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Amp and Cab modeling for beginners
The Axe-Fx is a guitar preamp. It models more than hundred real amps such as heads, combos and racked preamps. You can also use the Axe-Fx for effects only, for mastering etc.
Virtual power amp
It has built-in power amp simulation, which means that you can listen to the sounds of the modeled amps through headphones, and connect the Axe-Fx to a mixer or record it without needing a separate power amp.
Real power amp
To amplify the Axe-Fx through a speaker, you do need an physical power amp. This can be a standalone tube power amp designed for guitar, or a head or combo (through its effects loop) or a so-called neutral power amp. You can also use studio monitors and monitor wedges with built-in amplifiers ("active", "powered"). Depending on the amplification device you engage or disable the Axe-Fx's power amp simulation.
The speaker you use is very important for the tone.
You can use the Axe-Fx with a traditional guitar speaker, in combination with a power amp. This will give you the familiar amp-in-the-room tone. It does limit the possibilities of the Axe-Fx, because any sound will go through that speaker en therefore is colored by it.
You can also opt for FRFR (full-range flat respons) sound reproduction. This requires a FRFR speaker and an external or built-in neutral (power) amp. Studio monitors are FRFR, as well as some wedges / cabs. A FRFR setup requires power amp simulation and cabinet simulation to be engaged. Cabinet simulation means that the sound of a virtual speaker cabinet is added to the tone of the modeled amp. The Axe-Fx comes with many built-in cabinet models and also allows loading external cab models, known as IRs (Impulse Responses).
It's very important to realize that when you're using FRFR with cabinet simulation, you'll be listening to the sound of a mic'd speaker, as opposed to an amp-in-the-room. A modeled cab always represents the tone of a speaker that as captured using a microphone, mostly very close to the speaker. That's different from listening to a guitar speaker at some distance. FRFR has more lows, more highs and has the coloring of the used microphone baked in. It takes a while to get accustomed to FRFR tone, but it's the tone the audience hears too through the FOH system and when listening to recorded music.