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Hi
BIOS is executed on the CPU. So it may not run if no memory is installed. However, a motherboard will beep upon poweron if no memory is installed and it is commonly known as "BIOS beep codes". But BIOS may not run without memory and thus it can't activate the beeps.
Thus it is wrong to call the continous "no-memory" beep signal as "BIOS beep code" and it is controlled by the motherboard, right?

Well, I'm not too sure.
The BIOS runs in non-volatile memory which is part of the motherboard, so it can still execute. Possibly, there's a routine there somewhere that checks if there is available memory and then beeps if there isn't, thus making it a BIOS-generated beep code. Of course, if you took away the CPU though...
Why does it matter, anyway?

If I remember correctly the BIOS is memory mapped from CMOS or some type of NVRAM. The cpu jumps to F000:FFF0 and starts running the BIOS. It's later that the BIOS might be moved into your SIPP/SIMM/DIMM type RAM. This is why there is usually a setting in your BIOS config program for BIOS shadow. Without the BIOS shadow turned on it's never moved into RAM.

The BIOS starts running from ROM. That is why it will run when there is no RAM -- your BIOS flash ROM is always there.
As chase says, if some RAM is actually detected by the BIOS initialization code, then a small portion of the "run-time" BIOS will be copied into it, later.