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Clone an Amp:
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AxeFx:
In the AxeFx you depend on the amp models provided by the manufacturer.
The latest version of the AxeFx features some matching options to match real amp tones closer - but IMHO it's not this easy and accurate as the Kemper.
Kemper:
Allmost perfect match
EQ Match
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The AxeFx has a great feature to match the tone of a recorded guitar track.
KPA can't do it
Use of IR's
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Both units can use speaker impulses - the AxeFx out of the box - the KPA via a new toolkit.
Amp Tweaking
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If you don't like what you get from the AxeFx (or like to try something new) then your have A LOT of parameters to tweak - by this can you get closer to your dream tone - or spend a lot of time and tweak instead of playing the guitar.
The KPA has only a few but easy and powerful parameters to tweak an amp or cabinet sound.
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Effect quality and tweak ability
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The effects and the quality of the effects in the Kemper are great sounding and easy to use - there are even dedicated knobs for common parameters - to tweak while playing.
If you like to deep edit the effects - all fine with the Kemper.
The effects and even the amp parameters can not be tweaked this easy on the AxeFx - you need to dive (deep) into the menus and submenus of the AxeFx.
Effect routing
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The number of effects and routing options in the AxeFx are WAY more advanced then in the Kemper.
Kemper has max 4 effects before the amp and max 4 effects after it (the last two delay and reverb are at a fix position).
AxeFx has a 4x12 matrix, you can run effects in all kinds of combinations - much more options.
If you just like some wah, compressor, distortion, chorus, delay, reverb and stuff like this - then the KPA's effects are fine - but if you want more than one delay or reverb effect at once or strange routing options - then the AxeFx is better.
AxeFx can e.g. use a SpringReverb and a room effect at the same time.
Missing effects
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The AxeFx effect repository is almost perfect - all kinds of effects are available.
The number of effects in the KPA has increased but there are still some basic things missing:
1) No Spring Reverb effect
2) (Generic) Amp Stack (EQ) always after the amp (AxeFx has many stack types and can be set before or after the amp block)
3) No pitch effects
4) Only one delay effect (an this one can't be placed before the amp)
..... so if you compare the features - then the AxeFx is the clear winner.
I have owned an AxeFx Ultra and the AxeFx II - now I use the Kemper - and the reason is the IMHO much better amp sound.
... still hoping that some missing effects/options will be added in the future.
This post should be wiki'ed. Armin is an authority on all things KPA (including very very small nuances of the almost unhearable by anyone except dogs presence end of the spectrum) so I would trust him. And if you do get a KPA, be sure to check out some of his profiles.