Going back a couple of decades and more, the guys I worked with used Matlab to test and develop filters and filtering co-efficients to deploy to DSP's for almost-real time processing. That was the easiest way then, and is probably one of the easiest ways still.
Some DSP development boards we had also had audio codec hardware on board - early forms of sound-card AD-DA chips so could either do real time processing, or process/save, retrieve and playback etc.
One equivalent today is using a soundcard and Matlab or Labview, There is also a plethora of processing programs (even built in as a normal part of soundcard drivers/support), plus soundcard scope software that might be useful. There is very likely already software that would allow you to develop what exactly you are exactly looking for on a pc/laptop or raspberry pi without deploying to Arduino or more powerful board.
However, if the embedded system is a better option for portability, then a soundcard also seems an ideal way to test hearing response and develop filters/coefficients for spectrum/equalisation of real-time sound and playback of test files, to translate to code for a suitable platform.