Are there music keyboards with the possibility to import ANY sound onto each key and what are these keyboard / the function called?
Also any information regarding this would be helpful as well.
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It's called Sampling.
I think the very first was the Akai, or possibly the Bel BD80, but they had no keyboard.
The Emulator was the first widely available keyboard sampler.
They cost about 5 grand at the time & were really rather limited, compared to a modern structure. You might still be able to pick up a second hand one on eBay, but they still fetch a couple of grand.
There were keyboards through the 90's & early 2k's that could do it, Yamaha's PSR 8000, 9000 & later Tyros had a limited sampling capability.
These days, no-one would dream of doing it in hardware, when any PC can do it in software. Systems like Native Instruments' Kontakt are the kind of tool you'd use now.
As Tetsujin wrote, it's called sampling. I wanted to add to his answer because it's not all about software these days, as he concluded.
Hardware sampling may be having a bit of a comeback. Many DJs and electronic musicians like the features of looping and sampling software like Ableton Live or Fruity Loops, but don't like working with or gigging with computers.
Two product lines that spring to my mind immediately for hardware based sample playback are the Clavia Nord Electro keyboard line and the Elektron line of hardware samplers.
Clavia just upgraded the sample playback capabilities of the Nord Electro line with the Electro 6. What it lets you do is take one or more sound files on a computer, load them into a sample management software package and determine how you want the keyboard to play the samples back (what notes play what samples, do the samples loop, scale with key pitch, etc.) and then you plug the keyboard in with a USB cable and it loads the samples into a sample bank on the keyboard. Once the sample bank is loaded, you can play the samples with the keyboard without a computer. What you can't do without a computer is record or load samples into the keyboard.
The Elektron products, including the Octatrack, Digitakt, and Rytm products, let you record right into the hardware sampler, change the start and end points of samples, playback rates, etc. All like a traditional hardware sampler. They are not keyboard products, as they are geared more towards DJs and electronic musicians, but you can connect a keyboard with MIDI for sample playback.
Akai still makes at least one hardware sampler, the MPC X.
Finally, many of the instruments that were popular in the 80s and 90s are available on the used market for very little money, including sampling keyboards and rack sampler modules. If you can put up with lower resolution and bit rates, you can get into hardware sampling for not much money.