My goal in this series is not just an analysis using my method to uncover the sequences and patterns in Bach’s fugal masterpiece, but it is to function as a comparative analysis of the whole work comprising its twenty movements.
The analysis of every movement will be laid out and then once the final movement has been completed I shall present a comprehensive comparative analysis of the structures of all of the movements of The Art of Fugue.
I have no idea what secrets this endeavour will reveal but I am sure that reveal it will. This is a preliminary step towards analysing bigger pieces including the Concerti, Passions and the Mass in B minor as well as introducing analyses of music by other great composers, with works by Mozart, Beethoven, Schubert, Holst and others on the way. Can’t wait.
NB: In this analysis, the red notes signify a (+1) alteration and the blue notes a (-1) alteration. The practise of using two colours for altered notes in the score helps to highlight the direction that the music is moving, making it easily recognizable by sight. Red and blue alterations when mixed create a wave like effect whilst consecutive blue or red notes moves the music either forward or backwards through the circle of fifths. Plus, the type of sequence becomes more obvious, whether circular, symmetrical or something else completely different.
On with Contrapunctus I from BWV1080, or The Art of Fugue as it is better known. The key signature is F Major or D minor (just a reminder that I view D natural minor as D Aeolian and as such it is classed as mode VI in F Major, not as a scale in its own right. Whenever music is in D Aeolian, it is marked as F Major (mode VI) and bar 3 takes us into D Harmonic minor.
Bar 6 takes us back into F Major and then the first circular sequence occurs:
(F Major) – A H. minor – G Major – G Melodic – F Major – (D H. minor – F Major) x2
A circular sequence is one that starts and ends at the same place with no doubling back on itself in the modulations. You can see in this series of changes that first it moves up through the circle of fifths (+1) whilst changing to the relative of A minor and then up again but to the relative Major on G. Then the opposite side of the circle as the B is flattened for G Melodic (-1)and then so is F# for F Major with a little wave to end, (Major – H. minor) x2. Subtle but very effective in creating that other worldly sound. (In my opinion, Bach is a real mathematician in the way he approaches music, not just a musician of the highest order but a mathematician too.)
Bar 12 moves into C Major and then the next sequence occurs:
A H. minor – (F Major – G Melodic) x2 – G Major
This takes us up to bar 18. Now, look carefully at what is happening here. When compared to the previous sequence, it is clear that it is in a form of reversing the sequence. The A Harmonic minor is followed by the first scale in the first sequence and then it moves back again to the G Melodic and G Major, as if they were in a circle. I can show it like this:
Sequence 1: (F Major) – A H. minor – G Major – G Melodic
Sequence 2: A H. minor – (F Major – G Melodic) x2 – G Major
Look at A H. minor in the first sequence and then move back as if it was in a circle, G Melodic after F Major etc, that is the order of sequence 2. This I call a reversed sequence. Take a sequence, or a section of one and play it in reverse order, simple.
Next beginning in Bar 19 is another sequence, but this time it is symmetrical, starting out from one scale and then at the furthest point it returns via the same route:
A H. minor – A H. Major – D Melodic – C Major – D Melodic – A H. Major – A H. minor
You can see the C Major is underlined as it is the peak of the sequence which then returns to the same place in the reverse order. So far, we have identified two types of sequence:
1) Reversed sequence (bars 12 – 18)
2) Symmetrical sequence (bars 19 – 21)
F Major leads us into page 2 and the following sequence occurs over bars 22 - 27:
(D H. minor – D Melodic – F Major – C Major) x2
You cans see the simple idea, minor to relative Major then forward a fifth and repeat. The next has a surprise in store for the listener:
C Major – G Major – A Melodic – A Harmonic minor
When the same idea is stated again the repetition is abandoned after G Major and new ground is covered:
C Major – G Major – G Melodic – (D H. Major – D H. minor) – D H. Major – G Melodic –
(D H. Major – D H. minor)
Here, Bach moves into G Melodic instead of A Melodic in the first sequence and then the D H. Major change occurs 3 times, the first and third followed by its relative D H. minor and the middle time followed by G Melodic. So the wave like effect of ABA added to the first two scales of the previous sequence. It really is an ‘art’, as the title of the work suggests, but not just referring to the fugal form of the music. The use of sequences to such a detailed degree is an art in and of itself.
Bar 37 takes us back into F Major and then the music goes through:
F Major – (C Major – G Melodic) x2 – D H. minor - C Major – F Major – (C Major) - D H. minor – D Melodic – C Major
Here, Bach is using the symmetrical sequence again, with the C Major that would be after F Major omitted, shown in brackets. F Major is the peak, shown underlined, and then the direction is reversed all the way up to the C Major scale and the G Melodic is now changed to D.
You can see that sometimes a theoretical idea is applied to the music but then the actual sound of the music takes precedence and the sequence is altered to fit the composer’s taste. Obviously, the end goal is good quality music and not purely interesting sequences so the former takes priority over the latter. As any composing musician knows, sometimes a musical idea has a mind of its own and the original journey in sound that started as an idea morphs into a different one in reality. We can still see evidence of the idea though, even if it is altered for a musical reason.
Added to our list of particular sequences, we now have:
1) Reversed sequence (bars 12 – 18)
2) Symmetrical sequence (bars 19 – 21)
3) Reversed sequence (bars 37 – 48)
Next, bars 48 – 58 has this sequence of changes:
A Melodic – A Major – (D Mel – D H. min – D Mel – C Major) –
(F Major – D H. min) x2 – F Major – D Mel – D H. min
The first two scales are an echo of the previous two, Melodic – Major, and then a series of wave effects with D Melodic played twice, each time a different reply is heard, the first in H. minor and Major second. This is then repeated with an F Major wave, three this time, with the third having a Melodic in between the Major and H. minor.
Now the music introduces more unusual scales, ones that are far more difficult to use in a fugal setting as the specific alterations made in introducing theses scales are much more difficult to reproduce and repeat in the fugal form because the strict adherence to the structure is paramount, which is why they appear only once in a fugue. It will be interesting to see what unusual sequences the other movements contain.
F Major – C Major – D Melodic – D N. Major – G H. Major #4 - G H. Major – G H. minor
This is an unusual way to get to G Harmonic minor from F Major. The usual approach would be back a fifth to Bb Major and then F Mixolydian (+1) = F# Alt bb7 and we are in G Harmonic minor. Instead, Bach opts for moving forward a fifth into C Major and then C Ionian (+1) = C# Altered to get to D Melodic. Now he flattens the E for Neapolitan Major and then the unusual change of raising the third of the Neapolitan Major scale to reach G Harmonic Major #4 which is then resolves to the more usual Harmonic Major and then a parallel G Harmonic minor.
NB: The Harmonic Major #4 scale is used far more frequently than was previously known, and the main reason is that it is only one alteration away from a simple Major scale. For example, take C Major, flatten the 2nd and you now have F Harmonic Major #4:
C Major (flatten the 2nd) = C Db E F G A B = F G A B C Db E = F Harmonic Major #4
So you can see that the main use of Harmonic Major #4 is to get from Harmonic Major to a Major scale a fifth back in one move. Here, Bach is going in the opposite direction, Harmonic Major #4 to Harmonic Major to Harmonic minor.
For the final 15 bars, Bach summarizes the whole movement by revisiting all of the keys that the music has been through so far, like a much condensed version of the whole piece:
D H. min – D Mel – D H. min – C Maj – G Mel – F Maj – A N. min – A H. min – C Maj – C Mel – G H. Maj – G H. min – D H. Maj – D H. min – A H. Maj – D Mel
This is a great technical idea, like a fractal inside of itself. So, write a piece then towards the end, sum up everything that has happened in the piece in miniature. Genius at work again.
The final five bars are all over a pedal tone of D, the scales use are D Major, G Major, G Melodic, G Harmonic minor and G Hungarian minor and so the modes heard over the D pedal are:
D Ionian – D Mixolydian – D Mixolydian b6 – D Double Harmonic
A lovely section of dominant modes and the D Double Harmonic mode from G Hungarian minor is functioning as an A dominant, the A7b5 over a D bass surprises with its resolution to D Major, chord I from the D Harmonic Major scale.
Contrapunctus II is next in the series. Thanks for reading. Don’t forget my book on music theory is available here: https://www.bedwellmusic.co.uk/general-7
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