Next time you listen to jazz, you should be able to understand in more detail why that jazz player blows those notes.
Listen to 'Just A Closer Walk With Thee' on the 'Demos' Page. My six piece jazz band plays an excerpt of the song. See if you can hear the chord notes and melody notes being played at the same time.
A Look at Jazz Theory
Below you have a simple melody and chord progression. This is the first four
bars of 'When The Saints Go Marching In', a well know jazz tune. So, the first thing to do to play this tune in a
jazz style is to memorize the melody and the chord progression. Learning the melody is just memorizing what you see.
Below you have an exercise to learn the chord progression. The notes spell out each chord as it passes in time to the music. It's important to know the chord progression just as
well as the guitar player or bass player would know them. While the music is going by, you have to be thinking melody and
chords.
And below, you have a combination of the chord notes and the melody notes making
up a finished jazz tune called 'When The Saints go Marching In'. When studying the below four bars, you will find notes of
the chord and notes of the melody in every bar. And if you play this, you can hear the original melody coming through.

I know this is a real simple explanation of how jazz works, but it does cover
the basis of what a jazz musician might be thinking at the time of his improvisations. However, after years of playing
, the musician is not really thinking of the technical part of the music totally. He or she is only riding on the song using
the melody and the chord progression as an outline for the song. Speaking of the 'years of
playing', there are also years of studying attached to any jazz musician. Below is the cycle of fourths going clockwise
and the cycle of fifths going counter clockwise. This is a tool to learn the major chords in music.
Notice, how the
above tunes' chords relate to the below cycle. The tune is in the key of 'G' and the other chords used are right next to
the 'G' chord in the cycle. There are exceptions to this, notice the 'C#' chord. In the cycle, the 'C#' chord is on the opposite
side of the cycle. In general, however, the old jazz tunes follow the cycle in their chord progressions.

This is why this tool is so important to jazz players. Memorizing the cycle
forward and backwards allows the player to become accustom to the 'progression' of the chords in most jazz tunes. Playing the triad of
notes making up the chord would be playing the notes: C, E, and G of the 'C' cord. The 'F' chord is made up of the triad: F,
A, and C. Etc around the cycle. Below is an excercise to play the cycle of fourths. By reading it backwards, you would
be studying the cycle of fifths.
If you wanted to continue to study the cycle, replace the major triad with the minor chord, seventh chord, sixth chord, and even make up your on licks following the cycle left and right. I hope this brings you to a better understanding how jazz works.