The double fugue is apparently a type of fugue with two subjects. What I don't understand is what differentiates a second subject from a countersubject, and why a so-called double fugue cannot be called a fugue with several countersubjects(assuming there is another countersubject), why the second subject is not simply a countersubject. What differentiates a second subject and a countersubject, and what differentiates a double fugue from a fugue with several countersubjects?
A counter-subject is typically a continuation from the subject, in the same voice. So in a typical simple fugue, a first voice enters with the subject; then a second voice enters with the subject, and the first voice continues in counterpoint. Usually, this continuation (a) involves a distinctive motif and (b) recurs following the subject in other voices; these together establish it as the countersubject.
A second subject enters independently, in a new voice. So in a typical double fugue: voice 1 enters with the first subject; then voice 2 enters with the second subject; then voice 3 enters with the first subject (and voice 1 usually proceeds to a first countersubject); then voice 4 enters with the second subject (and voice 2 possibly proceeds to a second countersubject).
Here’s an example double fugue, the Kyrie Eleison from Mozart’s Requiem. The first subject is highlighted in red (“Kyrie eleison”, in the bass and later the soprano); its continuation as a (debatable) counter-subject is in orange. The second subject is in green (in the alto and then tenor).
A second subject is usually quite clear and unambiguous. The trickier and often debatable point is identifying a countersubject. Simple fugues usually have clear countersubjects, but double fugues often don’t have any (since the main subjects often supply enough density of counterpoint). In this Mozart example, there’s certainly no second countersubject — the alto states the second subject, then stops and rests while the tenor states it in turn. And the first countersubject is debatable: After stating the subject, the basses do continue while the sopranos enter with the same subject. But each voice that has the first subject has a different continuation after the first few bars — there’s no distinctive independent motif that consistently follows the first subject. So personally, I would analyse this fugue as not having a clear countersubject. But I think it can also reasonably be analysed either as (a) the basses’ line from the end of bar 4 onwards is a countersubject (it recurs sometimes, though not consistently); or (b) the motif in bars 3–4 of the basses is the countersubject, not part of the main subject (since it sometimes occurs independently of the opening “Kyrie” motif).
A normal fugue begins with the subject being stated (in the tonic) by one part (say A), which goes on to take the counter-subject when another part (say B) begins the subject (in the dominant). If there are two subjects they are stated together at the start of the fugue. They may well work well as counter-subjects for each other, but there is no way in this situation to assert which one is the subject.
To put it briefly: A second subject receives an independent exposition of its own of some sort; whereas a mere countersubject only appears as a counterpoint to the main subject, never being exposed or stated on its own and always showing dependecy to the first subject.
There are different ways how to execute the exposition of a second (third etc.) subject:
- There can be a full-fledged exposition in several voices following the exposition of the first subject (or even followning the first development of the first subject) [example: the second subject of the triple c# minor fugue from the Well Tempered Clavier I].
- The second subject can be exposed simultaneously, or rather almost-simultaneously, with the first subject, in a common exposition [example: Kyrie eleison from Mozart's requiem].
- The/a second(ary) subject can be introduced later in the fugue as an innovation/part of development [example, of a sort: the third subject of the c# minor fugue from WTC I, slowly emerging from the accompaniment of the second subject, and gradually gaining prominence and independence].
Of course, border cases may occur - it is all matter of the degree of independency of the countersubject/secondary subject. The more independent it is, the more it deserves to be called a second subject and not a mere countersubject. It definitely is a mere countersubject in case it:
- only enters together with the main subject;
- is never stated independently of it;
- is never independently developed;
- always combines in the same way with the subject;
- is not significantly more prominent than the main subject.
But should the melody which started as a countersubject (or something else, such as an episode motif) sometimes be stated without the main subject, or combine with it in different ways (thus showing independecy), or be developed independently of the main subject, or be so prominent as to effectively overshadow the main subject -- then it approaches the status of a second subject.
By "prominence" I don't only mean a nature contrasting to the main subject - most countersubject have such a contrasting nature; rather, I mean that the countersubject must distinguish itself both from the main subject and from the rest of the material of the fugue in order to be able to claim the status of a true second subject. For example, the countersubject of the f minor fugue from WTC 1 contrasts strongly with the main subject and is generally extremely prominent in the fugue - so much, in fact, that it provides all the rest of the material of the fugue. Consequently, there is nothing against which it might stand out, and so it is not a second subject: it is simply serves as the overall "background" material of the fugue which provides a contrasting scene for the only true subject of the fugue.
Another example of a typical, completely dependent countersubject is the one from Fugue g minor from WTC I - which is remarkable by being obtained from the subject by retrograding it (which also underlines its dependent nature).
On the other hand, in F# major fugue in WTC I, the disitnctive semiquaver motif introduced in the top voice in measure 7 and quickly dominating the fugue can indeed be argued to be a second subject of a sort (in addition, the fugue also has a regular, completely subservient countersubject, which is only later supplaned by this figure in its function of accompaniment of the first subject):
- it enters quite independently, in a contrasting manner, after the exposition of the main subject in all three voices has been completed;
- it is distinctive and prominent;
- it combines with the subject in various ways;
- it is developed/reworked in various ways;
- it does not sereve merely to accompany the subject.