Why does Bach sometimes end with flat 7ths?
I've been studying the Bach inventions, and I see a common patterns in the first 3 inventions: They tend to include the flat 7ths in the very end. Examples:
C major:
D major:
E♭ major:
From a music theoretical standpoint, it's confusing, but in my ears this harmony "sounds like the piece is ending". Does anyone here have an analysis on what's going on?
Edit: I understand that the music is modulating towards the subdominant but that seems counter-intuitive to do towards the very end of a piece. Bach left nothing to chance, and I'm looking for a way to motivate this choice from a theoretical point of view.
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In addition to what's been mentioned above, the motion to the subdominant also has a plagal sound. What's more final than a big "aaaa-men" plagal cadence at the end of a hymn? Sometimes Bach's flat-7s come after the authentic cadence. Sometimes before.
The use of the flat-7 was much more widespread in Renaissance polyphony and early baroque music. It was the natural function of some major modes to lower the 7th during motion downward in the scale. Baroque composers from the early 17th century would continue to pare down use of the flat-7 as music became less melodically- and more harmonically-driven. By Bach's time, you see he saves it for a special moment near the final cadence. (This is not to answer your question of WHY, but just to add some history.)
I honestly can’t believe the length and Amount of philosophical persiflage of these answers. They’re just plain common, garden-variety secondary dominants to add interest to the harmony, which may be one reason most people who play such music play Bach and not Georg Böhm or Johann Adam Rein[e]ken. Also, the remark about dark flat keys and bright sharp keys, while having some truth, is deeply dependent, more so than personal emotional perceptions, on what tuning is being employed. Bach alone, that is, as opposed to the tunings used in places where Bach did not work, used three different tuningS in his lifetime. They were called the Chor-Ton, the Kammer-ton, and the Tief Kammer-Ton. These were not necessarily his choice. They varied as much as a major third, one from the other, not to mention how the split-key tunings were handled. So! Easy on the deep-significance philosophisizig and a little more of old-fashioned musicology better answers the question. By the way, this is the first time I have heard of widespread bVII in renaissance music. There ‘s not one renaissance theorist who speak of chords, certainly not in a function way. Johann s Tinctoris would have seen what we call a simple c-major triad as three chords, one built on c, one on e, and one on g. As some of you clearly would know, it was Jean Philippe Rameau in 1722 who first explained that these three sounds were actually one chord, or, put another way, he was the first to formulate the invertibility of chords.
Peace
L
Actually, I would not even call it a modulation (e.g., into a subdominant in the first example--the one I consider here) because for a progressiin to be called a modulation it needs to follow the cadential formula T-S-(D 6/4)- D-T in the new tonality. What we see though is a progression that stays in the original tonality, except for the non-harmonic tone B flat. I think that the explanation for the use of this tone lies in modal scales, which Bach has used all the time. In the mixolydian C scale, the seventh degree is B flat, and this is the only tone that makes it different from the C major scale, and so on its approaching to the final cadence, Bach uses the progressions in the mixolydian scale (T I 7 -- S IV--SII--DIII), and then employs T-S-T in C major 6 chords before concluding with the conventional T-S-DT6/4-D7-T.
This is, in my opinion "what" Bach is doing. As to the "why," I think the only answer is: It spunds beautiful. In the beginning of the 20th century many composers turned to using modal harmonies in addition to the tonal ones.
The C major example looks like it could be a modulation to the sub dominant (F Major), that would explain the Bb flat. The D major example, again looks like a modulation to the sub dominant G major, which is the only key that has a F# and a C natural. The last example looks like a quick modulation to Ab major for half a bar, that explains the D flat.
He ends with the key the piece was written in, it just modulates often and sometimes just for a measure. Modulations or the movement between keys is how you give movement to music. It is also how you make music interesting. Classical music rarely stays static in regards to keys.
It's the same cliche as the 'Saints Go Marching In' ending. Not sure if Bach ever went the whole hog with a iv as well as the I7 though! And he generally did it over a tonic pedal, the modern(ish) version walks the bass line down. Same idea though. Decoration. No functional analysis required.
C, C7, F, Fm, C, G7. C.
I think this is closely related to a widespread device known as "backdoor" cadence: Bb7 resolving to C rather than the obiquitous G7 resolving to C.
This could be preceded by a F chord, perhaps. Which would be a subdominant.
Or maybe it would be a previous tonal centre from which you return. In which case, the Bb7 works much like the Fm in When the Saints Go Marching In as described in a post above.
It's not a modulation. It's only a secondary dominant for IV. An addition for an authentic cadence
Well spotted! This is very common. Bach often uses a brief modulation to the subdominant key near the end of his fugues, preludes and inventions (presumably other pieces, too). Sometimes this is so brief, that we feel like we are just travelling through this key, without really modulating to it. Sometimes this is over a final tonic pedal, which is really “bringing the harmony home”, although not in your examples.
The reason he does this, is related to how much of Western common practice music is harmonically structured. Although we may modulate to a wide range of related keys, by far the most commonly used structural modulations are basically an arch: Tonic -> Dominant -> Tonic (again). If we imagine this as a “swing” one place to the sharp side around the circle of fifths and then back again, the move to the subdominant helps to “balance” this, by swinging “a bit too far” to the flat side, as we return to the tonic, with the final settling on the tonic feeling even more like a resolution as a result.
Another way to think about the same structural harmonic movement, is considering the brightness of related keys. As we move to sharper keys around the circle of fifths they sound comparatively brighter; as we move to flatter keys they sounds comparatively darker. [Thanks to Tim for commenting on this, hence the edit for clarity...] We start at the tonic, our home key, move to a brighter key, the dominant, and then back to our home key. To emphasise this movement back from a brighter key to our home key, we go "a bit darker" than our home key before finally settling there.
We could come up with numerous analogies for this kind of movement: tuning a sharp guitar string down below the note we want, before coming back up to pitch; taking a clipping recorded volume down below the volume we want before gradually raising it to the optimum position etc.
This is nothing else than an extended perfect athentic cadence (the b7 appears as the Ib7 is used as V7/IV)
I - IV - I46 - V7 - I
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