We can think of minor keys as relative minors of major keys. To find a relative minor of a major key, go three note-names down (or a minor third, if you like.) Thus, the relative minor of C major is A minor.

Here's a list.

0                       1                      2                      3                      4                     5                      6                      7

C+ /  A-	G+ / E-	D+ / B-	A+ / F#-	E+ / C#-	B+ / G#-	F#+ / D#-	C#+ / A#-
	F+ / D-	Bb+ / G-	Eb+ / C-	Ab+ / F-	Db+ / Bb-	Gb+ / Eb-	Cb+ / Ab-

There are 3 types of minor scales: natural, melodic and harmonic.

To create a natural minor scale, write the key signature followed by the scale from tonic to tonic (i.e., in D- start and end on D).
Melodic minor: Make a natural minor scale, then raise the 6th and 7th by 1 semitone with an accidental going up, and lower them back as you go down.
Harmonic minor: Make a natural minor scale and raise the 7th by 1 semitone both going up and down.
Examples: 

What is an interval? Simply put, it is a distance between two notes. There are 2 kinds: harmonic (where the 2 notes are played together) and melodic, played separately.

The distance between the A and C is still three notes. 

Let's look at this interval: 

To find the number of an interval, simply count the note names. E, F, G, A, B, C. Six.
(Remember to count both the beginning and end notes as part of the total.) We must then be dealing with a sixth. But what kind of sixth?

There are several kinds of intervals: major, minor, perfect, augmented, and diminished. We'll get to this later.

To find the quality of the interval depends on the key we are in. You might get a key signature, as above, or you might have accidentals. There may be nothing at all... In any case consider the bottom note as the tonic (or first note) of a theoretical key. Here (and here's an example with no key signature) we have E as the tonic and consider E + as the key.
Is the top note C  sharp, natural, or flat in E major? It's sharp (F, C, G, D are the sharps) so in the absence of a sharp here, this must be a minor sixth. In the case of the example with a key signature (above) the C is sharpened by the key signature. 
 

If we wanted to have a major sixth, and had no key signature, we could do this, alone: 

There are also augmented intervals. Let's make an augmented sixth: in this case, because the key signature includes C sharp, we have to add another sharp (C double sharp, represented by an x).

major 6th       augmented 6th

Although the augmented sixth sounds like a 7th, it isn't, because the notes are still written as E and C. 

A diminished 6th is a semitone down from a minor sixth: here, we only have to use one flat (because the note itself is C natural) but in some cases would have to use 2.

Here's a fifth. Is it a major fifth? No: unisons, fourths, fifths, and octaves are called 'perfect'. If they have been enlarged by either a sharp or flat, they become 'augmented'; and if they are made smaller by adjectives, they also become smaller.

Just for fun, the sequence of accidentals goes like this:

doubleflat    flat     natural   sharp   doublesharp

How do you find the key or keys that contain a given interval?
- Write down the major keys that contain each note
- Cross out the ones that don't have both notes
- Check the relative minor (down 3 letter names) and the tonic minor (i.e., A minor for A major). If they contain both notes, add them to the list.
- Done!
 

INVERSIONS
We can also invert intervals. Think of this as Jenga (tm) except with notation. You can literally "take one from the bottom, and put it on the top." Our friend the sixth:

The figure at right shows how it has been inverted: the E from the bottom has shifted so that it is now in the first E spot above the old top note. You will note that it has formed a third. 

When you invert intervals,
- the numbers of each intervals (i.e., 6 and 3) equal 9 when added.
- major intervals become minor and vice versa. Augmented intervals become diminished intervals, and perfect intervals remain perfect.

(see example above.)
 

Download the intervals worksheet » (Acrobat Reader required)