Improvising Over Giant Steps

The chords in Coltrane's "Giant Steps" alternate between three keys:

B Major
G Major
Eb Major

These three keys are (non-enharmonic) major 3rds apart. Between them, the keys utilise each of the 12 notes, allowing a variety of improvisational methods. These are the chords that are present in this chart and the notes they are built from:

Bmaj7 - B D# F# A#
D7 - D F# A C
Gmaj7 - G B D F#
Bb7 - Bb D F Ab
Ebmaj7 - Eb G Bb D
Am7 - A C E G
Gb7 - Gb Bb Db Fb
Fm7 - F Ab C Eb
Dbm7 - Db Fb Ab Cb

The chords can be divided into small groups, each being either a V-I or a ii-V-I. Coltrane used dominant 7th chord substitutions to allow easy transitions between these keys. (D7 pulls to G, Bb7 pulls to Ebmaj7, Gb7 pulls to B). The corresponding keys for each chord are as follows:

Bmaj7 = I in B major
D7, Gmaj7 = V-I in G major
Bb7, Ebmaj7 = V-I in Eb major
Am7, D7, Gmaj7 = ii-V-I in G major
Bb7, Ebmaj7 = V-I in Eb major
Gb7, Bmaj7 = V-I in B major
Fm7, Bb7, Ebmaj7 = ii-V-I in Eb major
Am7, D7, Gmaj7 = ii-V-I in G major
Dbm7, Gb7, Bmaj7 = ii-V-I in B major
Fm7, Bb7, Ebmaj7 = ii-V-I in Eb major
Dbm7, Gb7, (Bmaj7) = ii-V-I back to the B major at the beginning. 

Arpeggios
Arguably the simplest method of improvisation is to arpeggiate each chord. This means you would just play the chord tones and nothing else. For example if you were soloing over the Ebmaj7 chord, the exclusive tones you would use would be Eb, G, Bb and D. For musicians who are familiar with their chords, this is one possibility, however it may become repetitive if it's the only technique used. 

Ionian and Lydian
Now that each chord has been categorised into a specific key, the major scale for each key would be an obvious choice for improvisation. This means that within each V-I or ii-V-I grouping, the corresponding major scale for the "I" will work. Theoretically, Lydian may also be used because each "I" chord is a major 7. The intervals in a major 7 chord are: 1, 3, 5, 7. Lydian and Ionian are the only two modes of the major scale that include each of these intervals. All the others include at least one "outside" note.

Minor Pentatonic
Using solely major scales for improvisation may not only be difficult to keep up with but also may be difficult to phrase with. There are three unique minor pentatonics for each key that can be used diatonically. These are from the 2nd, 3rd and 6th intervals of each key. (The 6th interval being the relative minor).

B Major - B C# D# E F# G# A#
2nd - C# Minor Pentatonic - C# E# F# G# B
3rd - D# Minor Pentatonic - D# F# G# A# C#
6th - G# Minor Pentatonic - G# B C# D# F#

G Major - G A B C D E F#
2nd - A Minor Pentatonic - A C D E G
3rd - B Minor Pentatonic - B D E F# A
6th - E Minor Pentatonic - E G A B D

Eb Major - Eb F G Ab Bb C D
2nd - F Minor Pentatonic - F Ab Bb C Eb
3rd - G Minor Pentatonic - G Bb C D F
6th - C Minor Pentatonic - C Eb F G Bb

Harmonically, the minor pentatonics from the 2nd degrees of each major scale (C# pentatonic, A pentatonic and F pentatonic) work particularly well with the minor 7th chords (which act as the "ii" chords in each key). This is because the tones that are present in the minor 7th chords all appear in the corresponding minor pentatonic:

Fm7 - F Ab C Eb
F minor pentatonic - F Ab Bb C Eb

Am7 = A C E G
A minor pentatonic - A C D E G

Dbm7 = Db Fb Ab Cb 
Db minor pentatonic = Db Fb Gb Ab Cb 

Dbm7, Gb7, Bmaj7 isn't an enharmonic ii-V-I to B major. If you are using the 2nd degree minor pentatonic from each major scale then the 2nd degree of B major would be C# rather than Db. This means that the resulting minor pentatonic would instead be C# E F# G# B.

Focusing on minor pentatonics allows totally different phrasing to major scale phrasing and with three to choose from there are many shapes that can be utilised and combined.

Problems
There are a number of problems that I have encountered whilst attempting improvisation over "Giant Steps:"

Losing the form - Not only is the sequence long and difficult to remember as a whole, it is also difficult to recover if you lose concentration for a split second. The ambiguity of the chord groupings means that you can't just revert to one scale and remain playing in it throughout. A thorough knowledge of the structure piece and the melodic sequence would be beneficial.

Phrasing - It is almost impossible to improvise coherent melodies over this piece if you don't know what you're doing. This is because the keys alternate at a fast pace and rarely stay in one key for long. However, during the second half of the chart, the time spent in one key is extended to two bars at a time. Although this is still fast paced, it is slightly easier to fit in melodies and appropriate phrasing.

Changes - The key changes are never on the first beat of the bar, which means that it is more difficult to feel the rhythm. The only way to improve your improvisational skills in this piece is to familiarise yourself with it to a point where the chord structure is ingrained.