# Chromatic diminished 7ths in Beethoven rondo, how does this harmony work?



## caters

Here is the passage I am talking about:









So starting at measure 291, there is a passage where 7th chords go up chromatically, and the majority of these 7th chords are diminished 7ths. The seventh chords only last for 2 bars until there is a break from them with a scale. So let me try to figure out the harmony here.

A, C, D#, F#, this is an enharmonic respelling of Adim7 to better fit with the key signature of G major. And then that goes to an A#dim7. What scale comes after these 2 seventh chords? Well it starts on B and then goes up and it has 5 sharps. It must be B major. C and C# are skipped. D, F, G#, B, it is a Ddim7, again respelt to fit with the key signature better. That goes to D#dim7 which then lands on an E major scale. E and F are skipped.

G, Bb, C#, E, it is a Gdim7 though here, it doesn't really fit with the key signature better than if the C# was written as Db. But that would mean spelling the E as Fb which doesn't fit with the key signature at all. As predicted it goes upwards to G#dim7. But here, Beethoven breaks the pattern of 2 diminished 7ths followed by a major scale. If he were to continue the pattern, it would end up on an A major scale. But it doesn't. We get our first resolution. G#dim7 goes to C minor. Beethoven sure loves his C minor.

The resolution doesn't last for long though, he keeps going with the 7th chords. F, A, C, Eb, this is F7. I would expect it to resolve to Bb. So what does Beethoven give us? The unexpected is what he gives us. He gives us an Adim7 in third inversion. The Gb then goes back down to F giving us another F7. This time, it does resolve. Not only that but Bb is the temporary tonic note. It resolves to Bb minor.

This then goes directly into Eb7 the same way that C minor went directly into F7. I would expect it to resolve to Ab. But, of course, Beethoven has to give us the unexpected. He gives us a Gdim7 respelt with an Fb instead of an E like the previous Gdim7. This then lands back on Eb7 and Beethoven gives us a scale. However, it isn't a major scale like the previous ones. No, it is Bb dorian where I would expect Eb major or Ab major. Then he finally resolves Eb7 to Ab without going to another 7th chord right after, ending this 20 measure long, chromatic passage. He then has to go back to G somehow but that isn't my focus here.

So, yeah, it starts off with a pattern of 2 diminished 7ths a half step apart followed by a major scale. Then it deviates from this pattern with the first resolution to C minor. This then starts another pattern of root position dominant 7th, third inversion diminished 7th with root a third above the previous chord root, same root position dominant 7th, resolution chord which then ends with Ab major. So this way, Beethoven has practically brought us all the way up the chromatic scale using these 7th chords.

But why does this work, this chromatic scale in the harmony? I think it has to do with not only the pattern and how it is deviated but also the circle of fifths.









In both cases where it lands on a major scale, the second diminished 7th has the leading tone as the root and the first diminished 7th has the note 2 spaces on the circle of fifths flatter than the major scale as its root. And A to D itself is a circle of fifths motion. Likewise so is D to G. G# is where things deviate. But this still works because every note in the G#dim7 except for F is a half step away from a chord tone in the C minor chord. The C minor, as an unstable tonic sets us up for the second pattern of 7th chords. Via a motion to the major subdominant, we get F7, 1 space sharper on the circle of fifths than C minor. This then moves to the minor tonic, 3 spaces flatter on the circle of fifths. Again the tonic is unstable. This continues with another movement to the major subdominant. Then we finally get to a stable tonic with the Eb7 to Ab motion that ends the passage, a cadence in Ab. Summing this up, we move 9 spaces flatter on the circle of fifths than where we started.

Is the circle of fifths the reason why this progression of 7th chords works?


----------



## EdwardBast

Could you please tell us what the piece is with an opus number or other reference?


----------



## Guest

_Rondo_ a capriccio, Op.129


----------



## Guest




----------



## EdwardBast

Here is what's going on harmonically:









Note that the harmony in mm. 10-11 of the example is a pivot chord, starting life as the vii7/V in D minor, becoming the vii4/2/ii in Bb minor.

The Gb and Fb in mm. 14 and 18 respectively are just neighbor tones - the harmony really doesn't change.

There is no Dorian mode. That scale is just Ab major, which is the scale of the local key.

Caters, you tend to get bogged down in details in your analyses rather than looking at the big picture. The passage is mostly four measure units which are then transposed and repeated. If you have any questions don't hesitate to ask.


----------



## caters

Why isn't the last scale of the passage an example of the Dorian mode? It starts and ends on Bb, which suggests to me that Bb is the root of the scale. Given that Bb minor has 5 flats and the scale in question only has 4 flats, it would make sense to me to call it Bb Dorian. I mean, if it was intended to be Ab major, wouldn't Beethoven have started the scale on Ab, like how he starts the B major scale on B and the E major scale on E? 

And what do the n's mean in your harmonic analysis? Does it mean that it is a chord based on the Neopolitan? Because those usually have a capital n in the analysis with 6 being the bass figure. Or does it mean that it isn't all that important harmonically and that the third inversion diminished 7th is really just a neighbor tone in the bass over a dominant 7th harmony?


----------



## Guest

^^^^ EdwardBast: may I ask what sofware you are using that enables you to take Cater's PNG image and figure it?
That could be useful to me !!


----------



## EdwardBast

caters said:


> Why isn't the last scale of the passage an example of the Dorian mode? It starts and ends on Bb, which suggests to me that Bb is the root of the scale. Given that Bb minor has 5 flats and the scale in question only has 4 flats, it would make sense to me to call it Bb Dorian. I mean, if it was intended to be Ab major, wouldn't Beethoven have started the scale on Ab, like how he starts the B major scale on B and the E major scale on E?


The progression is just V7-I in Ab major. The V chord has an Ab major scale over it. Doesn't matter what note it starts on. The root of the scale is Ab. The simpler answer is: It's Beethoven! He's not writing in Dorian mode. There is nothing modal about this piece.



caters said:


> And what do the n's mean in your harmonic analysis? Does it mean that it is a chord based on the Neopolitan? Because those usually have a capital n in the analysis with 6 being the bass figure. Or does it mean that it isn't all that important harmonically and that the third inversion diminished 7th is really just a neighbor tone in the bass over a dominant 7th harmony?


The n just means neighbor tone. The harmony in each case is V7 for three measures with the root moving to a neighbor tone for the second measure. The neighbor motion doesn't change the harmony. It is a linear phenomenon.


----------



## EdwardBast

TalkingHead said:


> ^^^^ EdwardBast: may I ask what sofware you are using that enables you to take Cater's PNG image and figure it?
> That could be useful to me !!


I'm just really fast copying in Sibelius .


----------



## Guest

EdwardBast said:


> I'm just really fast copying in Sibelius .


Via MIDI keyboard input?


----------



## EdwardBast

TalkingHead said:


> Via MIDI keyboard input?


Not in this case. Given the sequences involved and repeated figuration, it didn't take long: Cut-paste-transpose. 

Not sure how meaningful all the Roman numerals were, given the modulating sequential stuff.


----------



## Guest

EdwardBast said:


> Not in this case. Given the sequences involved and repeated figuration, it didn't take long: Cut-paste-transpose.
> 
> Not sure how meaningful all the Roman numerals were, given the modulating sequential stuff.


Well, that would be a lot of work for me (I use Finale), so thank you for making the effort.
I think your Roman numerals were useful, though I might have a couple of little quibbles which I'll address later, nothing to get into a lather about, more to do with US/FR figured bass conventions, no sweat.


----------



## EdwardBast

TalkingHead said:


> Well, that would be a lot of work for me (I use Finale), so thank you for making the effort.
> I think your Roman numerals were useful, though I might have a couple of little quibbles which I'll address later, nothing to get into a lather about, more to do with US/FR figured bass conventions, no sweat.


Please give a critique! It was a fast job.


----------



## Guest

EdwardBast said:


> Please give a critique! It was a fast job.


For a fast job it was pretty good, I must say !!
My quibbles really are very minor (you will call me a nit-picker !!).

1) *Figured bass conventions*: you obviously are schooled in one way (VII°7) and me in another (VII7 with the 7 crossed out); sorry, I can't represent that in this forum programme, here is an image of what I mean:









2) Concerning your point: _The 'n' just means neighbor tone. *The harmony in each case is V7 for three measures with the root moving to a neighbor tone for the second measure*. The neighbor motion doesn't change the harmony. It is a linear phenomenon_. 
I think they are more than just neighbour notes in bars 14 and 18; I see them as VII2, though I agree with you that the harmony doesn't _really_ change fundamentally, they are just "dominant prolongations", if I can put it that way.

3) At first I was going to take issue with your figuring at bar 11 (b-flat: vii4/2/ii) but on looking closer I see you are absolutely right! It took me a while to realize that Beethoven is using the chromatically altered supertonic II7 chord (in B-flat minor, with sharpened 3rd & 5th), but I was looking for an A-flat which is in fact present, though spelled as G# !! Bravo for that, I wouldn't have spotted that as quickly as you did.

So finally, my only really little quibble is how we analyse the neighbour notes as explained in Point 2 above.


----------



## EdwardBast

TalkingHead said:


> For a fast job it was pretty good, I must say !!
> My quibbles really are very minor (you will call me a nit-picker !!).
> 
> 1) *Figured bass conventions*: you obviously are schooled in one way (VII°7) and me in another (VII7 with the 7 crossed out); sorry, I can't represent that in this forum programme, here is an image of what I mean:
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 2) Concerning your point: _The 'n' just means neighbor tone. *The harmony in each case is V7 for three measures with the root moving to a neighbor tone for the second measure*. The neighbor motion doesn't change the harmony. It is a linear phenomenon_.
> I think they are more than just neighbour notes in bars 14 and 18; I see them as VII2, though I agree with you that the harmony doesn't _really_ change fundamentally, they are just "dominant prolongations", if I can put it that way.
> 
> 3) At first I was going to take issue with your figuring at bar 11 (b-flat: vii4/2/ii) but on looking closer I see you are absolutely right! It took me a while to realize that Beethoven is using the chromatically altered supertonic II7 chord (in B-flat minor, with sharpened 3rd & 5th), but I was looking for an A-flat which is in fact present, though spelled as G# !! Bravo for that, I wouldn't have spotted that as quickly as you did.
> 
> So finally, my only really little quibble is how we analyse the neighbour notes as explained in Point 2 above.


As for figured bass conventions, I wasn't going for anything so ambitious as figured bass.  I was doing whatever could easily be done with a normal typeface.


----------



## millionrainbows

I see caters struggling, but I see him as searching for some underlying principle of diminished chords rather than floundering, and the analysis, while I'm sure EB's is correct, doesn't explain the issues I think caters is struggling with, as evidenced by caters' zeroing-in on diminished 7th chords and the circle of fifths.

EB, could you speak a bit more on the two diminished chords in measures 1 -2, the e:vii4/3, and viii˚V, and how they are related harmonically to each other and to B:I?


----------



## EdwardBast

millionrainbows said:


> I see caters struggling, but I see him as searching for some underlying principle of diminished chords rather than floundering, and the analysis, while I'm sure EB's is correct, doesn't explain the issues I think caters is struggling with, as evidenced by caters' zeroing-in on diminished 7th chords and the circle of fifths.
> 
> EB, could you speak a bit more on the two diminished chords in measures 1 -2, the e:vii4/3, and viii˚V, and how they are related harmonically to each other and to B:I?


Sure. And thanks for putting the focus back on Cater's inquiries. The way to approach this passage is not to wonder first about the dim 7th chord pairs. They are like an adverb and a verb hanging in the air, meaningless without the noun that follows. One needs to look at the (relatively) stable points first and then work backward to the dim 7th chords. In the 20mm example I posted above, those stable points are mm. 3, 7, 12, 16, and (not shown) 21.

3 - *B major*
7 - *E major*
(11 - ? pivot chord; *A* major expected, but the pattern is broken) proves to be the vii 7/ii in Bb Minor)
12 - *C minor*
16 - *Bb minor* (cadence on - we've been in that key since m 11)
21 - *Ab Major*

So first (mm. 1-11) we have a cycle by 5ths, B-E-(A), with the final resiolution to A denied. Then (mm. 12-21) it descends by whole steps: Cm, Bbm, Ab.

The first three of those stable-ish harmonies are preceded by their vii°7 chords, the last two by their V7 chords. Everything after measure 11 is straightforward.

So given that the mysterious part of the passage falls into a pattern of tension-resolution in 4 measure units, the obvious question to ask is: Do the three chords in each of those units make grammatical sense within a key? The answer is yes. The chords in the first four measures are standard vocabulary for the key of E minor: vii4/3, vii°7/V, V, the next four measures just repeat the pattern in A minor. Now because we've just heard the same pattern twice, we assume that the dim 7 chords in mm. 9-10 are going to function the same way, as vii4/3 and vii°7/V in D minor. But that's where the pattern is broken. We don't get the A major we've been set up to expect. Instead the harmony repeats as a pivot chord to C minor - which is where the mystery ends. From there it's descent by whole steps. The Bb minor chord in m. 16 is the tonic in Bb minor becoming (another pivot chord) the ii chord in the key of Ab, our final (non)-destination.

What's interesting about the pairs of dim 7th chords is that the leading tones of the vii4/3s resolve correctly to the tonic note in each case (D# to E, G# to A, C# to D) but the lower voices don't resolve down, instead sliding up to the root and third of the vii°7ths of V.

In general, I think Caters is conceiving the analytical problem as a matter of taxonomy rather than understanding it as one of function. Her strategy seems to be to start with the key signature and try to compute the identities of the chords within that key. That is never going to work in unstable, continuously modulating passages like this because the signature has nothing to do with the key in these cases. That's why one must find the local points of stability and use them to anchor the 7th chords, both diminished and dominant. Also, Caters, you seem to expect unicorns and other crypto-zoological marvels, like Dorian mode. 'Fraid it's mostly just cows and horses. 

But her final observation about the circle of 5ths makes sense:

B, E … Cm, F7, Bbm, Eb7, Ab.

Progressions by 5ths are a standard way of spinning out sequential progressions in this style.


----------



## Guest

caters said:


> Here is the passage I am talking about:
> 
> View attachment 118884
> 
> 
> [...] Bu*t why does this work* [...]? I think it has to do with not only the pattern and how it is deviated but also the circle of fifths.


I think you're on the right track in that the "drive" in this passage (what makes it _work_, as you say) are the modulating sequences which are patterns built on the circle of fifths.

Maybe it could be more useful to say that the satisfying sense of progression is more to do with the basic academic rule of "*roots rising a fourth*" in the sequential treatment of secondary sevenths.

This is shown in the overall harmonic framework that EdwardBast has pointed out at the end of his latest post just above: B, E … Cm, F7, Bbm, Eb7, Ab.


----------



## Guest

@ Caters:
Thank you for raising these issues, they're very relevant and interesting!
That said, we mustn't let the analysis get in front of the music too much!!
If I'm not mistaken, this Rondo was subtitled "Rage Over a Lost Penny" (maybe by Schindler, maybe by Beethoven himself, I don't know).
With that image in mind, I can see these fast sequences and scale passages as Beethoven himself running desperately after his lost penny as it rolls away from him !!
Pure speculation, of course, but that image pleases me !


----------

