Harmonic rhythm is the rate at which the underlying harmonic chords change in a piece of music. If you’re listening to a piece and the chords change at the rate of once a measure (from the tonic [I] to the dominant [V], or whatever), that piece has a faster harmonic rhythm than another in which the underlying chords change at the rate of once every four measures.
In other words, the faster the chord changes occur, the faster the harmonic rhythm.
If you’re still with me, let me dump one more definition on you. In Chapter 11 of his book Harmony, the American composer Walter Piston writes that, “Harmonic rhythm looks at the root changes (chord changes) in isolation from the notated rhythm. There can be multiple root changes within a measure, and roots that hold across measures (syncopation). Changes can be very irregular to add interest.”

The Harmonic Rhythm of Gluck’s Overture to Iphigénie en Aulide
You might expect that a piece with fast harmonic rhythm would be more exciting than a piece with slow harmonic rhythm. But the interesting thing is, that isn’t necessarily so. You will discover this for yourself when you listen to the remarkable overture that Christoph Willibald Gluck wrote in 1774 for his opera Iphigénie en Aulide. This overture has a very slow overall harmonic rhythm, yet it is an electrifying piece of music.
As you’ll hear, Gluck has the guts to have the orchestra just sit on the tonic, the dominant, the subdominant, and elsewhere, for extended periods of time. And then there are those repeated, highly energetic returns to the tonic that are so remarkable. It’s an extraordinary effect – the composer lets the music almost scream, “Here comes the tonic again!” And do you know what? The tonic can be goddam exciting. You’ll hear it happen in the video below at timer marks 1:30, 2:38, 4:00, and 5:00, and then again later on.
There’s also the sheer length of this overture, at over 10 minutes.  I find it nearly hypnotic. That could explain why in 1847, Richard Wagner revised the opera and gave it a production in Dresden. Let’s enjoy a video of the overture in Wagner’s orchestration, conducted by Riccardo Muti.
If this is the first time you’ve heard this piece of music, you are in for a treat.