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This can, with the appropriate programming, most definitely be done on an ATmega328P (depending somewhat on the drum-loop's complexity. I'm assuming ~<50 drum events in the loop. Is that reasonable?).

Note that I said ATmega328P, not necessarily an Arduino.

The Arduino environment has a lot of default stuff going on in the background, that makes extremely deterministic programming (as you will need for something timing-critical) challenging.

The real question you need to ask here is how interested are you in programming, vs how interested are you in developing a instrument?

While I'm quite confident it's possible to do everything you want on a single ATmega (drum loop, multiple analog inputs, LCD, buttons, MIDI interface), the real question is how much work will it be to squeeze everything in? Again, do you want to learn to optimize embedded MCU code, or build instruments? It's quite easy to simply go to a faster MCU if needed, but you need to determine the MCU performance you need now, so six months of work in, you don't realize you can't quite get everything to work as fast as you need.

If I were you, the first thing I'd do is get it working without arduino stuff (basically, treat it as a raw ATmega, and use AVR studio or similar). Then, you can much more effectively analyse what kind of performance you need, and if the ATmega can manage it.

Once you're free of the arduino stuff, you're much more free to use different MCUs (they're generally more similar then different. If you can figure out one from it's documentation, you can probably do the same for others).

I've been working with the ATxmega devices a LOT recently, and they're really nice. You get three interrupt priorities, which make managing time-critical stuff waaaay easier. They're also really nice to work with (Sane peripheral designs! Convenient port structs! Etc...).

There are also the LPC devices from NXP, which are ARM based, as well as some of Atmel's ARM devices (as used on the Arduino Due), or the STM32 MCUs from ST. Any of these will have significantly more performance then an ATmega, or even an ATxmega.

The primary downside of a bigger, more powerful processor is price, but unless you're making thousands of those units, the per-unit assembly and manufacturing costs are so greatly going to exceed the cost difference (which will likely be only a few dollars) that it's basically irrelevant.