With the Falcon, Atari added specific Xbios system calls to drive the sound hardware. So now it is possible for different hardware to be used by Atari applications, provided they use the system calls. Under MiNT, an audio device driver should be used when it will be done/debugged/stable.
ARAnyM has some DMA audio emulation, but it may not be sufficient for programs to operate properly. Well-written applications use Xbios, and could use the ARAnyM audio driver.
Run 'zmagxsnd.prg' to install this compatibility layer, and you should have a better audio experience under ARAnyM (only with compatible applications of course). This program is TOS resident, so it must be run before MiNT kernel.