I'd like to ask what defines improvisation. I know that it revolves around playing without a score, but is the absence of a score the sole condition for improvisation? If I come up with a piano arrangement of a pop song, practise it on the piano repeatedly to figure out the right chords/inversions to play, without ever producing a music score (i.e. I know what chords to use and how to use them, thereby creating small variations in the music I play each time), does it still count as improvisation? Does the fact that I practise the piece beforehand go against the very definition of improvisation? Thank you!
Google's etymology of the word:
from French improviser or its source, Italian improvvisare, from Latin improvisus ‘unforeseen’, based on provisus, past participle of providere ‘make preparation for’.
Things that are (to some extent) improvised are things that (to that same extent) you did not prepare beforehand. This doesn't include mistakes - i.e. when you did prepare something beforehand but it got away from you in performance!
I know that it revolves around playing without a score
That's not necessarily true - you could have a score in front of you, but improvise variations on what you see on the score. (Though I agree that's not usually what will be happening when someone is playing to a score).
If I come up with a piano arrangement of a pop song, practise it on the piano repeatedly to figure out the right chords/inversions to play, without ever producing a music score (i.e. I know what chords to use and how to use them, thereby creating small variations in the music I play each time)
It depends if the variations come from mistakes, or because you have deliberately left yourself room to make up some changes or additions while playing. If the latter, then you are improvising (to an extent).
Does the fact that I practise the piece beforehand go against the very definition of improvisation?
If you practice with the intention of playing the same thing that you practiced when you perform, that's not improvisation. If you practice your technique to allow you to make things up 'on the fly', and then you do so while you perform, that is improvisation.
It all comes down to this : if you know exactly what you're going to try to play when you start the piece, you're not improvising. To the extent that you don't, you are. It's not a black-and-white thing ("this is improvisation", and "this isn't") - it's a question of the degree to which you are improvising. Almost no performances in any genre could be said to be entirely improvised, but in many genres, almost all performances will be improvised to an extent.
There's a long scale of 'improvisation', from copying a piece accurately 'by ear', making minor changes to a learned piece, inventing a new melody over the same chords (as in classic jazz improvisation) right up to extemporising a completely new piece out of thin air (as Bach was reputed to be able to do with complex contrapuntal music).
Then there's the question of whether a learned performance loses 'improvisation' status. We know the stories of how Louis Armstrong played identical solos on recordings of the same song many years apart. But can an improvisation ever escape from your familiar repertoire of phrases, 'licks' and techniques? And it's often quite easy to play 'spot the player' or 'spot the composer'.
From the listener's perspective (and, don't forget it's about them, not about YOU) is an improvisation still valid when recorded and heard for the 100th time?
So, as so often (and to the annoyance of those who crave neat answers), I can discuss the question, I can mention each end of the range of what can be considered improvisation, but you'll have to decide for yourself whether there's a cut-off point where 'improvisation' kicks in!
I am going to suggest that you are trying to sort out an issue in your mind but that you have not asked the precise question that is going to clear it up.
Improvising and soloing are two different things.
I think that a helpful question is this: When you do a solo, are you free to do whatever you want or do you have to reproduce the solo in some well-known version of the song you are playing?
In a jazz environment you will be able to do whatever you want, and that includes making something up on the spot or going with a general theme that you have prepared in advance. Anything goes.
However, I have actually had people tell me that my solo in "Hotel California" is not "correct" - but this was in a "cover-version" scenario.
So I would say that it is not a good idea to get philosophical about improvisation. In a solo, just play something fluent that the current audience will appreciate.
However, maybe the extreme case is worth mentioning - namely that at a jam session you get persuaded to participate in a song that you don't really know. What you do then is going to have to be improvisation!