Scale

(1) Pentatonic scale comprises 5 notes (1, 2, 3, 5 6) which simple fractional relationship with each other in term of frequency.

1, 1+1/8, 1+1/4, 1+1/2, 1+2/3, 2 or
1, 1+1/8, 1+2/8, 1+4/8, 1+6/9, 2

There is no semitones within the pentatonic scale that make the notes sound good together.  The scale has the following interval pattern:

1 1+1/8 1+1/4 1+1/2 1+2/3 2

The interval between the third and forth notes is more than 1 tone.  Likewise between the fifth and the next octave note.

(2) Major scale is built from the pentatonic scale plus two notes.  Two note was added in the interval higher than 1 tone - sub-dominant and the leading note.  The pentatonic notes have the follow relative frequency to the tonic:

The two additional notes are 1+1/3 (which is 1 semitone above 1+1/4) and 1+7/8 which is 1 tone above 1+2/3 and 1 semitone below 2 (octave).

1, 1+1/8, 1+1/4, 1+1/3, 1+1/2, 1+2/3, 1+7/8. 2

Major scale has the following interval pattern:

WWHWWWH

A drawback of the major scale is the tendency towards definite, complete statements.

(3) Minor scale has less definite full stops. Minor scale takes the same set of note as a major scale but use a different tone as the root.  For example, C manor and A minor uses the same set of note but the root (tonic) is C and A respectively.

There are 3 different forms of minor scale. 

Natural minor scale has the following interval pattern:
WHWWHWW

Comparing to the major scale, 3 notes (including the leading note) move down 1 semitone.
Comparing C Major to C Minor, the 3, 6 and 7 notes (E A B) are flat in C Minor.

C, D, E, F, G, A, B, C
C, D, E♭, F, G, A♭, B♭, C

The relative frequency is:

1, 1+1/8, 1+1/5, 1+1/3, 1+1/2, 1+3/5, 1+3/4, 2

Natural minor lacks the leading note.  So it does not sound good when the melody going up as the seventh and the octave note is too far apart.  It sounds good for melody travelling down in pitch (descending melodic minor).

For the ascending melodic minor, 2 of the notes are re-instate to their major positions.  The leading note is restored and its neigbouring note also to smoothing out the big gap.  The sixth note also bumped up a semitone so that it does not sound too far apart from the leading note too.

Ascending melodic minor scale has the following interval pattern:

WHWWWWH

C, D, E, F, G, A, B, C
C, D, E♭, F, G, A, B, C

The relative frequency of ascending melodic minor are

1, 1+1/8, 1+1/5, 1+1/3, 1+1/2, 1+2/3, 1+7/8, 2

For composing melody, we use the natural minor for the part going down and melodic for the part that going up.  However, the two minor scales pose a problem for creating chord which sounds good for both directions.  The harmonic minor is a compromise by only restoring the leading note from the natural minor.

WHWWHW#H

The 6 and 7 note is now 1.5 notes apart.   This creates a melodic awkward.

Classical musician practice minor scales by playing melodic minor as the scale ascend and natural minor in descend.  Harmonic minor scale is rarely practised.  This is more a harmonic tool and not a melodic one.