Important facts regarding the Fugue

Lately, I've been trying to improve my knowledge (I don't know if my writing as well) about the Fugue.
Apart from observing the intervals and parallels, which I need to be more aware of, I've been asking myself some questions that I think are important and I've been doing some research and thinking about it.
In this thread I want to point out two things. As an example, I put the expositions of the fugue in Gm and the fugue in Cm, both by Bach.

1. Why do many fugues have an ACAPHALOUS beginning? That is, they start with some silence and the notes in weak part.

2. Why in the answer to the subject, which occurs a fifth above, is there a note that Bach transposes a fourth?
In the first (fugue in Gm) the first note is a D, and it is raised a fourth (G), while the rest of the answer is raised a fifth.
In the second (fugue in Cm), the fourth note, a G, of the subject, is raised to C = a fourth. The rest is raised a fifth.

I have my theories and answers, but I would like more opinions. 

From my understanding:

1) This helps with the independence of both voices. For example, by having the subject enter on a weak beat, when the answer enters in it to will have that silence against the first voice which would have some free material against the answer. This helps with the overall independence of the voices. When looking at counterpoint texts, it is recommended to have rests against a C.F. when doing exercise for this reason of having a more independent voice.

2) This has do deal with keys. From my understanding, for some subject if you were to transpose the answer exactly it would take the answer too far aways from the given key. These are called tonal answers.

I haven't analyzed that many fugues so I might be wrong, but from the few I did analyze, my conclusions regarding point 2) were the following: 

"Most fugue subjects start in the tonic chord and end in the tonic chord (at least the first time they are presented). Since the fugue subject ends in tonic and this coincides with the beginning of the answer, the answer also starts in the original tonic harmony (even though the answer is restated a fifth above). This is why when the fugue includes the fifth in the melody during its beginning, that note will go up a fourth instead of a fifth up during the answer." 

For example, if the first two notes of a subject in C major are  C-G, the answer would start with the notes G-C, instead of G-D. This is caused because of the fact that both subject and answer will have a C major harmonic background during its beginning (so, even though the answer is restated in G, it does not start in a G chord, and that is what causes the change in the melody).

  (subject in C)                                      (answer in G)                          

      C-G                                                       G-C

 [C harmony]  .................................. [C harmony again]  

9 hours ago, Guardian25 said:

From my understanding:

1) This helps with the independence of both voices. For example, by having the subject enter on a weak beat, when the answer enters in it to will have that silence against the first voice which would have some free material against the answer. This helps with the overall independence of the voices. When looking at counterpoint texts, it is recommended to have rests against a C.F. when doing exercise for this reason of having a more independent voice.

2) This has do deal with keys. From my understanding, for some subject if you were to transpose the answer exactly it would take the answer too far aways from the given key. These are called tonal answers.

 

1) I don't see that fact, since when a voice begins, there is often no previous silence, precisely because it coincides with the end of the previous subject or with filler free notes.

2) Possibly it has to do with that but why is everything transposed a fifth but that note up a fourth?

12 hours ago, JorgeDavid said:

I haven't analyzed that many fugues so I might be wrong, but from the few I did analyze, my conclusions regarding point 2) were the following: 

"Most fugue subjects start in the tonic chord and end in the tonic chord (at least the first time they are presented). Since the fugue subject ends in tonic and this coincides with the beginning of the answer, the answer also starts in the original tonic harmony (even though the answer is restated a fifth above). This is why when the fugue includes the fifth in the melody during its beginning, that note will go up a fourth instead of a fifth up during the answer." 

For example, if the first two notes of a subject in C major are  C-G, the answer would start with the notes G-C, instead of G-D. This is caused because of the fact that both subject and answer will have a C major harmonic background during its beginning (so, even though the answer is restated in G, it does not start in a G chord, and that is what causes the change in the melody).

  (subject in C)                                      (answer in G)                          

      C-G                                                       G-C

 [C harmony]  .................................. [C harmony again]  

 

 

I am not very sure about this.
In the first example, it would have been very easy for Bach to put that first note of the answer on the A and respect the transposition to the fifth. But he didn't do it.

But in the case of the fugue in C minor, the note that "fails" is not the first note. Nothing would have happened if the A had been transposed to the fifth. We could think that Bach lowered this note to a C to fit better with the counter-subject that the left hand is doing (Eb-Re).... Since Eb-C forms a sixth.
However, when the tenor enters with the subject (last staff), it is the same (identical) as the initial subject, in the same key, and the counter-subject now above and in transposition is also the same as when it appeared the first time, so now there is a dissonance between the note G and the Ab above. Ok, it can be considered a passing note with the B flat above. So why didn't he do the same in measure 3?

Rephrasing everything, why in the transposition to the upper fifth of the subject there is ONE NOTE that, SOMETIMES, is transposed to the fourth?

Hey @Luis Hernández,

That's an interesting question!!

On 7/17/2023 at 2:03 AM, Luis Hernández said:

2. Why in the answer to the subject, which occurs a fifth above, is there a note that Bach transposes a fourth?
In the first (fugue in Gm) the first note is a D, and it is raised a fourth (G), while the rest of the answer is raised a fifth.
In the second (fugue in Cm), the fourth note, a G, of the subject, is raised to C = a fourth. The rest is raised a fifth.

Actually in music history, a tonal answer with the fourth and fifth interchanged in the answer is a default rather than a strict real answer in Bach's time. (I remember it's from the book "Theories of fugue from the age of Josquin to the age of Bach" by Paul Walker, but I'm not sure whether it really does come from there). It may not be a theoretical reason, but rather a historical reason Bach choose to write with tonal answer.

Regarding the third staff of the fugue in C minor, I think you're mixing the subject and answer. It's the initial appearance of the subject for the tenor voice, and I don't think Bach will change the notes for harmonic reason in expense of the integrity of the subject. Actually in all the subsequent G minor answers Bach retains the tonal change of that particular interval, that is changing the 4th to a 5th. I think it's more the historical reason for him to do so.

On 7/17/2023 at 2:03 AM, Luis Hernández said:

1. Why do many fugues have an ACAPHALOUS beginning? That is, they start with some silence and the notes in weak part.

Maybe there's no reason for it except Bach wants to. But I think with a weak beat in the later entries it makes the music more moving forward instead of having the metrical and rhythmic accent happens at the same beat. It fits more to the Baroque fashion with its driving forward. The fugues with the subject in the first beat, like the one of C# minor, D# minor, Bb minor in WTC I are more serious in their tone and more "stile antico" in general.

Henry

@Guardian25 

@JorgeDavid

@Henry Ng Tsz Kiu

I give you my thoughts after studying some things and checking them out.
Of course there is no single answer but one thing is quite clear to me, and that is that Bach did not do anything without a reason. Or to put it another way: if he did something out of what we expect, it was for a reason.

Why many leaks have an acephalous start has to do with what @Henry Ng Tsz Kiu was saying. But being more specific, we have to take into account that the resolutions (or cadences) must end in strong beats. If you start a fugue with a "free space", you have a simple way to build that cadence, since it will provide just the right space, in strong time, for the corresponding tonic. Of course there are fugues with other types of beginnings, but think about it, it is more difficult to do all this when the beginning is anacrusic or thetic.

 

@Guardian25 

@JorgeDavid

@Henry Ng Tsz Kiu

Regarding the other issue, we can find in the books what was previously said: "Actually in music history, a tonal answer with the fourth and fifth interchanged in the answer is a default rather than a strict real answer in Bach's time. (I remember it's from the book "Theories of fugue from the age of Josquin to the age of Bach" by Paul Walker, but I'm not sure whether it really does come from there). It may not be a theoretical reason, but rather a historical reason Bach choose to write with tonal answer."

Specifically, the degree that is affected by this change is the 5th of the key. Thus, in the fugue in Gm, the note that changes is D (the 5th) and in the fugue in Cm it is G (the 5th).
In fact, what we can also read in the books is that "when the 5th appears noticeably in the subject, especially at the beginning, a 4th (just that note) will be transposed"......
But why or what for?

I think there is a tonal or, rather, a tonal transition reason.
As the beginning of the fugue begins in Gm, the answer "must be in the key of the dominant". If the first note of the fugue (D) had risen to a fifth (A), the dominant harmony would have already been almost defined by the fifth and the fundamental. However, Bach did not want the modulation to be so abrupt, in fact in measure 3 the F natural still appears and not the sharp. So it is prolonging the dominant. In this way measure 4 can be interpreted as having modulated to Dm or is still in Gm. When the third entry occurs in measure 5, it returns to Gm by means of a clear dominant.
This is why it is the "rule": when the fifth a appears noticeably in the subject, a tonal response is appropriate, and the fifth is transposed a 4th.

The same applies to the leakage in Cm. The 5th (G) appears early in the subject.
When it is transposed in measure 3, if it had been transposed as a 5th, it would have been a D, and the key of G would have been Definite.
But transposing it a fourth (C), the key is still Cm, according to what sounds underneath. Yes, even though there is an F#, but this is a passing and a tonicization, insufficient to establish the new tonality. This tonality is not established until the end of measure 4 and the beginning of measure 5, in that cadence. It has modulated to Gm, but in these two transitional measures 5 and 6, the dominant G7 appears for the new entry in measure 7.
In this third entry, the note remains as at the beginning, because the interest now is to reinforce the key of Cm again, so it makes more sense to leave the note as G than as F.
 

We have another example in the Fugue in C# major from the i book of the WTC (Fugue nº 3):

Here's a very basic crash-course I made which discusses the structure of fugal answers from a slightly different point of view.

7 hours ago, Fermata said:

Here's a very basic crash-course I made which discusses the structure of fugal answers from a slightly different point of view.

 

PDF

• Fugal_answers_

Thanks! 

16 hours ago, Fermata said:

Here's a very basic crash-course I made which discusses the structure of fugal answers from a slightly different point of view.

 

PDF

• Fugal_answers_

Thank you so much, really useful information and well arranged!

As a curiosity, I was following the score for the first solo violin sonata BWV 1001, and the second movement, the fugue, goes like this: 

I immediately realized that, in here, the answer is not in the dominant key, but in the subdominant (from Gm to Cm). I believe it might have something to do with what we have been talking since, in this case, transposing the subject to the dominant while modifying the D (dominant note of G) for a G (tonic of G) at the beginning of the answer would have resulted in an answer with the following notes: G-G-G-G-G-F-G-E-F. Maybe Bach though this answer would have been extremely dull and decided to answer in the subdominant instead. 

Thank you all for all the useful information!

47 minutes ago, JorgeDavid said:

I immediately realized that, in here, the answer is not in the dominant key, but in the subdominant (from Gm to Cm). I believe it might have something to do with what we have been talking since, in this case, transposing the subject to the dominant while modifying the D (dominant note of G) for a G (tonic of G) at the beginning of the answer would have resulted in an answer with the following notes: G-G-G-G-G-F-G-E-F. Maybe Bach though this answer would have been extremely dull and decided to answer in the subdominant instead. 

Yeah I also remember the fugue in that famous Bach's Toccata and Fantasia BWV 565, when Bach has the answer in subdominant G minor rather than dominant A minor:

And the default of having the dominant as the answer is getting weaker after Bach too.

Or as @Fermata pointed out, only the first portion of the answer is in the dominant but the latter half is instead transposing back to the tonic.