Bsrgvty

'Gain' just means 'amount of amplification'. It can be applied at various stages in the signal chain between musical source and final playback.

The topic we're talking about is called 'Gain staging'. It basically means that at EVERY point where amplification takes place we should strive to keep the signal above the noise floor but below the overload level. In the old days of analogue recording (and noisy preamps/mixers) we erred on the high side. Noise floors were high, moderate overload didn't hurt much - sometimes we even LIKED the effect and called it 'valve sound' or 'tape saturation'. Now we tend to keep levels safely 'under the ceiling'. Noise floors on today's equipment, even the cheaper stuff, are pretty low but hitting the digital ceiling just sounds nasty.

Perhaps your question is about the 'gain' knob at the top of a channel strip on a mixer? Setting this low but amplifing later would normally mean MORE noise. Unless, perhaps, you were setting it TOO high and causing overload distortion at that stage. Although the overall volume can subsequently be modified, the distortion will still be there.

There's also the special case of guitar amps, where a 'gain' control may be used to deliberately push the signal into overload, causing an interesting-sounding type of distortion. In this case you have the choice of setting 'gain' low for a clean sound, higher for a distorted one. The main volume control comes later. But you say this question is about a mic input.

The old Mackie mixer manuals were very informative. Read 'Set the levels' on page 5 here. It may not be directly applicable to your setup, but the principle applies.

https://mackie.com/sites/default/files/PRODUCT%20RESOURCES/MANUALS/Owners_Manuals/1604VLZ4_OM.pdf

In amplification jargon, all volume changes are described by the term "gain", where gain is the ratio of the input and output signals. In a lot of modern amplifier design, there are effectively 2 separate "gain stages" where signal amplification is performed: the input, or "preamp" stage, and the output or "power" stage. The input stage amplifies the incoming signal for use in the amplifier's circuitry, and the output stage amplifies the signal coming out of the amp for use in a speaker or similar. Turning the gain up too high on a given stage introduces distortion on the signal.

A fair amount of mid-20th century amps (especially for guitar) had just one adjustable gain stage, simply labeled "gain." Musicians and producers found that by cranking the gain very high, the resulting signal distortion actually sounded kinda cool and so they used it as their sound. The term gain gradually became synonymous with distortion, and "volume" or "master volume" was used on output stages to differentiate the output level from the distortion level.

This also holds true in non-guitar amplifiers, like those found in mixing/recording consoles and emulated in DAWs. In these cases the input gain is more for setting the input signal level high enough to get a good noise floor without introducing distortion, so that the signal can be routed to various EQs, effects, and the like. However, a lot of classic microphone preamps like the old Neves and APIs weren't capable of providing noise-less amplification, so the gain on them (and any emulations) is also a distortion level. It's a completely different-sounding distortion, but a distortion nonetheless. The output level is used to set the desired loudness of the processed signal. in the case of a console or multi-track project, the loudness is relative to the master volume of the system, and is used to set the relative signal volume compared to the other tracks.

In my recording experience, I've found a good rule of thumb to get a good-sounding track is to roll up the input level to the point where you just start to get distortion on the highest peaks of your incoming signal, and then back off a touch. The output volume is then set to get your track as loud as you need it, relative to the other tracks in the project. However, the ultimate judge of "good tone" is your ears, so do take the time to find a way that works and sounds best for you.

Electrical engineer here.

Gain is the ratio of the output signal magnitude to the input signal magnitude of an amplifier. In idealized circuits, as you turn the gain up, you increase this ratio and the signal's gain increases and it gets louder. Or you reduce the ratio and it gets softer. No distortion occurs.

However, in the real world, even well-designed amplifiers can only reproduce the waveform up to a certain magnitude before the tops and bottoms of the waveform get chopped off. This is called clipping and depends on the maximum upper and lower voltages available to the amplifier.

The "Volume" knob, in terms of circuit design, often routes some ratio of what would be the full output signal (0 to 100%, based on knob setting) to the output of the circuit (jack, speaker, etc.). In this case, max volume represents no (or negligible) attenuation of signal, while 0 volume represents the would-be output scaled down to near-zero amplitude. This knob can only reduce the overall gain of the circuit.

Basically, if you have a well designed piece of equipment and you crank the "Gain" knob, you're more likely to get clipping distortion (sometimes this is intentional, as in tube guitar amplifiers and distortion effects pedals), which you may or may not want. But if you turn down the volume, you're typically just scaling an already gained-up/possibly distorted signal. Sometimes high-headroom amplifier circuits that can be gained up massively without clipping will be called "Volume" on the user interface, but ultimately this will clip if you gained it up without bound too. (This last bit is the difference between high-end $4 apiece opamps and consumer-grade $0.25 apiece opamps, as well as relatively high-voltage supplies for the amplification circuitry.)

So the answer is, as with all things engineering, be it audio or electrical, "it depends".

You really have to follow your ears.

-- EDIT
As you increase gain, you also amplify circuit noise. The general level of this noise is your noise floor, which in layman's terms basically means that any signal of magnitude lower than this noise is indistinguishable from the noise itself and contributes nothing of value to the recording. I typically adjust my gain up to the point where the noise becomes (subjectively) unacceptable, and then back off a bit.

It's as I said in the comment:

Gain is referring to the input, volume to the output.

Volume is how loud the OUTPUT of the channel or amp is. It controls loudness, not tone.
Gain is how loud the INPUT of the channel or amp is. It controls tone, not loudness.

https://www.musicianonamission.com/gain-vs-volume/#targetText=Conclusion%3A%20Gain%20Vs%20Volume&targetText=Volume%20is%20how%20loud%20the,It%20controls%20tone%2C%20not%20loudness.

Edit:

My question is which one is recommended in music production to make music louder without distortion or other bad side effects? Gain amplification or volume increase?

If the gain-level is in the red area it means the input signal is to strong. Thus the best sound will be resulting from a good balance of input and output. That's what will be called a good mix.

I see answers all over the map on this with some DuckDuckGo-ing, so here is my as-yet-undisturbed understanding:

Gain is a measure of amplification, whereas volume is a measure of limit. Gain is a control on a circuit which actually amplifies a signal (and its noise, ATBE) whereas volume is a control on a circuit which limits a signal.

Implication: increasing volume too much can distort a signal because the downstream components cannot handle it, whereas increasing gain can distort a signal because either downstream components or the amplifier itself cannot handle it.

I could well be wrong -- I do not have any of my 1980s circuit construction textbooks anymore :-(

The term "gain" tends not to means nothing specific in the labeling of knobs on commercial audio equipment for music. I've seen plenty of "gain" knobs that actually performed simple attentuation of a previous fixed-gain amplification stage: i.e. simple volume controls.

"Gain" doesn't exclusively refer to amplification in electronics, either. In radio, antennae are said to have different amounts of "gain". An antenna is a completely passive device; it only captures energy.

However, in amplification circuitry, "gain" does refer specifically to an amplifier's amplification factor. A honestly labeled device uses "gain" only for amplification factor controls, and "volume" (or similar, like "output level" or such) for passive attenuation of a signal voltage with a variable resistor.

Gain control influences an amplifier's action by altering the amount of impedance in its feedback circuit. It can affect the signal quality in complicated ways compared to a passive volume control. The frequency response, distortion and noise characteristics of the amplifier are affected by variations in the gain.

With respect to an audio mixer, gain refers to the pre-amplification of the instrument or microphone signal and volume refers to the volume fader or knob, which is normally a resistor.

Usually volume faders or knobs have a detent, marking unity. They are designed to give the best signal to noise ratio at unity gain, where the signal input equals the signal output. Because different instruments and microphones will have different signal levels at the input stage, you often need to adjust the amount of gain to achieve unity gain to maximize the input's signal to noise ratio. If you have the gain too low, you will need to adjust the volume fader up above unity, raising the noise floor. If you have the gain too high, you may need to adjust the volume fader below unity, again raising the noise floor, but at the gain stage.

Ideally, you want the signal to stand out from the noise floor to achieve the highest signal to noise ratio. If the gain is way too high, it can also clip the signal or distort the signal before hitting the volume fader, so no amount of adjusting at the fader will clean up the signal. It can be a bit overwhelming, but with audio, your ears are your best guide. Different microphones or instruments will perform better with different gain settings, but the best way to find that is to set the fader at unity and start with the gain all the way down, then increase it until the signal sounds clear. Good luck!