Hacking a toy keyboard - how to output polyphonic CV?

I'm learning electronics by making synth modules and messing around with associated circuits. I have a lot of programming experience, and a bit of electronics now.

I found a cheap toy synthesizer at a thrift store and have figured out how to scan the button matrix with an Arduino to recognize when keys are pressed and released. I think it would be fairly simple to output midi messages, though I've never done that before.

What I'm really curious about right now: how would I output CV if I wanted to control a polyphonic euro rack module system? I know I'd need all the modules for multiple voices (let's say 4 for example). And I've already read that that's not realistic, and it's expensive, and it's a headache to keep them synchronized and so on. So let's say I'm not going to do it, I just want to know *how* it's done.

I'm assuming I need four (or whatever number) of CV outputs from the keyboard (or the midi to CV module I would make). Those CV outputs could be based on DACs controlled from the Arduino.

The part I'm specifically curious about: when a key is pressed, how do you determine which DAC-output to send the note/CV through? If I'm holding down one key while jabbing quickly at three other keys, what does the routing from keypress to output look like? Is it just clever programming to keep track of what outputs are in use? Is there some other trick or chip that gets used?

Just to reiterate: I don't want to buy modules that would do this. I want to know how the output routing can be done. Thanks for your help!