The Aeolian Mode (natural minor scale)

The Aeolian mode starts on the sixth scale degree and like the corresponding chord is minor in nature.  It is probably the most commonly used mode over minor chords and can be found in rock, jazz, blues and metal and is much more commonly called the natural minor scale.
In the Key of C :  C D E F G A B C  the sixth scale degree is A so we have an A Aeolian in this Key.
A Aeolian:   A B C D E F G A  The intervals are therefore  T S T T S T T
If we compare to A Major : A B C# D E F# G# A we can see that the Aeolian mode has a b3, b6 and b7 compared to the corresponding major scale.
An example pattern starting on the A string is shown below.  I’ve started on the A string to keep the fingering pattern fairly uniform.  As with all the modes, you definitely want to experiment a lot with playing this pattern in as many different positions as possible (audio here):
If you need a diagram showing the notes on the fretboard, one is available in the Ionian mode (Major scale) post.
The Aeolian mode works over minor chords or minor chord progressions because of the b3.  The best thing as always is to experiment with voicings and phrasings to find the sort of sounds that you like.
As an example to get you started,  I’ve created a simple pattern using an Am Am Dm Dm Am Am Dm Dm (vi –  ii) progression just to give you an idea of the sound.  (Audio here)
The other important point to note is that unlike the major scale, there are in fact 3 minor scales.
The natural minor scale is discussed above;
The harmonic minor is a natural minor scale with the seventh raised a semitone i.e. A B C D E F G# A
The melodic minor is a natural minor scale with the sixth and seventh raised a semitone i.e. A B C D E F# G# A.  Usually the melodic minor is used with ascending notes and the melodic minor with descending notes.
Good luck, and the most important thing – have fun!

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