Gain is the amount of additional energy a signal receives (gains) in the preceding amplification stage.
Volume is the total energy that the system output signal produces, measured by the area (2D *volume*) bounded by its waveform.
Gain describes by how much a pre-amplifier amplifies the energy of an audio signal between its source (detected directly from the media, like a cassette tape, phonograph record, or a CD digital to analog converter. A very small voltage) and the preamplifier output, which usually feeds the final amplifier that drives the speakers. This is what sound mixing boards so. They accept inputs of various magnitude and allow the operator to normalize and mix the audio before amplifying it and sending it on.
Preamplifiers normalize source signals of varying magnitude so that the preamp output/final amp input levels are the same (which is why stereo receivers have component selectors. Each has a different pre-amp gain). In other words you want to feed the final amp with signals of similar level, lest you need to turn the volume knob every time you switch sources. So pre-amp gain may change to keep the volume steady.
As described above, volume describes the magnitude of the energy that the final output signal contains (or can contain). If you look at the signal on an oscilloscope, energy magnitude is found by calculating the area between the reference voltage (usually 0) and the wave scribed by the signal’s oscillating voltage (+/- some amouunt).
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