Invention No 4 in D minor, BWV 775
Also called Praeambulum 2 (see note 1 at end), it is very easy to follow the motif of this one. It starts with the 6 notes of the scale, and instead of playing the 7th grade following the line, it jumps the VII down and comes down the original scale again.
I could not find any diminution in this piece. Some examples where Bach uses the other devices are the following:
Imitation
There is an exact imitation as a canon one octave lower, starting in bar 3; then another one octave higher from bar 5. There is a further development of the motiv after this third entry (stave 1, bars 7 and following), which is then imitated starting in bar 11, stave 2, transposed an octave and a fifth lower (I guess one could say “to the lower thirteenth”).
There are other figures imitated though. The higher voice in bars 11-14 is imitated by the lower voice in bars 22-25, to the lower eleventh. In bars 47-48, both voices imitate their rhythmical figures from bars 16-17, while the lower voice (47 & first beat of 48) is a transposition of the higher in bars 16 & 17 (first beat).
Inversion
The second half of the main motif looks already like an inversion, which makes difficult to find one. One can arguably start in bar 22, stave 1. Another, shorter one, can be found close to the end, in bar 49. None of them is an exact inversion, however the first is closer to being an exact one.
Augmentation
In bars 7, 8 and first beat of 9, there is an augmentation. At first sight, one could think that the second note (E) is missing; however, it could also be that it was hidden in the top stave (on that beat, there’s an E semiquaver). To support that idea, Bach himself writes in the second note a D that is one octave higher (something like “missing something? look up!”) This, at least, looks like the type of hints & hidden features I would use.
(1) The book mentioned is Clavier-Büchlein for Wilhelm Friedemann Bach, Bärenreiter, Kassel (Germany), 1962, pages 50-51.
The Art of the Fugue – VIII Contrapunctus 7 a 4 per Augment et Diminut
I could find 28 different entries of the main theme, whether as dux, comes or other, and several times they are augmented. My favourite version so far can be heard here.
Imitation
The first dux appears in voice 3, then it appears again in the same voice, in bar 6 (just changing the first note, E instead of D) – provide this is not a false entry! It won’t show up again until bar 28, voice 4, in the same octave as in bar 5; and then immediately in bar 29, 3rd beat, it appears again in voice 2. Its last appearance will start in the last beat of bar 54, changing again the original first D by an E.
The augmented dux appears in bar 36, voice 3; bar 38, voice 1 (some of the notes, including the first note, are not augmented, but most are); and bar 50, voice 1.
Needless to say, there are quite a few examples of imitation based in other tonalities rather than the original D minor. The ones I could find were:
- F major: bar 23, voice 2 (1st beat) and voice 3 (2nd beat, augmented)
- G minor: bar 13, voice 1; bar 45, voice 2; bar 55 (all three with first note changed into A); and bar 47, voice 4
- A minor: bar 7, voice 2
- Bb major: bar 17, voice 3; repeated in bar 20, voice 3 (with first note changed into C)
- C minor: bar 16, voice 4 – it is not an exact imitation though, e. g. there are some transitional notes also between the first and the second notes.
Inversion
The first inversion can be found right in the second bar, augmented, and apparently based on an A minor. Being the A minor chord the dominant one, this makes me think that we actually have the comes here (see note 2 below). It appears again in bar 9, voice 3 (we could consider it the original dux speed, but we will consider this a diminution of the comes); bar 35, voice 2 (augmented again); bar 41, voice 1 (with a shorter 1st note); bar 43, voice 2 (the first note is the last of the previous voice 2 comes). The inversions in other tonalities I could find are:
- C major: bar 24 (3rd beat, diminution)
- D minor (inverted dux), always with a diminished II: bar 46, voice 3; and bar 14 (3rd beat), voice 2 – in this one, the first three notes are rhythmically those of the comes, and then does a diminution which lasts another three notes; it augments three notes again, then goes back to diminution and stays there for the rest of the theme
- E minor: bar 3, voice 2 (diminution); bar 5, voice 4 (augmented). In both cases, there’s a diminished II. Both start on D instead of E
- F major: bar 20 (3rd beat), voice 4 (with minor VII – diminution)
Combinations
There are relatively few bars in which neither the dux nor the comes are sounding. At bar 38, we even have the dux in the 1st voice, an augmented comes in the 2nd voice, an augmented dux in the 3rd voice, all accompanied by the development in the 4th voice. A similar situation occurs again in bars 55-56.
Bach tries many different combinations of dux and comes, transposed or not, augmented or not. Some examples can be found as follows:
- dux with comes: bar 2
- comes with itself in diminution: bars 3-4
- augmented comes with dux and transposed dux: bar 7
- augmented comes with comes in diminution: bars 9-10
- in bars 13-15, I reckon we have the dux transposed to the dominant and the comes transposed to the tonic, if that is possible!
What is the effect of inverting the main theme? The dux (bars 1, 2 and first note of 3) sounds fairly finished: it hits its peak in the very second note, comes back to the tonic, attempts a second peak and again back to the tonic. End of the game. However, the comes starts on a fairly high note and reaches its peak in the middle – which happens to be a higher peak than that of the dux. With this technique, Bach increases the dramatism of the piece.
The augmented lines can sound a bit heavy here and there, but they also give a solid ground for the rest of the music to happen on top of it, whether it is the lower voices or in the higher ones. This may be especially true from bar 35 on, when both the comes and then the dux are augmented after a fairly lighter section of the development (quick 1st and 3rd voices, no 2nd voice, 4th voice quite light as well), announcing a new entrance of the dux in bar 38.
I can actually imagine Wagner listening to this for inspiration before he wrote some of his pages.
The passages using major scales (see above) give a grandioseness of a different type, of an almost fanfare taste, ceremonious, celebrating.
I couldn’t find any new entry in the conclusion (bars 58 to 61) or any occasion in which the dux or the comes would be played backwards; neither could I find a point in which the notes of dux or comes appeared in alternation with other notes.
If we consider the exposition is the presentation by each voice of the theme (dux, comes, or altered in whatever other form), then it lasts until bar 13 (the forth voice). It is quite unusual to introduce the themes tonally as tonic-dominant-submediant-tonic. According to L. Zenkl (see note 3 below) fugues tend to have a structure T-D-T-D (T= tonic, D= dominant). However, the same author admits that the fugue structure is quite open to changes.
Notes
(2) According to Zenkl (see note 3), the comes appears for the first time in the dominant. I will follow that idea, and consider the comes as the augmented inversion of the dux. Therefore, when I refer to the augmented comes, it will in fact be twice augmented in relation to the dux, and when I refer to a diminution of the comes, the duration of the notes will be that of the original dux.
(3) Luděk Zenkl, ABC hudebních forem (ABC of musical forms), Bärenreiter, Praha, 1984.
The Art of the Fugue – VI Contrapunctus 10 a 4. alla Decima
This is a double fugue. There are two subjects (or themes, duxes). The first one starts in bar 1 (let’s call it Dux A) in D minor. The second one (Dux B) is the inverted dux from the last analyzed work, in D minor as well, starting in bar 23. The devices used in this work are mainly imitation and inversion, or at least I’ve been unable to find others.
Sometimes the first note is shortened (e. g. bar 85, first voice), or the interval is changed (e. g. bar 52, voice 4, an interval of descending fourth instead of fifth; and again bar 85, an ascending third instead of a minor second). This is done to avoid dissonances or to form concrete chords. Below I’ve pointed out some of those imitating Dux A.
Imitation – Dux A: it appears imitated and transposed to the following tonalities.
- D minor: bars 85 (with a shortened 1st note), 115.
- Eb minor: bar 103.
- F major: bars 42, 103 (augm. 1st interval).
- G minor: bars 3, 44, 103 (major second interval, instead of minor one).
- A minor: bars 14, 75.
- Bb major: bars 66, 85 (augm. 1st interval), 115.
Imitation – Dux B:
- C major: bar 31.
- D minor: bars 44, 66, 75, 85, 115.
- F major: bar 75.
- A minor: bars 26 (after a false entrance), 34, 52.
- Bb major: bar 103.
Inversion – Dux A: I guess the defining note for these ones should be the third note, because it would make harmonic sense to me that the grade 1 of the chord would be a forth before the previous one. Considering that, I found the following inversions:
- D minor: bar 14.
- E minor: bar 8.
- A major: bar 7 (turning to minor in bar 9).
False entries: Dux B, bar 25, in A major.
There is a pattern in bars 95-97, voice 4, that looks like inspired in Dux A, in a retrograde way. Instead of the rest the fugue starts with, there is a prolongation of the note next to it.
On the structure and tonality
In terms of how the themes are presented, we can appreciate three different sections. . Bars 1-22 belong to Dux A and its development, 23-43 to Dux B and its development, and from bar 44 the piece is dedicated to the combination of both themes (and the subsequent development).
Having heard repeatedly and separately the themes, the third section allows as to follow the combinations and enjoy them much more. This slightly contradicts D. F. Tovey’s point of view of the fugue as a musical texture rather than a musical form – or at least if we consider, like M. Weeks, that in a texture “it’s difficult (impossible, actually) to hear all the individual lines“. It may be that other Bach’s works or other authors’ fugues are textural, but this one is so cleverly done that the lines are possible to follow at once separately and together in one listening (not necessarily the first, though!!).
As for the tonal relationships, Dux A is presented in the tonic, at the lower fifth, then comes inverted in the dominant and at the higher twelfth of the dominant. After some more development and a rest in the first three voices, Dux B appears in the tonic (bar 23), then in the dominant, the relative major of the dominant, and again the dominant. In the combined part, sometimes two lines carry the same theme, separated by a third (bars 75-78, 85-89, 103-106 and 115-118). The combination of notes in the three voices either build a chord (G minor in bar 85) and / or the chord is build with the tonics of the triads used by each of the voices (D minor in bar 75). I’m afraid this is already beyond my capacity at this point, but the result is simply superb.