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Improvisation is essential to jazz music. Different styles of improvisation define different artists and help them distinguish themselves. To improvise means to compose and perform simultaneously and spontaneously. Improvisation can be based around a written piece of music, but is never written down beforehand. Improvisation is such that an artist will never improvise in a given context in the same way twice. One may hear the same piece by the same artists several times and it will never sound exactly alike. Improvisation is important for jazz, but jazz is all about the balance between improvisation and composition. Jazz era's can be differentiated by the different balances that existed between improvisation and composition. The Swing Era involved pieces that involved very little improvisation while free jazz and bebop were eras where improvisation played a very large role.
Most of the time when an artist is improvising, he or she does so over a prearranged chord progression. The chord progression dicates which notes the musician may use at any one particular time. When improvising a musician may choose notes that are dissonant or don't sound like they fit completely with the chords. However this is only done occasionally for effect. The freedom to improvise also carries the responsibility of avoiding too many "wrong" notes. Also a good improvised solo will also convey a mood or message that is consistant with the artists interpretation of the song he or she is performing. In a sense each improvised solo is a new composition.
This page serves as an introduction to jazz improvisation. Here are some links to sites where you can learn anything you want to know about jazz improvisation.
A
Jazz Improvisation Primer : Everything you'll ever need to know
Jazz Improvisation: a series of
lectures about improvisation
The Jazz Improvisation Forum