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It may be considered common knowledge that frequent root changes, i.e., a fast harmonic rhythm, usually cause a piece to be perceived as being more purely "harmonic" and less polyphonic in nature. (See for example, Piston's Counterpoint, p. 63.) Yet, many allegro-tempo instrumental pieces of the late-Baroque period that were written in the fugal style feature extremely melodically active basso continuo lines, implying a fast harmonic rhythm, the thing just mentioned as being "anti-contrapuntal". (Even if one takes into account the fact that in such pieces usually not all bass notes are harmonized.) A brilliant example is the third movement from Bach's fourth Brandenburg concerto, in which the continuo occasionally appears as one of the fugal voices.

How did composers of the era solve this apparent paradox? Were the root changes implied in their continuo lines in a sense illusory? Or did they employ special tricks and devies to make their fugal music sound properly polyphonic despite the fast harmonic rhythm?

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Melodic activity in the bass doesn't necessarily imply frequent root changes. The melody may jump from one chord tone to another, or it may move from one to another by way of passing tones. Both techniques appear in the example you've chosen.

How did composers of the era solve this apparent paradox?

It's only a paradox requiring a solution by the composer if the composer is concerned with how the piece is perceived, and if the composer believes that contrapuntal writing implies a particular harmonic rhythm. Remember that modern theorists are not generally concerned with describing the composer's thought process; for example, Roman numeral harmonic analysis was invented some time after Bach's death. The idea of harmonic rhythm as an analytical tool was developed by Piston himself, roughly two centuries after Bach died. It is rather more likely that Bach set out to write a fugue with a particular affect, in this case a happy one, and chose the melodic material and harmonic rhythm accordingly.

Were the root changes implied in their continuo lines in a sense illusory?

You could look at it that way. I would speak of "passing harmonies." If you look at the example fugue, the first two measures basically have one chord each. The second measure is a vii diminished chord, and the B and G on the fourth eighth note of that measure don't change that.

Or did they employ special tricks and devies to make their fugal music sound properly polyphonic despite the fast harmonic rhythm?

Don't get too hung up on the idea that counterpoint can't have a fast harmonic rhythm. Look at the Amen from the B minor mass. It is definitely imitative counterpoint, and it starts of with a harmonic rhythm equal to the beat. The dichotomy between harmonic and polyphonic is perhaps useful for analysis, but it is hardly absolute, and the rule of thumb given here is probably most useful to people who are unfamiliar with baroque style -- it appears rather early in an undergraduate textbook, after all.

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I was not with them but I can tell you from my experience of playing Euphonium in the brass band. This instrument has usually the c.p. to the melody of the cornets. I started at the age of 10 years and after A few years I was able to play this kind of variations (broken chords, motifs, passing tones, change notes and contrary voice leading without knowing or having heard the term counterpoint.

Certainly the great composer of the Baroque era were able to write the c.p. without minding the theory, they surely also had the the musical turns in their blood and just could let it flow.

We know about Bach and Händel that they were able to improvise and play a fugue on a given theme, like a beginner on piano is able to accompany broken chords (Alberti Bass) and passing tones.

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