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Here's measure 12:

https://drive.google.com/file/d/17zMs-wAC27w8y2FhDwL2a_bzRDEpSQPJ/view?usp=sharing

Here's measures 1-12 (maybe image is too small): https://drive.google.com/file/d/1zOIxZBVelDGRg_OmLYYdBkzUcNipEre1/view?usp=sharing

I'm curious if anyone has any interesting observations about the harmonic development of the fugue as it passes through the highlighted chord, with the function of the G# paid particular attention.

While I don't want to make my brain ache by taking apart the fugal engine to washer-and-nut level (yet), I want to have a sort of high-level awareness of what Bach is doing. To me it seems like a slight dip towards reciting the theme in an E-maj modulation while balancing the previous C-maj and G-maj iterations. So the choice of G# is obvious, given the motif and the root of E, but was it necessary--or in any event, is there a concise way to give it useful context?

The chord is a D minor with the G#, not sure what that is other than a minor chord with a b5, unless there's something else implied by the surrounding context.

Well, thanks for any insight.

The A is a non harmonic tone. The chord is a G# dim ( vii) in second inversion as this part in the piece is establishing A min.

This squares with my intuitive feel--the A min moment right after is very strong, so a G#° to A is a great description of the 'feel' of that moment.

At the same time, in the moment, it feels like that D min wants to be known and the G# is the first moment of great tension and uncertainty. I love it, how it can be both this and that, parts of both yet neither. Defined more by its movement through time more than what specie it is when pinned like a butterfly.

Bar 12 and 13 are modulating and cadencing which lead to the PAC on A minor in bar 14. You have to be carefull that Bach is not using the modern harmonic theory with the fundamental root. He is working on harmony based on thoroughbass progressions and chords. You are free to interpret harmonies in modern sense but that was not his approach. The basis of the harmony is in the bass line.

In addition, in a fugue not all harmonies are designed as such, many are just the result of the mouvement of the voices, as long as they remain within the rules of counterpoint, in particular on weak beats. So one has to be carefull not to overinterpret some progressions. You have to read both vertical harmonies and implied harmonies in the bass line. Some vertical harmonies have a real meaning, others being mostly passing non structurally meaningful ones, that is particularly true during sequential or modulatory progressions. The 2 bars 12 and 13 are particularly dissonant which may be stylistically intentional. The whole piece has some connotation with some older type of fugal compositions of the 17th century and the stretto technique is being used often. There is not much of episodes here.

The typical cadential mouvement is to approach melodically V by IV in the bass, in this case D. Bach is using the dorian minor with its major VI, F sharp and the raised VII, G sharp in ascending mouvements. Bar 13 is clearly based on an alternation of D and E in the bass. In bar 12 the bass lines goes B to D to F natural, which is the harmony of II in A minor. This leads to V which is the basis of the cadence in bar 13. Thus II, V, I. This ends the first large section of the fugue, after which we return straight to C major.

^ this is a lot for me to unpack, I appreciate it @Sidokar.

I'll go through the passage with close attention and your words to get the full meaning of them. For now, what stands out to me are two things: 1) the structure and discipline of the fugal form can lead to less 'intuitive' harmonic choices, that's ok--just go with it, and 2) Bach might play bass in your rock band...

I was doing a google search to look up PAC and found this website that has some fugue analysis, maybe it's interesting to someone?

http://www.algomus.fr/fugues/bach/index.html

Yes. Baroque harmony should be evaluated horizontally as well as vertically. The A resolves to a B after which D, G#, F, B together is a diminished 5-7 chord.

Sorry, I should have expanded. It means Perfect Authentic Cadence. Though the term is better suited for classical period and less relevant for baroque music. Lets say the piece has a strong formal cadence on A minor with a typical V-I.

The actuel details of how Bach does the harmony shifts are probably more of a second priority vs the overall harmonic architecture and the style of the piece, the rythmic organisation and the counterpoint work on subjects. There are probably plenty of materials available on this piece. Most important is how you choose to play it ! Good luck.

Sorry, I should have expanded. It means Perfect Authentic Cadence. Though the term is better suited for classical period and less relevant for baroque music. Lets say the piece has a strong formal cadence on A minor with a typical V-I.

The actuel details of how Bach does the harmony shifts are probably more of a second priority vs the overall harmonic architecture and the style of the piece, the rythmic organisation and the counterpoint work. There are probably plenty of materials available on this piece. Most important is how you choose to play it ! Good luck.

Also to clarify what I mean here. It is a bit complicated for someone unfamiliar with baroque continuo, but the purpose is to explain the logic used by Bach.

On that second part of beat 2 in bar 12, in modern theory it could be interpreted as a second inversion of a diminished chord Gsharp-B-D-F, which then gets resolved on a first inversion of A minor. But in fact Bach is not using the fundamental root theory which is anachronistic for baroque composers. Though it is factually correct, it does not explain how Bach was organizing the voice mouvement and why he used certain harmonies.

In a thoroughbass approach and notation (counting intervals from the bass), the initial chord D-G sharp-F-A is a 5-4(sharp)-3. That harmony is extremely dissonant with both a triton with the bass and a minor 9th between the tenor and the soprano voice. It is obviously a harsh passing dissonance, though representative of baroque music. That dissonance gets partially resolved when A moves to B and completely resolved on the strong beat 3.

There we have a very frequently used chord which is a 6-4-3 with a raised 4 fourth (quarta irregolare). That dissonant chord is allowed with a prepared or unprepared fourth in particular when the bass line is coming descending stepwise which is the case here coming from E to D. That chord can be used with a perfect fourth as well or with a diminished fourth in a 6-4-2 configuration. When used with the augmented fourth, the sixth has to be major which is the case here. It is frequently encountered on a subdominant bass (D) but other scale degree can work as well.

That chord typically resolves to a 6-3 chord, a consonant chord, the bass goes down stepwise, the raised fourth up half a step and the major sixth (B) up a step. That explains the various counterpoint mouvements used by Bach.

All professional baroque composers learned all the various possible chords based on bass mouvements and their resolution since childhood. That was necessary in particular when playing continuo. You will find all these chords and how to use them in various treatise of the time where the logic of voice progression is explained (Heinichen, CPE Bach, .....).

This explanation greatly enhances my appreciation of what Bach is about, and gives me a glimpse into 'intention' if that makes sense. It is still a lot, but I asked for it. I do understand what you said. To get into the mind of Bach, pay attention to the fundamental tone more than whatever chords that all the notes happen to form. You can still make sense of it using that more modern lens, but that's not what was going on in his mind.

There is something that I don't understand though. When you say the "D-G sharp-F-A is a 5-4(sharp)-3." So the 5 is the interval between D and G sharp? No, that would be the 4(sharp). So the 5 must be the D-A. I would have thought it would be called a 4(sharp)-3-5 if it is sequentially upward from the fundamental note.

Maybe I don't need to get deeper into the intricacies of thoroughbass--your explanation is already sufficient for what I need right now--but I figured I'd let you know that part was still a bit confusing.

I am grateful that you shared your knowledge.

You are right. The notation does not specify how the intervals are placed in the chord. The 5 is D-A and the augmented fourth is D to G sharp. In the thoroughbass notation, for example 5-4-3, the 5, 4 or 3 can be anywhere, though the 4+ is often between the bass and the tenor or the alto (a triton between the bass and the soprano would be even more audible and is therefore to be avoided).

When playing in the continuo, with a figured bass, the player would decide how best voice the chord (ie how to place the various intervals) and also which notes should be doubled, based on the context. When supporting a simple voice, the continuo must be fairly discrete, but with an orchestra, the player could use both hands to create as much volume as possible. It was really a notation system designed for professionals with a high degree of training.

Yes , you do not need to get into the intricacies. It is interesting only if you want to compose yourself. Many people say they have a hard time understanding the logic of certain voice progression in Bach, but they make a lot of sense if you understand how Bach was using the chords and the counterpoint rules.

It may not explain Bach's thought processes when he composed the piece, but music theory today is a perfectly adequate tool to analyze music of earlier eras, and enables comparison of music across eras.

I think our modern separation of harmony and counterpoint is a more powerful analytical tool, and was able to explain the usage concisely in one sentence.

You have no idea how long I've waited for an argument like this.

It depends what you put under music theory. If you refer to the classical tonal music theory, with the fundamental root approach, i would disagree. It allows to approach roughly a subset of classical music. It makes no sense to use it for music before 1650 which is not composed in a tonal harmony framework and it does not work for atonal compositions. It actually does not even work well for composers like Faure. For Bach and other composers of his period, it is more complicated because they were in between 2 worlds. So some of it works well and some parts not at all.

Of course one can always use the modern theory to analyze any composition but it only leads to misinterpret and give a false sense of understanding. There isnt any universal theory that works in every case and that is because music is not a science and there isnt any absolute truth. Besides the classical tonal harmony, there are plenty of other analytical methods. They all are more or less relevant and more or less suited to certain types of music.

In addition very frequently the modern theory is not even teaching the voice behavior. Composers of the 19th century also studied counterpoint, so that they knew how the diffetent voices should move, how to double certain notes, how to resolve suspensions, and so on. A pure harmonic based analysis can not explain classical music. Learning the basic harmonic theory provides an all around foundation which is good enough for navigating at high level in mainstream classical compositions, roughly from Bach to end to 19th century, which is what most amateurs plays.

Bach himself when he read Rameau theory of fundamental root was in complete disagreement with it. If you read the various composition manuals of the time, you will see that it is impossible to understand why the music is written in a certain way if you dont know the theoretical context of the time. So for example fundamentally for a baroque composer a sixth chord on E is not an inverted C chord it is a sixth chord on the harmony of E and that is how they would hear it. You can always decide that it is a C harmony, and to some extent it is, but then it would potentially lead to subsequent false interpretations.

But i guess, it is barely a topic for a beginner pianist who is learning to play the piano for his enjoyment.

Nonetheless, hawgdriver's question was answerable in one sentence with our current treatment of harmony. The A in the melody is a dissonance that resolves to the B, which then forms the diminished 5-7 chord of the key the piece either was in or was transitioning to at that point.

I dont deny that it is how you would interpret it in usual theory, which is perfectly correct. I am saying that in this case it is not explaining what Bach was doing. Just like using tonal harmonic theory to interpret the music of the 16th century wouldnt explain how the music works. In this case it works reasonably well, but in many others it would just be misleading. The purpose of any theory is to explain the compositional process and to give a framework to compose, and for that it must be aligned with what the composer is doing. Applying tonal harmony to Berg would make no sense for example. Similarly to understand the composition process of Bach it is necessary to use the same tools he was using, just like to understand Berg, it is necessary to use his theory. But i certainly understand that for amateurs activity, Simplicity is always a winner.

Berg and Webern would consider many of their works to be a failure if conventional harmony were applicable. Atonal music is not particularly relevant to a discussion of analysis by conventional harmony.

This thread:

Hawgdriver: so measure 12, what was Bach thinking?
People: dissonance and resolution, ABC123
Sidokar: Bach focused on bass, here's a helpful primer to get into his head
Still other people: a lot of this stuff was transient dissonance, you'd get these fleeting moments of strong dissonance, it's almost like they were aware that their audience didn't want dissonance to last for very long
Sweelinck: Yo Sidokar, your novel is cool but seriously bro, it's just some dissonance
Sidokar: yeah, but ... *sigh* nevermind. (these people)

And I really love pretty much every response so far