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There are many different ways to go about describing jazz improvisation. Improvisation is the most important aspect of jazz. Many People have different beliefs or views on how to go about learning how to improvise. Basically, improvisation is composing on the spot and coming up with melodies off the top of one's head. There are several techniques to do this effectively.

Modes
See main article Musical mode

A mode is musical scale derived from various chords or chords. Musicians can use these "modes" as a pool of available notes. For example, if a musician comes across a C7 chord. The natural mode to play over this chord is a "C mixolydian" scale.

These are various chord derivations: C7 → C mixolydian C-7 → C dorian Cmaj7 → C Ionian (natural major) Cmaj7 → C Lydian mode Csusb9 → C phrygian C- → C Aeolian mode (natural minor) CØ/C7b5 → C Locrian

Targeting
One of the first concepts of jazz improvisation is targeting. The main idea of targeting is basically landing on the important tones of a chord. A chord is built up of the 1,3,5,and 7th degree of the scale. These are the "most important" tones in the chord. There are a number of ways to target a chord tone. The first is by ascending or descending chromatic approach. This means playing the note above or below one of the chord tones. In the key of c, the notes in the chord are C(1),E(3), G(5), and bflat(7). So by playing an eflat at the end of a line then resolving to an E, this would be one basic example of targeting and would be targeting the third of the chord(E).

"Flat 9" Theory
Another approach to improvisation is known as the three to flat nine theory. This is a very Be-Bop approach to improv, just like targeting. This technique is just using the third of the chord (in c it would be E), and playing the flat nine of the chord with it in various ways (in c the flat nine would be D flat). This technique has a very distinct and unique sound to it.

Pentatonics
Pentatonics are a more advanced method to improvisation. John Coltrane used pentatonics extensively. Most scales are made up of seven notes: (in the key of C - the major scale) C D E F G A B). The pentatonic scale is comprised of only five notes of the major scale (C pentatonic scale is C D E G A). Pentatonics are useful in pattern form and that is how they are usually played. One pattern using the pentatonic scale could be 3 6 5 2 3 5 (in C: E A G D E G).

Cells and Lines
Almost every jazz musician has cells or lines, although they don't like to admit it. Lines (also known as licks) are pre-planned ideas the artist plays over and over. Lines can be obtained by listening to Jazz records and transcribing what the professionals play during their solos. Transcribing is putting what you hear in a record onto music paper. Cells are basically the same things as lines, but they are shorter.