Harmony/Modulation: Learning from Mozart et al.

It's my first post in this part of the forums!

I'm a very experienced amateur pianist but a very newbie composer. In my first little composing attempts I found that my main weaknesses are:
1. finding interesting harmonic progressions
2. learning modulation techniques

I thought it might be a good idea to learn from the masters directly rather than only from method books. I have started to look at Mozart piano sonatas with a focus on these two topics (because they might be easier to start with than e.g. some late-romantic music).

Would anyone be interested to do this jointly by discussing here? It could be fun and very instructive. If so, how about starting by discussing the harmonic progressions and modulations in the exposition of K.332? (it's just a suggestion to make things concrete; I'm open for all alternatives!).

Ok let's see if anyone is interested!

Sounds like a good idea. Anything interesting in that piece you'd care to bring up?

I'm game, and I think K. 332 is a great choice!

I'm in.

I'll be happily lurking.

Wow guys, I'm happy to see others interested (as I said, K332 is just a suggestion - one of potentially more items we could look at). Let's hope we can make this a good little learning experience for all of us.

I'll post later today the rudimentary observations that I could make (along with a number of questions) and others would be invited to either post their own observations (so we can try to harmonize them [pun intended ]), or help to answer some of the questions from me or others!

I guess most people have a paper copy of the Mozart sonatas, but for those who don't, here is K.332 on IMSLP

This is a good idea, I try this sometimes to get an idea of how former composers structured their music.
I'll bite first with my view of the exposition.
Granted, my view of harmony is not authoritative but it works for me when I'm trying to understand a piece of music.

I view bars 1-4 as essentially static F major harmony with an implied F pedal-point (the first LH quaver of each bar) which underpins the harmony and establishes the F major tonality.
Mozart embellishes this architecture by changing the implied harmony on a one bar rhythm: I - V7/IV - IV6/4 - viio6 and back to I at the start of bar 5.
I guess bar 4 could be considered to be some kind of chord derived from the dominant eleventh but I rather consider it a simple leading tone triad in second inversion over the persistent F pedal.

Bar 5 picks up with a cadential bass pattern outlining an I-V-I harmonic movement but states it in the treble.
The bass imitates it two octaves lower right away in bars 7 and 8 with the treble filling out the harmony this time.
The F in the treble at bar 7 is suspended into bar 8 (which is dominant harmony) where it resolves ornamentally unto the leading tone then completes the movement back to I in bar 9.

Bars 9-12 represent a straight forward march to the first conclusive cadence of the piece.
The implied harmonies are I-IV6-IV-I6-VI-VI6-V7-I and the whole thing could be reduced I-IV-VI-V7-I without loss of function.
Though the bass notes that give rise to the inversions make for a more interesting line and more musical interest.

Bars 13-22 introduce a new idea that seems like an embellishment of the previous cadence and prolongation of the sense of stillness and closure.
The first leg from 13-16 represent the first I-V-I motion.
The idea and harmonic gesture is then repeated at bars 17-20 with a little semiquaver embellishment and a close that's almost straight from bar 12, emphasising the subsidiary nature of this phrase.
Bars 20-22 then imitate the phrase ending and reduce the previous phrase to its bare essentials: repeated V7-I oscillations.

The unequivocal reiteration of the F major tonality may be due to what Mozart intends to do next.
Maybe he wanted to contrast the placid stillness of F major with the sudden plunge into d minor that now occurs.
Because at bar 22b the closing figure quickly becomes a transition into a vigorous D minor theme that begins in bar 23.

The harmony of bar 23-24 is i of D minor followed by the leading tone diminished 7th (viio6/5) in bars 25-26 which 'leads' back to i in bar 27. After bar 23's material is repeated in 27-28 (though harmonised instead with i6), it alights on the leading tone diminished 7th of C minor in bar 29-30 by displacing the consequent idea (bar 25's analogue) down a tone.

This resolves, as expected, on the c minor triad in bar 31 (c: i6).
Bars 31-34 outline a falling third progression from i6 to VI6 of c minor.
Motivically, the treble doggedly emhpasises the new c tonality, embellishing an octave rise with semiquaver passage-work derived from bar 23.
In 31-32, the semiquavers arpeggiate the c minor triad and in 33-34 they do the same for the Ab major triad.
In bar 35 the bass-line again falls by a third but the root motion does not.
The harmony this time is an augmented sixth and while the treble motive remains the same it plays a more essential harmonic role of providing the important f#.

At bar 37, the augmented sixth resolves on the dominant of c minor: G major.
The half-cadence occurs at bars 36-37 and the following material up to bar 40 just prolongs this important arrival on G.
Motivically, the treble now takes up material from bars 25-26 while the harmony oscillates between dominant and tonic of c minor (though the tonic is the subsidiary chord here) before ending the phrase on octave Gs (G major is V/V when viewed from F major's standpoint).

The next theme takes up the C tonality, thus resolving the half-cadence, but starts off in the major mode.
So at this point, Mozart has accomplished the customary modulation to the dominant of the movement's key and it all started with the plunge into d minor at bar 23.

The new material seems very reminiscent of the material from bar 13 foll.
The commonalities include thrice repeated notes, the dotted rhythm in descending pitches (though augmented here) and the way bar 44 closes out compared to bar 16.
I often find these little cross references in Mozart once I look closely.

I wanted to get to the end of the exposition but wow, I'm tired.
So I guess I'll end at this point where Mozart has modulated do the dominant of F major.
I hope I didn't miscalculate any chords, got the bar numbers right and others will have more to add for this movement.

I think it would be helpful to take a high-level view of the form that Mozart is using since this relates directly to the tonal structure of the piece. In very simple terms, sonata form for a major key is this (Roman numerals indicate tonal areas):

Exposition: I - V (repeat)
Development: V (with various modulations)
Recapitulation: I

Before Mozart composed a single note, he had this general form in mind.

To drill down a little to the exposition, we get this basic outline:

Principal tonal area (I)
Principal theme(s) concluding with PAC or HC

Transition modulating to the Secondary tonal area (V) concluding with HC

Secondary tonal area (V)
Secondary theme(s) concluding with PAC
Closing Theme(s) concluding with PAC

So for Mozart's K332, the exposition is laid out like this:

Principal tonal area (F)
Principal theme 1 mm 1-12 concluding with PAC
Principal theme 2 mm 13-20 concl. PAC (21-22 are cadential extensions)

Transition mm 23- 40 modulates to C concl. HC

Secondary tonal area (C)
Secondary theme mm 41-56 PAC
Closing theme 1 mm 56-67 HC (mm 67-69 are extension; m70 is linking material)
Closing theme 2 mm 71-86 PAC
Codetta mm 86-93 PAC

Please note that I purposely withheld many interesting details, and the above does not constitute a full phrase analysis. The goal here is just to address the "big picture" of the form/tonal structure, especially since modulation is an item that pianovirus specifically wants to address. This will give us some context to get more granular (which Allazart has begun doing already).

I think a phrase analysis is a logical next step, but we can go wherever you all would like.

Wow this is almost totally over my head but it's the kind of stuff I'm really interested in learning. Great topic!

Amazing! I'm very tired but at least want to start contributing right away, more to come from day to day. It would be great if we would converge to a consensus view from our individual postings over time. Not sure if someone will bother to read and comment on this long post, but it would be very nice! (I also plan to give Harmosis and allazart's posts a more thorough reading)

Harmosis, good idea to approach the topic from different angles: Allazart analyses "bottom-up" (exactly what I thought of), so your top-down view is a good complement. Sorry what are the acronyms PAC and HC?

In this post, I'm just looking at measures 1-41:

-------------------------
Part 1: Summary of tonic centers and modulation techniques (please correct if wrong!):

1-22: F
-> From F to d minor without modulation, just bei using the Leitton c# (is this technique of changing keys called a displacement/Rückung??)
22-28: d minor
-> From d minor to c minor by using the diminished seventh chord B-D-F-Ab which can be heard as V7/9 without basis G in the new key of c minor. But does this chord also have an interpretation in the "old" key of d minor?
29-40: c minor
-> From c minor to c major without any modulation whatsoever! Is it easy to use this in own compositions or is there anything to consider?
41-: c major

-----------------------------------
Part 2: Harmonic analysis:

bars 1-4:



I think my view is a bit different (please challenge or correct!):
bar 1: F, heard as the tonic

bar 2: Now it's F7 which suggests the ear to hear this as a dominant. That's a pretty rare harmonic start isn't it?

bar 3: Bb with 5 in the bass (and sixte ajoutee on beat 3). Is it intended that this is heard ambiguously? I mean "ambiguously" in the following way:
For the tonic F, this bar is a clear subdominant.
However, when we hear F7 as a dominant function, Bb would be the tonic. So, the ambiguity in this bar is that it can be heard either as a tonic or as a subdominant function.

bar 4: this resolves the ambiguity in that C7 is heard (even though the C itself does not appear in this bar). The F on beat 1 in the RH is just an appoggiatura (I mean "Vorhalt" in German, hopefully the right translation?). The F in the bass on beat 1 I would agree as a pedal point (to be honest, I don't know how pedal points fit in with harmonic analysis). In summary, I'd think the harmony for this bar is a dominant 7.

Given that the dominant of bar 4 resolves to the tonic F in beat 1 of bar 5, the tonic center of F is now established.



Sorry, I don't understand this paragraph. It sounds like we have some differences in understanding bars 1-4?

--------------------
Because I'm tired, let me jump directly to bars 22 to 41, because I don't really understand these and have some questions.

General remark for bars 22-37: It seems to me that in these bars, the use of leading tones (Leitton in German) in the RH melody is quite important for making sense of the harmonic progressions in two places (specifically, the c# in bar 22 and the b at the end of bar 30). I haven't tried so far, but I guess it wouldn't work without the leading tones (?).



Sounds good. Until bar 22, we are firmly in F. But in bar 23, we don't hear the d minor as the parallel of F, but rather as the new tonic center (do we?) - even though there is no formal modulation at all. It's just mediated by the use of the c# which is not proper to the F scale and is heard as a leading tone.

After two bars of d minor (as new tonic), bar 25 brings a diminished seventh chord (abbrev. "Dv"). The "natural" Dv chord in a given key/scale builds from the VII of the scale upwards in minor thirds. In d minor it's C#-E-G-Bb. The Dv chord can also be considered a dominant7/9 chord without the basis (in this case, missing A). After two bars of diminished seventh we go back to tonic d minor (with 3 in the bass) in bar 27.

From bar 29 on I'm uncertain and have several questions. In 29 and 30 we have again a diminished seventh chord, but this time NOT the "natural" diminished seventh that comes from piling up minor thirds from the VII (C#). So this chord would NOT be seen as a dominant7/9 relative to d minor without the basis!
So questions regarding this are:
(1) Can I use any arbitrary diminished seventh chord in any inversion, no matter which is my current tonic center?
(2) Does this chord also have an interpretation in the key of d minor (for example, could I use it in d minor without modulating?), or is its use only justified retrospectively after the new tonic center c minor appears in bar 31?

Bars 31 and 32 more precisely: we are in c minor (as new tonic center for a short time) with 3 in the bass. This shows in hindsight that the previous two bars should be heard as the "natural" Dv chord (G missing)-B-D-F-Ab of the key c minor. The leading tone at the end of 30 helps to appreciate c minor as new tonic (?).

Bars 33/34: Ab with 3 in bass: parallel key to the subdominant (relative to c minor as tonic center)

35/36: I don't get this - it sounds like Ab7, but is notated with F# instead of Gb, and I'm sure Mozart has a reason for writing it like this. Moreover, since Ab is parallel to the subdominant, it seems very uncommon to have a minor seventh in a subdominant chord. Help?!
Or is it just what Allazart writes? However, I'm not familiar with augmented sixths chords as a subdominant function. It sounds like a seventh chord, so I'm not convinced yet.

37-41: G7 as dominant sept chord still relative to c minor. Half-close ("Halbschluss") on the dominant.

41: Without modulation, use of C major, instead of c minor, as new tonic center!

That's it for today. I'd be really happy about any kind of feedback or help on my questions.

PAC = Perfect Authentic Cadence
HC = Half Cadence

Anyway, what I was getting at about bars 1-4 is that the shifts in harmony are fundamentally embellishment of a persistent F and are not really structural.
In other words, if you harmonised all 4 bars as bar 1 is harmonised it wouldn't sound wrong, just a whole lot less interesting.
Compare that to the harmonies of bar 23 foll. which are of structural importance.

By the way, there is no real written out pedal-point but the low F on each 1st beat implies a pedal point and it continues even through bar 4 which is dominant harmony (viio6).
In performing on the piano you'd probably pedal in a way that sustains the F or actually sustain the F with 'finger pedaling'.

The subdominant 6/4 in bar three is tonicized by the seventh in the previous bar but it wouldn't often be heard as a new tonal center because:

1) It's too brief
2) The Bb chord is diatonic in F major
3) The chord is in second inversion

The key observation is that everything Mozart chooses in bars 1-4 allows F to remain as the lowest bass note (even in the bar of dominant harmony, he retains the low F).
So all four bars are really emphasizing the F major tonality despite the surface harmonic movement which is only 'skin deep' so to speak.

I hope I was able to explain what I was getting at.



The chord in bar 29 is the leading tone seventh of C minor.
He wants to emphasize the following arrival on C minor harmony so he treats us to two bars of C minor's leading tone seventh.
You can introduce chromatic chords quite freely if you resolve them sensibly.
Mozart did this at the outset with the I7 in bar two resolving on the subdominant 6/4.
The rightness of his move to the c minor leading tone seventh is also supported by the motivic material of bar 29 which is the material of bar 25 moved down a tone.
Note that despite the emphasis on C at bar 31 we never feel firmly in the new tonality until the cadence.



An augmented sixth and a minor seventh are enharmonically equivalent. However A-F is always some kind of sixth and since Ab-F is a major sixth the larger Ab-F# is an augmented sixth.
Note that both tendency tones want to move to G (The Ab down and the F# up) so this chord has a strong predominant function when used as Mozart uses it here.
In contrast, when Ab7 resolves the 7th (Gb) will almost invariably move down to F.
Which is precisely why Mozart here writes F# rather than Gb and we call the chord an augmented sixth.



Yeah, C minor and C major share the same dominant.
Once you arrive in C you've arrived regardless of mode!
You'll see that later in the exposition he introduces Eb again for color.

Also, look at how the Bº7 is approached from a Dm triad - the two chords share two notes (D, F) so it's a pretty smooth transition.






Again, look at how this chord is approached - from the Ab with three common tones. Another interesting thing to notice here is the parallel 5ths between the Gr6 and the V (mm 36-37) - often called "Mozart 5ths."

pianovirus: If it helps your understanding any, you can think of the Gr6 as an altered iv7 chord in 1st inversion:

F--F#
Eb-Eb
C--C
Ab-Ab

The F# exerts a stronger pull towards G.

One thing to remember about Mozart is that he is a master of voice leading. When you see a chord progression that doesn't make sense to you, look at how the chords are approached.

Interesting point Harmosis, about the common tones between the d minor triad and the c minor leading tone seventh.

It's funny, I was listening to Mozart's second quartet and realized he used the exact same approach in starting it as this sonata.

The progressions are the same, the tonic pedal is explicit this time ( a left hand tremolo on E flat) and there is not even a melody to speak of.
As here all four starting bars ground the piece in the tonic while allowing us to hear a variety of tone colours.

Second quartet score:
http://imslp.org/wiki/Piano_Quartets_(Mozart,_Wolfgang_Amadeus)

Performance:
http://www.youtube.com/watch?v=kz01WmRGVxc

Not to distract from the sonata but just found the similarity interesting.

Mozart, IMHO, is one of the easiest composers to analyze. Why? Because, his chords are mostly diatonic, and unlike many Romantic composers, he doesn't stray off the beaten path with many non-diatonic chord changes. Analyzing this piece is very simple, and doesn't need too much thinking.

I cannot analyze the whole piece of music, since it is way too much for me to put on one post. So, I shall proceed with my method of analyzing this piece of music:

The key is in F Major, because it has one flat. Hence, Mozart writes his first chord as an F Major chord. In the second bar, we have an F7 chord, which is the V of the Bb Major chord in the 3rd bar and it serves as a secondary dominant chord. The Bb Major chord is the IV in the key of F Major.In the 4th bar, we have a C7 chord(Or G minor chord, which can easily be substituted for the C7 because the chords share 3 common chord tones - G, Bb, and D. D is the 9th of a C7 if we add the 9th onto the C7, thus C9.) The C7 serves as the V in the key of F Major.

Then for a little while, we just have a series of V-I changes in the key of F Major, until in the 22nd bar on the 4th beat, we have a A7 chord, which modulates to the key of D minor, which is the VI(or the relative minor key) in the key of F Major. Then not long after, we have the C# diminished 7th, which is simply just an A(b9) with the root omitted. In the 29th bar, we have a B diminished 7, which is just a G(b9) with the root omitted, which modulates to the key of C minor. Then we have an Ab chord, which is the VI in the key of C minor. Mozart then takes the Ab chord, and adds a 7th to it, making it Ab7. The G Major chord in the next bar is just a half step below the Ab7 chord, thus making it a good choice for the Ab7 chord to lead to. We then have a series of Ab Major(or C minor chord as substitute, because it shares common chord tones) to G Major changes. Then in the 41st bar, we have the C Major chord, which was preceded by the G7 chord in bar before it(bar 40). This G7 serves as a modulation to the key of C Major.

In the next bar after the C Major chord(bar 42), we have the G7 chord, which is the V of C Major. This chord progression continues until we get to bar 45, which is an F chord, which is the IV in the key of C Major. The for a little while, we have the same kind of changes, G7(V) to C(I) and F Major(IV) to C(I), until we get the G7 modulator to C minor in bar 57 and 58. The key shifts a 4th up to an F minor chord in the next bar, and another 4th up to Bb7, and another 4th up to Eb Major, and then another 4th up to Ab Major. This is just the circle of 4ths changes. Then we have a ii-V-i in the key of C minor. Then we modulate to the key of C Major with the G7(V), and follow up with IV to I, then V(G7) to the VI(A minor, which is the substitute for a C Major chord because it is the relative minor of C Major and has common tones C and E). This V to VI is called a deceptive cadence, when the V goes to anything other than the tonic.

I shall put an end to this harmonic analysis, because I can only go so far with this, and it would probably take all day for me to write out this whole analysis for you. While a musical analysis may take up to a few days, it is possible to make the analysis in only a couple of minutes, that's if you have perfect pitch and you know your chord changes really well. I play jazz, so that's why I know all of this.

Oh, and though I mention the word "modulation", I don't mean key changes. Modulation is almost synonymous with the concept of secondary dominants. Back in Mozart's day, any dominant 7th chord leading to the tonic was considered a modulation and NOT a secondary dominant. But in our days, if a 7th chord leads to a diatonic chord in a certain key, it is considered a secondary dominant. Otherwise if a 7th chord leads to a non-diatonic chord in a different key, it is considered a modulation.

Romantic changes are MUCH MUCH harder to analyze, because they are usually all non-diatonic. Ex: C7, Bb7, C# dim 7, D min, E min, Bb7, F# minor 7th b5, Eb min.

(Key of C Major modulating to the key of Eb Minor)

The deeper context behind the chord changes is what the OP is getting at too, I think.
As a simple example, a cadential 6/4, though consisting of the tonic notes, does not have tonic function but is a rather decorative appoggiatura to the dominant.
So in a sense, one has to look beyond the chord-to-chord note changes to gain insight.
It is in that spirit, I was drawing attention to the purpose of various phrases (as betrayed by pedal points, voice leading, motivic content, etc) and Harmosis was outlining the overall structure of the exposition.

No, it's a Gr6 chord. If Mozart meant Ab7, he would have written Ab C Eb Gb, not Ab C Eb F#. Yes, they sound the same but they aren't the same. Just like the words "two" and "to" sound they same but they are not. The leading tone of G is F#, not Gb.




That's a little misleading to say that. Some theorists of the time would have called a secondary dominant a "modulation" (modified by the words "imperfect" of "half-digressive"), but they still thought of it just as we do today - a brief tonicization and not a general change of tonality.



This statement is hard to believe when the only time you used the word "modulation" prior, you made this statement:



I think we would all welcome your continued participation in this discussion but try not to be so careless.

No, actually I didn't even bother looking at the notes in the score. The only thing that I looked at was the bars, and I used a recording off of YouTube to help me in writing out my harmonic analysis. I have perfect pitch, so I really don't need the score to help me out. I didn't know how Mozart actually wrote it out.


http://en.wikipedia.org/wiki/Secondary_dominant

From wikipedia:

"The concept of the secondary dominant was not recognized in writings on music theory prior to 1939. Before this time, in music of Bach, Mozart, Beethoven, and Brahms, a secondary dominant, along with its chord of resolution, was considered to be a modulation."

Your perfect pitch (very handy at times) will enable you to identify which keys are pressed on the piano, but that's not the whole story. How it's written is significant for the harmonic analysis as I'm sure you're aware. Actually, given the starting chord or note my relative pitch enables me to do exactly the same, but I still need to use the music to fully understand what Mozart is doing.

Well I would advise you to look at the score as it would surely help you in making a more accurate analysis.



Please tell me that you're not really going to Wikipedia for this!

Gottfried Weber (1779-1839), in his Versuch einer geordneten Theorie der Tonsetzkunst (1st published in 1817), most certainly recognized the secondary dominant and gave an example of it. He even refered to other writers' concept of the subject. It was discussed under the overall heading of modulation, but as a certain species of modulation "as a slight allusion to a foreign key, as a transient digression, a mere momentary stepping aside into a foreign realm of tones..."

After the example (I V65 I V65/IV IV V7 I in C), he states that the modulation "is so imperfect that it does not expunge the impression of the key of C, but, on the contrary, one eventually feels himself perfectly at home again in this key."

The term, "secondary dominant" was not used but the concept was clearly understood in the same way that we understand it today.

Frank Shepard used the term "attendant chord" in his book "How To Modulate" (http://books.google.com/books?id=EcUPAAA...vzaFJbcMZOwhbUJ), and he refers to other works describing the concept of secondary dominants.

The term "modulation" seems to have been used in the 18th century sort of like it is in "frequency modulation" rather than to mean "change of key." The meaning of the term has drifted over the years.

Are you saying that most of my analysis is not accurate?

So if I am not correct, how would you describe this harmonic change:

the G7 to the C Major chord

So the key is in F natural. C Major is in F natural but the G7 isn't. So is this a modulation or is it a secondary dominant? From what I know, a secondary dominant is a dominant chord that leads directly to a diatonic chord degree other than the tonic and modulation is changing the key by changing the key signature.

Key signatures may be changed to change key, but that's only for the reading convience. Just playing a secondary dominant is not sufficient to effect a key change (as such a change is understood in more modern texts.) One must generally "neutralize" the original key (see Schoenberg's "Structural Functions of Harmony") and then establish the new key. Normally this takes some time; one must treat the new key-center like it was a tonic.

The expositions of Haydn, Mozart, and Beethoven illustrate various methods of establishing a new toinc.

Please be specific as to the measure(s) and beat(s) your are referring to.

I am not saying that most of your analysis is inaccurate, just some of it. Specifically:

1] The Gr6 vs. Ab7 already addressed.
2] Your analysis of the viiº7 chords in the transition as V7b9 chords. Since the mode is minor here, viiº7 chords are entirely appropriate. Also, there is no sol-do motion (or even the presence of the 5th) so the argument for dom7 chords is weak.
3] Your confused use of the word "modulation" already addressed.

That's about it. Your perfect pitch is obviously a blessing, and it seems to me that you have a pretty good grasp of what's happening here tonally.

Ok, so moving along...

Some other things to notice here in the exposition:

1] Transition (mm 23-40): Sometimes composers (including Mozart) will "sneak in" the transition so it happens almost without us knowing it. In K332, Mozart is making the transition very conspicuous with the sudden shift to d-minor, starting with that unprepared C# note played forte; with high-energy 16th note figuration, all played forte and sforzando. It's like Mozart is saying, "Look, here's the transition; you can't miss it!" It has already been pointed out that the transition moves into c-minor, but the standard modulation would've been to C-major (V). Mozart is playing with people's expectations here by going to v-minor instead of major (he does this a lot). But we still get the half cadence at the end of the transition as expected (I know some of this info overlaps with what Allazart has already stated).

2] Secondary Theme (mm 41-56): The secondary theme in an exposition is often called the "singing" theme because it is usually more lyrical, softer, with less vigor than the principal theme. Even though the principal theme in K332 is quite lyrical, it still has more energy in it than the secondary theme. Compare the 8th-note accompaniment (mm 1-4) of the principal theme 1 with the sparser quarter-note acc. of the secondary theme. Also, after the "sturm und drang" of the transition, the secondary theme, which is in the expected tonal area of V-major, seems even more song-like. The phrase structure of the secondary theme is a parallel double period which, again, emphasizes the song-like character of the theme with its antecedent and consequent phrases.

3] Closing Themes: The first closing theme (mm 56-70 incl. extension), with its reversion back to v-minor and half cadence (notice how measure 66 spells out the Gr6 chord), serves as a kind of "transition" to the second closing theme. The motivic and melodic material of the second closing theme (mm 71-86) is reminiscent of the secondary ("singing") theme; and the rhythmic motives of the last measures (mm 82-86) are similar to first closing theme.