Leonard Bernstein describes the history of the 12 tone scale in 6 minutes.
The main point of the following discussion is that classical era composers like Haydn and Mozart used "closer" harmonies and key modulations than did Mozart. This made their music sound more "pleasant" than Beethoven's. Far more than his predecessors, Beethoven wanted to express "disturbing" emotions like surprise, melancholy, apprehension, anxiety, sorrow, grief, depression and the like.
Note: The discussion below is described in the key of C because it is the easiest to understand. The C scale has no sharps or flats (on a piano, which is in the key of C keyboard, the C major scale is all white keys and no black keys). But the theory described below applies to all keys as harmonies are relevant to the particular key you are in.
The "perfect" harmonies, the 4th and the 5th
In the first three centuries of western classical music there were no harmonies. In medieval chants (plainchants), c. 500, all religious chants were sung in unison (mostly by monks). A century or so later, boys were added, and they sung exactly one octave above the older monks. But singing in octaves (e.g. singing a lower C and a C an octave higher) is not harmony.
Sometime around the year 800, the first harmony, the, 5th, was added. That means that if someone sang a C, someone else would sing the 5th note up the scale from C, the G, at the same time. Next the 4th was added (singing the C and F at the same time). These were the only two harmonies used until c. 1400 when the 3rd began being used. From there it is about 200 more years until the triad (playing the 1st, 3rd, and 5th together, or what we commonly call a chord) came into use.
"Close" and "distant" harmonies
The 4th and 5th were used first because they are "close" harmonies to the tonic (in this case, the C). In the case of the 4th (F), their is only one flat (the Bb in the F scale), and in the 5th (G) there is only one sharp (the F# in the G scale). A little further away are the keys D (2 sharps) and Eb (2 flats). This pattern progress up the scale in 5ths for sharps (C,G,D,A,E,B,F#,C#) and in 4ths for flats (C,F,Bb,Eb,Ab,Db,Gb,Cb). As you can see, the keys of C# and Cb are the furthest from C. So striking the C and C#, or B and Cb (which is B natural, the note one key down from C) produces the harshest harmony. The "closer" the harmonies, the more consonance a harmony has, the "further" the harmonies, the more dissonance a harmony has.
The upshot of all this is that the closer a harmony is to the tonic, the more "natural" (or pleasant, or non-disturbing) it sounds to the ear, while the further away a harmony is in the cycle of keys, the more it sounds in some way "unpleasant" or disturbing. An extreme example is the 4-sharp (or 5-flat) harmony. Take F#, the note between the closest harmonies to the tonic C. If you strike a C and an F, or G, together on a piano, the sound is very pleasant. But if you strike a C and an F# together, the sound is very grating.
The relations between major and minor scales and harmonies
While there are many different types of scales, by far the two most used ones are major and minor scales. So far the discussion has been about the harmonies of major scales. Minor scales work in relation to each other just as major scales do, but the relationship between major and minor scales is different.
The common western scales have eight notes (e.g. the C scale is C-D-E-F-G-A-B-C. Each successive note on a keyboard is a 1/2 tone above the preceding one (there are 12 1/2 tones in the western chromatic scale). As can be seen on the keyboard pictured above, there are two half steps in the C major scale, one between the 3rd and 4th notes (the E and the F), and one between the 7th and 8th notes (the B and the final C).
Minor scales work the same except that their half notes are between the 2nd and the 3rd and 6th and 7th notes instead of between the 3rd and 4th and 7th and 8th). So a C minor scale is C-D-Eb-F-G-A-Bb-C. The C major triad/chord is C-E-G, the C minor chord is C-Eb-G.
While major scales and chords have a "bright" sound, minor scales and chords always have a "darker." Theev most famous example in western music of this "darkness" is the first eight notes of Beethoven's 5th symphony. The first 4 notes, G-G-G-Eb are in the symphonies main key of C minor. The second four notes, F-F-F-D, are in the G minor scale (the "perfect" fifth of C minor).
Now, the major point is that every major scale (and chord) has a "relative" minor one. That is, they have the same number of sharps and flats. The relative minor scale is three half tones below the major scale. So, A minor is the relative minor to C major (neither has sharps or flats, they are both played on only the white piano keys).
Because they both use the same notes, they are very close to each harmonically (while C major and A major are pretty distant (C has no sharps or flats, A has 4 sharps). So while a change from C major to A major sounds pretty strange to the ears, a change from C major to A minor sounds more "normal" (despite the "darkness" of the A minor chord).
So classical era composers were just about as likely to move from C major to A minor as they were from C major to F major, or C major to G major both within a scale or chord pattern, or when modulating (moving) from one key to another). This not only happens within a composition movement, it is common to do this between movements.
For instance, Beethoven's Symphony No 5 in C minor's first movement is in C minor, the second movement is in Ab major (C minor is the relative minor to Eb major, and Ab major is the 4th of Eb major), so it seems pretty "normal" to the ears. However, in the final movement Beethoven goes to C major, which is a fairly further harmony (from two flats to no flats) but the move from the"darker" C minor to the "brighter" C major allows Beethoven to give you a more positive resolution to the "doom and gloom" of the first and third movements. Typical Beethoven!
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