Every musician should learn jazz.
As most musicians today understand it, jazz is a musical style. That is not wrong, but it's not really the most important thing because jazz is also a process. It's the process of making music in the moment, an essential skill that was mostly lost in classical music towards the end of the nineteenth century.
A long time ago, musical notation was developed out of the necessity to make a piece of music permanent. Without recording technology, it was necessary to write the music down if it was to be played more than once. Music, up to the end of the eighteenth century, was improvisatory in a very high degree. Even in the romantic period, there was still a great deal of improvisation serving as the basis for concert music.
Musical notation allowed composers to write works of increasing complexity, to the degree where a composer could take months to write a single minute of music. It also marked a new division between composer and interpreter. Today we have composers that can't play a single note of music on any instrument; some of the really bad ones are also practically tone-deaf. There are also interpreters out there that can't pick out a melody on their instrument, let alone deviate from their precious score-- even when the musical style demands it!
The necessity for the ability to improvise goes beyond being able to pluck out a cadenza in a classical work or playing a jazz standard by ear, it affects everything we do as musicians.
It affects our interpretation. When we hear or play a piece, being able to follow along with the "jazz process" of the composer is something that gives us depth and insight. It gives an interpreter the ability to see what a composer didn't write and what he could have written instead of what is on the page. This ability also allows to understand the "why" and the "how" better. I've been to too many classes where the lesson the teacher is able to give with his limited insight does not go beyond correcting the note values and marking the tempo by clapping along. There is no talk about the musical process the composer followed and there is no deeper meaning to what is being done. Too many musicians go through their whole careers thinking that an interpreters job is just counting , measuring and following along with what is written like a good little bureaucrat.
Being able to improvise also affects the way we teach and learn. I believe that the best way to learn is by playing around. In many languages, the word for playing an instrument and the word for playing with a toy is the same. I have found that the best way to teach a child is through improvisation. By letting them write their own songs and trying out alternatives to what is written down. In the same way, good practice is really the process of teaching ourselves. In that sense, one of the best ways of learning is by making variations on the music and playing the things that are not there.
Most important of all, the jazz process allows us to be real with what we play. Particularly with the piano, it is easy to play a note without feeling it, without thinking it. You drop the finger, and the instrument makes noise. You could just as well have hit the piano with a pencil, or thrown something at the keys. It is absolutely essential to listen in your mind to every thing you play or it isn't real. This is more apparent when making jazz. It is easy to play scales up and down, or have a few formulas that sound good almost anywhere, without really feeling the different harmonies or thinking out the melody that you are improvising. The sense of actually doing what you are singing in your mind is quite hard to do, but being honest and real with your playing really makes a difference in the end result and in what you, as an interpreter, get out of your music.
In The Universal Mind of Bill Evans, Bill Evans sits down and has a talk with his brother. I consider this video essential viewing for any pianist, specially those of us who play mostly classical music. Bill Evans was not only a great pianist, he was also a philosopher of music and, in my opinion, one of the greatest musical minds that the world has known. I don't compare Bill Evans to other jazz pianists, I think of him as a modern day Chopin or Schubert. Here is part one of the video, the whole thing is up on YouTube: