Analysis
Bach's Prelude in C major, and harmony without a melody
Four bars with no melody at all, only broken chords, and for that reason the harmony lies exposed as nowhere else. The bass steps down, the inversions make the path, and every chord is heard note by note.
The first prelude of The Well-Tempered Clavier has no melody. Each bar is a single chord, broken into a semiquaver arpeggio that repeats twice, and what is left when the melody is taken away is the harmony, bare. That is why it is the most analysed piece in the repertoire: the score is already the analysis, and all that remains is to name what you hear.
One chord per bar
The harmonic rhythm is as regular as it gets: one chord per bar, without exception, through all thirty-five bars of the piece. The rhythmic figure does not change either. Everything that happens, happens in the choice of chords and in the path of the bass between them.
In the first four bars: the tonic; the second degree with a seventh, but with C in the bass; the dominant seventh, with B in the bass; the tonic. Read by degree, I, ii7, V7, I: the commonest phrase in tonal harmony, the subdominant represented by the second degree, the dominant, the rest.
The bass steps down
What the inversions do here is the textbook example. If the four chords were in root position, the bass would go from C to D, D to G, G to C: two leaps. Bach holds the C under the D minor, the third inversion of the seventh chord, and puts B under the G seventh, the first inversion. The bass goes C, C, B, C: one semitone down and one up.
That is voice leading reduced to the smallest movement, and it is the voice leading, not the chords, that gives the opening the character of something breathing rather than walking.
The upper voices barely move
Inside each arpeggio the five notes are the voices of a five-part chord, and from one bar to the next each moves as little as possible. From C major to D minor seventh, the C and the E stay; the G goes to A, the upper C to D, the upper E to F. From D minor to G seventh, the D and the F stay; the A goes to G, the bass C to B.
The four chords blocked, as Bach would have written them in a chorale: this is what the arpeggio hides and what the ear, at the right tempo, reconstructs. No voice leaps further than a third.
The dominant that resolves inwards
The third bar is the dominant, and it contains the tritone, B and F. In the fourth bar the B rises to C and the F falls to E, exactly as the theory says it should. Play the third bar arpeggiated and then the fourth: the arpeggio makes every note of the resolution sound separately, and the pull of B towards C becomes audible as it rarely is in a blocked chord.
What the piece teaches
That harmony is a set of lines, and that an inverted chord is a decision about the bass. That the phrase I, ii, V, I holds up a whole prelude if the path between the chords is good. And that analysis does not need a melody: in this piece, it is the text.