Hi. Thank you for the pointers. But I should have mentioned that I'm an absolute newbie to FPGA design. Also there's no way I can solder BGA in my home setup, and with all the passives and regulators and stuff around an FPGA chip, there is a lot that can go wrong. Therefore I'll stay within the bounds of ready-made dev boards.

I have yet to begin, therefore I still don't know how many LUTs I will need.

So far I've read up quite a bit and I have decided for the following:

- Bluespec HDL, classic version (Haskell-like) because life is too short to be writing (and debugging) Verilog

- Altera / Intel FPGA, because their IDE runs on Linux and is somewhat user-friendly (Xilinx gives off way too much of an elitist vibe)

- At first I'll try to do everything in hardware, including the graphics primitives for the display, because this will allow me to use the display as a real-time oscilloscope / FFT, which is a functionality I'd like to have.
If that proves too difficult, or if I run out of LUT space hard, my next best option is to add a $10 Pi Zero W and do all the GUI using whatever programming language and library I find most appropriate.
Actually, another option would be to program the GUI and all the application logic on the Pi, but pass the display serial interface through the FPGA in a pass-through mode, so that I can "hijack" it when I want to use the display as a scope.

- As for audio, I will try to use some of the FPGA-based techniques. So far I like @zipcpu's "bit-reversed counter PDM" technique the most. Do you know if there are standard values for the audio-range RC filter? You were talking about a 10nF cap. Anyways, if that proves too dodgy, I might look into a discrete DAC chip.

Wish me luck...

Re: amplifier

That's cool. But I only need line output, so I think a filtered FPGA pin should be enough. Yes, I intend to use MHz PDM and a RC filter.

If open toolchains are important, I would go with a Lattice ECP5.  There's probably one that's big enough for your synth, and they're basically cheap.  The various RAMs, distributed and block, are all supported, as are the fast multipliers.  AFAIK, some of the chip's fancy DSP functions are still to be worked out, though, so if you need them, you might look elsewhere.  (My guess is that you won't. Audio is "slow" for FPGAs.)

I like your strategy of an external chip for the UI, mostly because you can crib existing UI code, and especially if you have used something before.  You could go with a bigger FPGA and a soft core, but you'd want to be sure that the UI would be there for you.

Yes, I will only need to use the various RAM and multipliers.

By the way, do you know if the open-source synthesis tools for Lattice FPGA are noticeably faster than, say, Altera / Intel tools for Linux?

I'm thinking that a $10 Pi Zero W would make my life incredibly easier as far as programming the UI is concerned. I could write the UI in any programming language and library I want. Also I could SSH into the Pi via WiFi to debug or update the system.

On the other hand, doing everything in the FPGA, including the display logic, would mean being able to use the display as a real-time oscilloscope / FFT, which would be very useful for my purpose.

In fact, I might be able to do both. (See my answer to @Sina Roughani  above)

Although above your pricepoint (and truthfully probably overkill) a Xilinx Zynq on some devboard might still be an option. An FPGA with an onboard CPU, linux capable but standalone (FreeRTOS) possible just as well. It is quite steep to learn but there is a lot of examples to find as well. Altera probably has similar offerings. A shortlist I found: https://www.joelw.id.au/FPGA/CheapFPGADevelopmentBoards

Although cheaper options are available, as a learning platform I'd suggest something many other people have used.

literally just plugging my own project, but I recently designed a simple board with an atmega328pb and a lattice lmcxo2-2000hc-5tg100i: https://hackaday.io/project/180005-a-digital-real-time-clock, github link to the board: https://github.com/ZakSN/z_1_rtc_pcb/tree/2000hc-version. Its definitely not the cleverest way to do things, but you can maybe use it as a starting point (or as an example of what not to do...)

Altera (now intel) and Xilinx both have synthesizable processors and the IP blocks to implement a complete processor inside the FPGA. This would let you do your display and user interface in software, while implementing the synthesizer in logic. The boards that I used for the custom processor implementation that I did were from Digilent and were around $100 US with Xilinx FPGAs.

A similar idea to @Bharbour came to mind for me. My affection for CPLD meant that I imagined a CPLD in coordination with a commodity CPU, using the CPU both for UI elements as well as advanced synthesizer functions. Of course, the market has proven that FPGA-based synthesizers are viable. FPGAs have  also progressed to the point that you'd have fairly limitless flexibility in addition to your pick of platforms and toolsets (check @Elliot Williams' post). Don't be afraid to jump in and find yourself with a few extra dev boards as you determine what mix of features works for your design goals.

I've linked below to what I would consider a more minimal version of what you're attempting(?). Good luck!
https://thetinysynth.wordpress.com/technical-details/

Wow, that is super helpful! Thank you!