There are plenty of classical (and other) musicians that can sight-read the most difficult music but cannot improvise.
There are people who can't read music that can improvise well but put a moderate piece of music in front of them and they can't play.
I often read posts by players who dis the readers. This saddens me. It's as if the non-readers want to feel better about themselves by diminishing the talents of the readers.
I read well and I improvise well. Both are handy skills but not being able to improvise does not make you an inferior musician by any stretch of the imagination. Ask Itzak Perlman and so many of the most famous classical musicians.
Like improvisers, there is a skill to reading music and making it come alive. To the non-reader it seems like all you have to do is mechanically read the dots on the page and voila, you have music. This is far from the truth.
If you read the notes as if you entered them quantized into a sequencer, you are not necessarily a good musician. There are markings in the music that give you an idea of how the composer wants it to sound, and you use your skills and the lessons your teacher taught you to put the nuances into the music. When appropriate things like dragging the beginning of a phrase and rushing the end to end on time, pushing the second beat of a waltz a little bit ahead of the beat, dragging 2-beat triplets, putting more emphasis on syncopated notes, adding some 'swing feel' to eighth notes, deciding how much space to leave between the notes, using various articulation techniques, when and how much vibrato to use, dynamics, and thousands of other expressive devices that you have to instinctively know where and when to apply while sightreading. This skill is every bit as difficult and perhaps even more difficult than improvisation. And in reality, the good sightreader is playing by sight AND by ear. You need both to be a good sightreader.
So don't ever dis anyone who can't play when you take the music away. You are sending the wrong message about yourself when you do.
I learned to read music first. I was in the school band, and even made it to first sax in the all-state band every year I was in high school. I did this on tenor sax, and the chair usually goes to an alto sax player by default. By high school, I also got into a rock band and at first used my ears to duplicate solos I heard on records, picking out one note at a time. This is true learning by ear and it is a laborious process at first. Then I tried my hand at improvising. I was pretty lousy at it at first, but as I learned more I got better at it to the point where I enjoy improvising much more than reading music. Take the music away, tell me what key, and I can still improvise better than I can transpose the melody in my head.
But I'm still blown away by a person who can sit with a Rachmaninoff concerto in front of him/her and sightread it. That to me seems much more difficult than improvising a jazz solo over a set of chords.
Insights and incites by Notes