A harmonic sequence is a short progression of chords which is then repeated, using different chords which are the same distance apart.
For example, in the progression of chords I-IV-V-I the intervals between the chord roots are: a 4th, a 2nd and a 5th.
C-F = 4th, F-G = 2nd, G-C = 5th.
If we start on a chord of D minor (ii) and keep the intervals the same, we have written a harmonic sequence:
Harmonic sequences help to give a piece of music a feeling of unity, without being overly repetitive. Notice that in the above harmonic sequence, some of the major chords (I and V) become minor (ii and vi), which means the harmony will sound quite different (this will depend on which chords are used, however).
Harmonic sequences based on the progression of 5ths (which looks like a sequence of rising 4ths) are particularly common, especially in Baroque music (but they are used in all eras and all types of music).
Bach’s Prelude in D minor (Well-Tempered Klavier Book 1) contains a sequence of 4ths (progression of 5ths), followed immediately by a sequence of 2nds. The key modulates to F major when this section begins.
First we see the progression vi-ii-V-I-IV (rising 4ths, or falling 5ths). This is followed by rising steps of ii-iii-IV-V.
Harmonic sequences can also be used as a way to modulate. We keep the same chord progression, but apply the Roman numerals to a new key. We could apply the original chord progression to A minor, for example: