Seems to be lots of slightly differing and confusing views here.
The term noise is usually used to represent the total sound present when no signal is being passed and how loud this is being the noise floor. A lot of pieces of equipment specs quote a signal to noise ratio.
This is at its greatest when the music signal is highest before clipping. If you run a mixer at 50% then it makes sense that the S/N ratio is going to be lower than at 100%. How much difference this makes is debatable in modern equipment.
With the advent of digital equipment, resolution comes into play, and lots of system techs now try to maximize the resolution being utilized by using as much of the digital range as possible (often by running very hot into the initial A/D stage).
It is correct to say that you should really set gain structure where all pieces of kit are hitting 0db at the same time (esp digi kit to make best use of resolution) and that amp gains/input sensitivity are adjusted accordingly so that the amp clips at this point, however in practical terms its often safer to sacrifice some of the perfect gain structure and run the amp gains on full, in order to prevent people being able to turn it up further.
Modern kit tends to have very good ways of limiting the signal at most points prior to the amps, whilst leaving simple gain knobs handily available to the uninformed. Ask most general public how turn up the volume and most will point to the amp....
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Anyway, off-topic question - I don't know how much of all this applies to digital OP mixers and digital IP amps...but if there's leads going from such an amp to the speakers, it can't really be 100% digital....can it?
No it can't, but the quest should be to try to minimize the number of A/D conversions that take place. If your mixer has digi outs and your crossover/EQ has digi ins then use them.
Imagine a relatively simple setup using mixer, EQ, crossover, amps, speakers.
digi to analogue from mixer
analogue to digi into the EQ
digi to analogue out of EQ
analogue to digi into crossover
digi to analogue out of crossover to amps
If you were using a digi mixer and say a Behringer DEQ2496 and DCX2496 you could go digital all the way until you had to go analogue into the amps.