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I am reading Alfred Mann's translation of "Gradus ad Parnassum" and found a footnote that says that Fux considers a fourth that comes from harmonic division the of the octave (fifth-fourth) a consonance and a fourth that comes from arithmetic division of the octave (fourth-fifth) a dissonance.

What about fourths in multi-part writing that do not have either of the notes of the fourth in the bass? For example, third-fourth? What does Fux or other theorists have to say about this case?

Also, is this the reason why in classical harmony triads in root position and first inversion are used quite freely while triads in second inversion are a special case?

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A fourth above the lowest note is a dissonance in species counterpoint. In Gradus, the intervals are always counted above the bass. The interval C-F is a fourth as is E-A; both are treated as needing resolution (which is the definition of dissonance: the composer treated the interval as needing resolution). (I couldn't get brackets or quotes into that sentence; perhaps translating to German and back using Google....)

Haydon's book "The Evolution of the Six-Four Chord" discusses the matter a bit. Still, I have not figured out if the fourth against the bass is considered dissonant because it may imply a 6-4 chord (sort of a harmonic viewpoint) or the 6-4 is dissonant because it contains a fourth against the bass.

To a great extent, the fourth against the bass and the 6-4 chords are classified as dissonances because composers wrote music treating them as such. Thus E-G-C is consonant as the fourth is not directly sounded against the bass, but G-C-E is dissonant. (In this style.) Similarly for E-A-C and C-E-A.

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    I’ve always held the view the 4th was considered dissonant because it was considered “unstable” likely stemming from contextual cultural emphasis on its use relegated to neighboring tones and suspensions.
    – jjmusicnotes
    Commented Dec 9, 2020 at 11:40
  • then how about G-E-C, is this config dissonant?
    – Someone else
    Commented Aug 1, 2023 at 7:26
  • It's often treated, in CPP harmony, as a dissonance. Especially with a doubled G. The 6-4 chord needs special care. In a cadential setting, it cab treated as a double neighbor to a chord with the bass as root:. G-C-E->-B-D. other uses are possible.
    – ttw
    Commented Aug 11, 2023 at 20:51

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