Important facts regarding the Fugue (III, and no more): invertible counterpoint.

The truth is that there are many, many topics that could be discussed about the leak, but in order not to be tiresome, this will be, for the time being at least, one last question that I would like to address.

It is about invertible counterpoint, or double counterpoint.

What is it? Well, that counterpoint in which the upper voice and the lower voice can be interchanged and still sound good. For this, it is necessary to make sure that the intervals (in strong beats) are consonant both in "normal" and "inverted" position.

The most frequent is to see invertible counterpoint at the octave. That is, two lines that can be inverted by raising one an octave and lowering the other an octave.
But you can also see invertible counterpoint at the twelfth (which is a fifth + octave).
More rarely we will see the tenth (third + octave).
And other intervals are not practical, or very rare.

WHY IS THIS FUNDAMENTAL IN THE FUGUE?
If we compose an invertible counter-subject with the subject, we gain two enormous advantages:
1) In following expositions or entries, we can safely change the order of the subject - countersubject voices.
2) This makes the fugue sound coherent and, for the sake of redundancy, makes it sound like a fugue. Because the fugue tries to reuse the initial material as much as possible.

If anyone is interested, I can post my own appreciations of how to compose these counterpoints. The truth is that mastering the invertible to the octave, you have a long way to go.

To illustrate this, I bring here a fantastic analysis of Bach's C minor fugue where there is not only one invertible counter-subject, but two. Thanks to that, during the development he is placing them up, down, in the middle..... 
With this technique you have a great part of the fugue done.
 

Hi Luis,

Yup invertible counterpoint is crucial to fugal writing.

I attach a numerical graph here for reference:

Source: https://lostandfound.fcsh.unl.pt/musical-types/presentation-and-texture

The grey I think it's the marking for the avoidance of parallel octaves and fifths in the inversion. So for invertible counterpoint at the octave, you have to avoid parallel fourths in the original pairing as this would result in parallel fifths in the invertible counterpoint. In an invertible counterpoint at the twelfth, parallel octaves and fifths are avoided in the inversion when your original one is ont in parallel fifths and octaves, and only the parallel sixth in the original one, which is often used, should be avoided since it will turn to a parallel seventh in the inversion. For invertible counterpoint at the tenth, both melodious parallel third and sixth has to be avoided in the original counterpoint as they will be turned into parallel octaves and fifths respectively.

Henry

6 hours ago, Henry Ng Tsz Kiu said:

Hi Luis,

The grey I think it's the marking for the avoidance of parallel octaves and fifths in the inversion. So for invertible counterpoint at the octave, you have to avoid parallel fourths in the original pairing as this would result in parallel fifths in the invertible counterpoint. In an invertible counterpoint at the twelfth, parallel octaves and fifths are avoided in the inversion when your original one is ont in parallel fifths and octaves, and only the parallel sixth in the original one, which is often used, should be avoided since it will turn to a parallel seventh in the inversion. For invertible counterpoint at the tenth, both melodious parallel third and sixth has to be avoided in the original counterpoint as they will be turned into parallel octaves and fifths respectively.

Henry

 

Hi, I think this information is misunderstood. At least in part.
The tables show what each interval becomes when the voices are inverted. That's why each type of invertible counterpoint has to be treated differently.
However, the problem is not that parallel octaves and fifths are formed (which depending on the case also happens). The problem is that by inverting consonant intervals they become dissonant.

Let's see the example of the inversion to the octave. What happens is that the dissonant intervals 2 and 7 are inverted into dissonant 7 and 2. The 3 and 6 become 6 and 3, so there is no problem. The caution must be taken with the fifths, which are perfect consonances, but when inverted, they become dissonant fourths. So we cannot use fifths in strong beats. In other words: we must treat the fifth as a dissonance, which we can use as a prepared note or as a passing note.

As you can see in the example, the fifth in measure 2, and also in measure 3 (not marked) are inverted as fourths. In the second part it is corrected. In this type of counterpoint, fifths and fourths can be used, but as passing notes, in weak parts.

It is possible, as Bach does, to compose three lines that are invertible and play with them by switching them around all the time.

I tried to do it (I don't think I uploaded this here, but it's relevant now):

49 minutes ago, Luis Hernández said:

However, the problem is not that parallel octaves and fifths are formed (which depending on the case also happens). The problem is that by inverting consonant intervals they become dissonant.

I don't think that's entirely misunderstood since preventing parallel octaves and fifths are also important, especially when it's the case with parallel intervals. According to the graph marked grey in the inverted counterpoint at tenth, a third is turned to an octave while a sixth is turned to a fifth. Third and sixth are imperfect consonances while octave and fifth are perfect consonances which they are both consonances. If parallel octaves and fifths aren't taken into account, that marking of grey will be futile.

For the case of a fifth inverted to a fourth it's won't be a big problem if there are more than two voices especially when there is faxbourdon texture. Of course inside a two voice counterpoint a fourth can be treated as a dissonance.

It is also important to note that you can always change some small details of your countersubjects for counterpoint rule, for example using your example of the C minor fugue of WTCI:

In b.9 of the C minor fugue. the top two voices form a 4th at the strong beat in b.9, but is compensated by the bass Eb which forms a faxbourdon. So the rule of a fourth as a dissonance can be remedied in a three or more voices counterpoint.

But in b.16-17,

Even though the blue one (CS1) and green one (CS2) are not inverted, they form an octave here instead of a fourth, since the green CS2 now goes to the bass and has to end on the root, and if it follows the original counterpoint it will instead end on a D, which will form a secondary inversion of a dominant and not fitting the counterpoint rule of preventing unnecessary 6/4 chords. So you can change some details to fit the moment.

P.S I think I've heard your piece before on YC!!!

P.P.S Yup I have heard this before!!!

Henry

1 hour ago, Henry Ng Tsz Kiu said:

According to the graph marked grey in the inverted counterpoint at tenth, a third is turned to an octave while a sixth is turned to a fifth. Third and sixth are imperfect consonances while octave and fifth are perfect consonances which they are both consonances. If parallel octaves and fifths aren't taken into account, that marking of grey will be futile.

 

Henry

 

The case of the inversion to the tenth is particularly complicated. 
Apart from the fact that it has less "musical utility" the result is much more monotonous.
As you pointed out, the problem is that the imperfect consonances (3 and 6) become perfect (8 and 5), so if you write parallel thirds or sixths you will get parallel octaves or fifths in the inversion.
The way to avoid that is that the initial counterpoint can contain thirds and sixths but always arriving to them by contrary movement. Even so, when you invert it, there are too many octaves and fifths, even if they are not parallel.

The case of the counterpoint to the twelfth is more useful, and less complicated. What is done here is to avoid sixths and play only with thirds and tenths (third + octave) and some perfect consonances.


In any case, if the counter-subject of the fugue is written with this in mind, you can then invert it in a multitude of voices.