Rule on doubling the second rather than the fourth when using ligatures

I was going through the ligatures in 4 parts exercises from Fux's Gradus. On page 133, Aloys says, "Finally, in the sixth measure of the same example the fourth is doubled, although as a rule one doubles the second rather than the fourth":

When Josephus asks for clarification, Aloys presents the answer as a matter of complete harmony (a third, fifth, and octave) in the second half of the measure. However, his correction (below) does not double any note. Instead, he understandably opts to use a C so that it leads up to D, forming a complete harmony in the second half:

I'm still a bit confused about the reasoning for doubling the second over the fourth since Fux did not provide any other detail.

• Why is doubling the second much more preferable than doubling the fourth? I know that when the bass descends by step a fourth will become a fifth and a second will become a third. This means that doubling the second would result in doubled thirds, which could be problematic if the thirds are major. Or does this rule only apply when the resulting thirds are minor?
• Are there any instances/examples where you would need to double the second?

Thanks in advance and sorry for the long question.

-Allen

Edit: After doing more research, I have found this section about 4-part ligatures in Louis Van Beethoven's Studies in Thorough-bass, Counterpoint and the Art of Scientific Composition:

From this I can assume that (correct me if I am wrong):

-Minor thirds on the upbeat (meaning second on the downbeat) can be doubled.

-Major thirds can be doubled only if

• The third is a third tone
• The doubled notes are are in the middle parts
• The third is not a leading tone

What is a third tone in this case? I don't think it is a tritone since that would mean it is a leading tone, which is forbidden.

• Welcome to Music.SE, and congrats one heck of a great first question! Commented Mar 13, 2023 at 0:36
• It appears that by "the third tone" they mean the third iterval up from the bass note harmonically Commented Jul 31, 2023 at 22:05

As regards the edited question re. Beethoven: as far as I can see from the original German language version, the topic is being discussed specifically in the context of dissonances arising from suspensions and their resolution (the examples at the bottom of p. 99 are clear enough even if you don't speak German). As to the specific sentence here

Die kleine Terz kann gut verdoppelt werden; die grosse jedoch [bloß] in der Mitte, als dritter Ton; niemals als siebenter (semitonium modi).

[The minor third can be doubled advantageously, the major however only in the middle as a third tone, never as the seventh (leading tone).]

"dritter Ton", by analogy with the "siebenter [Ton]" here most likely refers to scale degrees, as in the modern German compound "Tonstufe" (which does not appear in Beethoven's text; and which in any case does not appear to have been used much if at all before the 19th century, Grimm's dictionary giving as example only Heinr. Christoph Koch's Handbuch bey dem Studium der Harmonie, published in 1811; other sources give me similar results)

However, if you look slightly further up the page, the heading is "On the fourth species of simple three-part counterpoint"... (the musical example above does, however, show the third above the bass being doubled, FWIW, though it is 3rd species), so I'm not sure how relevant this is to four-part writing.

As regards the original question: doubling the fourth will lead to a doubling of the fifth in the following chord. This is of course not ideal for voice leading as fifths are trickier to deal with (and thus might require judgement in determining which rules are ok to break in the context, as Fux does in his example), while parallel thirds are perfectly ok so long you don't abuse them. In any case, the previous page makes clear that this is just general advice and that "it is not so much a matter of the second or the fourth as it is a matter of the complete harmony [following the resolution of the suspension]".

As to an example of "needing" to double the second, and I'm not sure this is the kind of progression you would expect in Fux, the best I could come up with is a cadential progression with the bass descending to the leading tone, where the fourth would "resolve" to the tritone, in which case you are however best just not putting any fourth (and, yes, doubling the second which becomes the third):