs - much less hardware tweaking and less chance of the input being out of harmonization range, and if you went the fast clock divided fully-digital approach to generating the tones, you just need a fast enough base clock so that when you divide down you can divide smaller than what sounds out of tune to most ears - if the steps are too coarse, you won't be able to hit notes exactly where you want them, depending on the division ratio.