Invention No. 3 F major

2 part invention.

Starts in F maj, goes to C maj, then to G mi, then to D mi, then back to F.

Feedback is greatly appreciated.

Please let me know what you think is good or bad or what i could have done differently.

Thanks,

Dan

Hi

In my opinion, this goes too fast, but it's a question of taste.

I think the theme or subject is not as good as could be. It sounds like a trill, I mean, many intervals of second up and down... That makes the theme a bit plain and..., when the other voice comes in, it's easy to fall in messy spots of sounds. Take a look just at ehe second part of measure 2: in the right hand you have A-G-F-G, and in the left hand F-G-F-A-G-F-E.... The problem is that this happens often.

The character of the motive and the fast tempo makes me feel the whole piece is a bit the same again and again. If you take a look at the score, you see that the subject is present in every measure in one hand or in the other one.

Some chords are notated oddly. How can there be in m. 4 Dm and Gm(m7) (which I don't see) at the same time?

The idea is good, but I think this kind of works are much difficult than one thinks.

-Invention is based on imitative counterpoint (double counterpoint is welcome). So, in this style classic counterpoint rules should be observed.

-That means you have to work on the manipulation of the motives (inversion, retrogradation, duplication, fragmentation, sequence..).

(I would go over the construction and name of chords, too).

Good comments by @Luis Hernández, especially concerning the counterpoint rules. Everything needs to line up as though it were its own line in a sequence, rather than strict canon.
m. 27 (B vs. Bb)
Chord markings in the beginning are way too much... letting the bar speak for itself might be better.
 

I'm not entirely sure if @Luis Hernández is thinking of the same 'classic counterpoint rules' as I am, because he then goes on to say, '[t]hat means you have to work on the manipulation of the motives (inversion, retrogradation, duplication, fragmentation, sequence)'. These thematic operations (I'm not sure what is meant by duplication here: is it augmentation or doubling?) are not integral to counterpoint as such, although they are often used in contrapuntal textures and pieces: that is to say the two concepts operate on different musical levels, although the metonymy is understandable.

To illustrate this, we can talk of sequences with good counterpoint, or we can say that the counterpoint in this stretto combining the subject and its inversion is a bit clumsy. What does counterpoint actually refer to, here? The etymological origin of counterpoint literally means note-against-note; thus, I would take counterpoint to mean a framework which governs (or suggests, really) how melodic parts could be stylistically (tastefully?) combined or even crafted out within an accompanimental texture, as one often finds in, say, Chopin and Schumann; this will involve some consideration of harmony in nearly all (pre-20th c.) styles, including pre-tonal polyphony, so I would take any attempt at driving a wedge between counterpoint and harmony with a pinch of salt. Therefore, thematic operations are of importance in counterpoint only when one chooses to explore how one could combine in various permutations material derived from these these thematic operations with the source theme, another theme or itself - the exact details are trivial; what matters here is the act of combining - while observing good counterpoint: that is to say, while remaining within the bounds of stylistically acceptable voice-leading and dissonance treatment and all the other style guidelines that inform counterpoint proper, some of which approximate rules more closely than others, such as the tendency to steer well clear of overt parallel perfect consonances (which, as a prohibition, first appeared in the writings of the 13th century music theorist Franco of Cologne; yes, you will find numerous parallel fifths in Machaut and even some Du Fay, but it will take many, many more words than I have the patience for to explain why).

But of course, this recherché aesthetic is in itself more proper to some styles than others: Bach is the first to come to mind (but we know that Bach was considered outmoded in his own day); even then, within Bach, we see so many species of contrapuntal writing that do not involve the painstaking labour of finding and rhetorically presenting (this is what distinguishes an excellent piece of counterpoint from the merely good) such arcane contrapuntal techniques of combination as stretti, triple invertible counterpoint, double counterpoint invertible at multiple intervals and the like, even within the WTC. Just take a look at the fugue in E minor from book I and the fugue in B flat minor from book II to see how extreme the contrast between different types of contrapuntal textures can be even within a single composer's style confined to a single genre and instrument.

I suggest that you read (and re-read; it takes time to sink in) a harmony and voice-leading textbook in addition to books on counterpoint if you really want to sink your teeth into writing contrapuntal textures. I 'grew up' with Aldwell and Schachter's Harmony and Voice Leading, but Laitz' The Complete Musician is pretty good, too. Or at least, it should be ...

@Monarcheon Actually, he can't be blamed for having come up with a strict canon; he must have been looking at Bach's invention in the same key (which is one).

Thanks Luis, Monarcheon and John for the feedback. It's really great because how else can you improve!

Sorry about the chord names, i forgot to get rid of a few of them when i changed some things such as in M4. I have just changed the tempo of the file now :)

I actually modeled it off Bach's invention no 1. He has the subject and then uses it in its inverted form throughout. He also uses the augmentation of the first 4 notes throughout especially during sequences where he modulates but also in other areas. His subject is in every bar throughout and i think in his invention 13 he does this too. However i can accept that his subject is more interesting :)

Some questions:
1. I tried to stick to the counterpoint rules as it is 2 voices i made the notes consonant (i think). Haven't written anything in a while so may have forgot some things.
2. Could you please explain why " second part of measure 2: in the right hand you have A-G-F-G, and in the left hand F-G-F-A-G-F-E.... " is messy as if i understand why i can avoid doing it?
3. My understanding of how to write an invention (roughly)
- introduce motive
- repeat motive in lower voice while playing counter motive in upper voice
- play counter motive in lower voice, while this occurs repeat motive perhaps in dominant.
- play counter motive in dominant in upper voice.
- use sequences to modulate to a new key (using alterations to motive)
- circle progressions leading to a cadence
- repeat for how many sections you have in different keys

I understand that of course you can vary this a bit but is that not the rough formula for it? If not, could you please point me in the right direction (a website maybe)?

Thanks again for your help!
 

What I was trying to say is that when G is in the right hand, in the left you have G-F, and with the F .., you have anoher G..

I think it's better to look for other chord tones (and non-chord tones, too).

Sorry, I shouldn't have jumped the gun - I saw @Monarcheon's comment and assumed that it was indeed a strict canon without listening to the piece or looking beyond the first two bars of the score.

There's a lot to address and I don't have the time to cover it all, but we'll start with your third question:

8 hours ago, dmc said:

3. My understanding of how to write an invention (roughly)
- introduce motive
- repeat motive in lower voice while playing counter motive in upper voice
- play counter motive in lower voice, while this occurs repeat motive perhaps in dominant.
- play counter motive in dominant in upper voice.
- use sequences to modulate to a new key (using alterations to motive)
- circle progressions leading to a cadence
- repeat for how many sections you have in different keys

I understand that of course you can vary this a bit but is that not the rough formula for it? If not, could you please point me in the right direction (a website maybe)?

I strongly disagree with the notion of there being any pre-conceived 'formula' for an invention (although it's possible to come up with one); it's not even like the sonata form, by which you can broadly expect any sonata-form movement to fall into five types and measure the specific movement's 'deformations' (not a pejorative term) against a number of generic and conventional norms - if you follow Hepokoski and Darcy's understanding of it, at least, as expounded in Elements of Sonata Theory. The fact that there are only twelve pieces labelled inventions in the entire corpus of Bach's oeuvre makes it difficult to generalise a genre out of them. This is, of course, if we make the (in truth, faulty) assumption that we should treat an invention as a genre to begin with. How would you even qualify it? Form? Only one (E major) is explicitly in rounded binary form, and while some others could be described as being in simple binary or rounded binary form (without repeats), a few others are more discursive and approximate thematic and tonal architectures typical of fugues. Compositional technique? Some are more-or-less strict canons (C minor and F major), although all involve some imitation. But imitation is hardly peculiar to inventions. Perhaps we would do better to remember Bach's purpose for these pieces: not only as didactic pieces for keyboard amateurs in both keyboard performance and composition, but also as compositional exercises for W. F. Bach. So, I think you would benefit from looking at inventions as two-part exercises through which you could get a better sense of how to generate form with limited material.

If this all sounds very vague, it, unfortunately, has to be: the best way to learn is to analyse each and every one of these inventions. But it is better to avoid the idea that you could somehow reduce composition to a recipe and to confuse general articulatory and rhetorical gestures in the style (such as cadences as sequences) for what they reinforce (closure and 'motion', respectively). In other words, you cannot simply rip these out of context, just as you can't put a full stop in the middle of a sentence without confusing the reader. Just try applying your 'recipe' to the next two inventions - C minor and D major - in the set. How well (or not) does it work? I do not mean to demean your observations, some of which are good: that the upper part usually has some primacy over the lower, or that there involves some form of imitation in the exposition, but they are too reductive, and in some instances, inaccurate - the imitation operates on the level of a theme in some cases (E-flat major, B minor) - or misleading - I'm not sure what you mean by 'circle progressions leading to a cadence'; a cadence is defined by a set of progressions, depending on the type of cadence it is (perfect, imperfect, interrupted), and these progressions include more than just the two 'defining' chords (e.g. V-I, iv6-V, V-vi); for example, the typical perfect cadence progression in the Baroque style is I6-ii6-V-I (and, mutatis mutandis, this holds for the minor). Sure, you could see that as I-ii-V-I, I suppose (hence your circle progression?), but such a reading has been inaccurate and inadequate for, oh, I don't know, two-and-a-half centuries now? The actual scale degrees of the bass matter, sometimes even more so than the 'roots' of the harmonies themselves.

Right: on to your piece now. I'll take a look at the first page up to the first 'cadence' in C, and I will make some comments about the general form.

• Bar 1: What harmony do you see this bar in (I presume you see it all as being in the tonic)? The last four semiquavers imply the dominant, actually, even if you might think that is counter-intuitive: sure, the tonic notes fall on the strong sub-beats, but the ear (I don't imply a biological universalism with this term, but rather an appropriately trained and culturally acquainted ear) has a tendency to hear these as appoggiaturas. It's not easy to explain why, but it is due in one part to expectations generated from familiarity with tonal music (i.e. that figure would normally be harmonised by the dominant), and in another because the E leads us to expect continued motion, either to F or to D, even if we interpreted it as falling under the tonic; hence, the bar does not sound 'closed'. I think you recognised this lack of closure, because you harmonise the lower voice E with upper voice G in bar 2. In any case, it's easier to demonstrate this by negation. Look at the opening motive of the C major invention. Change the last note from C to B. Do you hear how much worse the music becomes? The way I see it, there are two alternatives you could try. One, you could change the last four semiquavers to Bb-A-G-F, or you could rewrite the entire bar so it expresses (in essence) tonic and 'dominant', with each harmony occupying two crotchets (of course, the 'dominant' may not even involve the dominant chord, so long as the listener is ready to impute a dominant function to the harmony). You could do this by changing the last two crotchets to, say, G-A-G Bb-A-G-F, and then come in on the A instead of the C at the start of bar 2 (which fills in the third you are currently missing, anyway).
• Furthermore, you need to think about the larger lines in your music. It all sounds very haphazard at the moment: in the first three bars, the outlined motion seems to be from F (b.1) to C'-A'-F' (b.2) and down to C' (b.3). It doesn't make for a good line. Again, just refer back to the C major invention: can you see how elegant it is? Essentially, there is an octave leap from C to C', and then it makes a scalar ascent from C' through D' to E', the arguable Kopfton (a Schenkerian term that is okay to ignore at this stage) of the piece.
• Bar 3: Fourths between outer voices are not consonant in Baroque style; even so, the V4/3 harmony is used only very rarely in Bach, let alone V6/4. In any case, it sounds quite off to have the dominant foregrounded on the strong minim (your understanding, or perhaps Finale's understanding, of bar 3 is inaccurate; in the absence of the fifth, the ear would interpret this as I6 over iii).
• Bar 4: The leap from Bb to G in the upper part is ungainly, and likewise for G-Bb in the lower. Furthermore, the accented augmented fourth, while having an acceptable downward resolution, is too on the nose, and the last crotchet is essentially terrible counterpoint: the harmony here is IV, but you resolve the (in the larger context) dissonant C upwards, and just when we might be persuaded to retroactively reinterpret the harmony of this last crotchet as V/IV (and thus reinterpret the C as consonant, even if the 7-8 the lower voice forms with this C is ugly), you move from C to D. Make up your mind!
• It seems like you lack a game plan for these first four bars. The points of imitation seem to be arbitrarily chosen: why does the motive enter in the lower part in bar 4 on the subdominant? What larger harmonic progression do these four bars express? Where is the point of closure? Assuming that you keep the motive as it is for now, the simplest (and also most elegant solution) would be to change the lower part to enter on the dominant, as the upper voice does in bar 3, and lead on naturally from the Bb to A, thus reaching I6 in bar 5. I'm not saying that you can't have the entry in the subdominant - it can be made to work, but it also requires considerably more wit and thought than is advisable for a simple style exercise.
• Bar 5: I found the repeated unprepared sevenths excessive, and they certainly are unstylistic and unnecessary. I get what you are trying to do with the augmentation of the 'subject head', but it's not even strict, so why have the worst of both worlds?
• Bar 6: This is not the right place to begin the modulation, which sounds forced and quasi-modal (pun intended). The sequence sounds like it's trying to get back to the tonic (i.e. F) for a number of reasons (it begins on the 'subdominant' and the bass outlines a 4-3-2-1 descent), and when you try to use this to modulate, you are essentially sending the listener conflicting signals. If you look at the C major invention, the music closes convincingly on the tonic in bar 3, and it uses this harmony as the pivot chord.
• In any case, the use of four iterations in the sequence is one too many (not that you can be faulted for it: even Mozart succumbs to this pitfall at times). Remember Quantz' maxim: use at most three!
• Bar 9: It is good that you realised that you needed to start moving towards the cadence at this stage. However, this realisation comes across much more on paper than it does in sound. To begin with, the first harmony of this bar, as a point of arrival, should really be in the tonic (root or first inversion), not the mediant. Because the material in this bar is not made distinct from the sequence, it's hard to get a sense of a directed progression towards the cadence in bar 11. Once more, I refer you to the C major invention: at the analogous point (bar 5.5), the inverted motif is fragmented, and only the implicit ascending thirds (G-B A-C etc.) remain in the upper part, whereas the lower part launches into a strongly directed bass progression leading into the cadence . If you reduce the lower part, you get the very elegant G-E-B-C-D (G for the second half of bar 5, E for the first half of bar 6), a linear progression that would have made Palestrina proud (note the I6-IV-V-I cadential progression here!). I'm not saying that the intervening harmonies within the two halves don't count (of course they do), but they nonetheless express a larger breath of a minim's breadth. Do you see how the cadence is the most accelerated portion of the phrase?
• Bar 11: The above is one reason why the cadence doesn't sound as strong as it should, but there are others. The material for a cadence is often conventional (although one could adapt it to reflect the motives in a piece or to create other connections otherwise). However, your repetition of the same figure in the same bar in the upper part detracts from the cadential effect. Furthermore, the bass progression is very weak: yes, you rarely get 2-1 in the bass for a 'perfect' cadence in this style (but harmonised with a vii6, not a V6/4!), but it is always hierarchically weaker than a 5-1 cadence. In fact, even putting the figure into the bass part, changing the first note from C to E and the last note from B to G, would have made this so much better as a cadence.

I like the general tonal architecture of the piece: you close in the dominant for the first and the relative minor in the second (no doubt you would have taken this from Bach, but there is a good sense of key progression in the second section nevertheless, from the dominant through the supertonic to the relative minor). However, the sequence in the third section drones on for far too long: you would need to possess the genius of a Scarlatti, one of the very few composers to use overly long sequences to good effect! In fact, you could safely erase bars 30-35 to no detriment to the flow and proportion of your invention (again, with the necessary registral etc. changes made, even though the material is perhaps still too similar as the sequence as to tire the ear out). Obviously, the above detailed remarks apply to the analogous points of your other sections.

***

If I were you, I would attempt re-writing this piece after having carefully studied a handful of Bach's inventions (those similar to the C major one, preferably). I'm sorry to say this, but the piece is quite unsalvageable, although I hope you would take it positively as a good learning experience. After all, progress arises from failure as much as it does from success.

@JohnBucket

Thanks so much for taking the time to write as much as you did! I spent many hours carefully reading what you wrote and i think i I really gained some insights. I have got my hands on a copy of "Harmony and Voice Leading" which I'll be studying.

I understood most of what you were saying but just some questions:

1. Re bar 3 you say " in the absence of the fifth, the ear would interpret this as I6 over iii ". Does this apply to other chords to or just I6 and iii for example would you head V6 over viio?
2. In Bar 4 you say " Furthermore, the accented augmented fourth, while having an acceptable downward resolution, is too on the nose, " which notes are you referring to?
3. What do you mean by what larger harmonic progression do they represent? Could you please give an example of a harmonic progression they could represent? It seems a bit to me that because of the nature of counterpoint harmonies change constantly throughout a measure. Is it ok to have a  I - V - I progression in opening bars just to establish the key? I imagine any harmonic progression is acceptable as long as there is logic to it? What is typical? What sounds good?
4. Re first 4 bars how do you achieve a point of closure (V-I cadence????) and why is it important to have this?
5. Re: unprepared 7ths in measure 5.  How are they supposed to be prepared? I assume the 7th must be approached by step?  Will do some research on this :)
6.Could you please explain how in C maj invention M3 closes on tonic. I see it ending on an f in the upper voice and a b in the lower voice? I understand what you mean if you begin on the sub dominant of the new key then it will sequence naturally and lead to this new key. Excellent. I like that :)
7. Re: Quantz' maxim does Bach not use in Invention 1 bars 3 and 4 four sequences?
8. Should cadences be PAC or IAC or does it just depend what you want?

Thanks a lot for all your help, really appreciate it. I'm going to rewrite this. Perhaps it will have a new motive or similar motive but I'll just try and do essentially rewrite something similar in style but with your suggestions. Through doing this i hope it is easy to see if I have improved.

Thanks again!
Dan

Dear Dan,

You will have to excuse me if I am brief or sloppy in my responses, but I will try my best to answer your questions.

1. This is, of course, a matter of context. The matter of the fact is that, in this style, iii does not usually appear as an intermediary harmony in the major, and if we make certain reasonable assumptions, as Schenker did and Lerdahl and Jackendoff do, about how we listen to tonal music (again, whether this is culturally acquired - which is more likely - or psychologically-innate is a battle that does not concern us here in the least), such as the hierarchical priority of tonic harmony and the implicit filling in of missing tones, one would tend to parse the sonority as I6 without the root (which we nonetheless fill in with our ear) rather than iii without a fifth. Put it this way: there is incomplete information, and our mind tends to the more likely possibility (I6) over the less likely one (iii) - this despite the fact that the fifth of iii is generated in the overtones (these overtones don't count in harmonic parsing, just as minimal variations in pitch contour or vowel height should not hinder us from understanding another person's spoken English). Put another way, we learn to interpret tonal music within a certain framework of tonal syntax and grammar, and within this system, I6 is privileged over iii. As an analogy, try to think: why did it rain this afternoon? We would think it would be because moisture has evaporated from the ground into the air and has condensed into clouds, and/or because warm and cold air masses had met - in short, all-natural causes. But it may well be because of cloud seeding. The point is that we would tend to attribute our imagined rain to the former over the latter should we lack the additional informational, an announcement by the relevant governmental agency - or, returning to the matter at hand, the fifth of iii - that would have incontrovertibly settled the dispute. Mutatis mutandis, this holds for the minor, too. I'm sure those who actually know a thing or two about psychology and evolutionary biology would be able to put this against a more general backdrop and provide better allegories.
   
   However, that is not to say that it may not sometimes make more sense to interpret this dyad (or other ambiguous cases) as iii or III over I6 or i6. Take a look at the following extract from the D minor invention:
   
   
   
   Leaving aside the second harmonies of each iteration of the sequence (which I have understood as vii7 and vi, because the motif is harmonised at the outset such that we hear the semiquavers falling on the second and third quavers as accented passing notes, although that means that the implied seventh in the 'vi' bar is resolved in the upper voice - but then again, we don't have to be so pedantic and enslaved by dichotomies, and so we could view the B-flat as a 'passing resolution' of the C on the lower part's way to A), we are faced with the question of reading the boxed harmonies as one or the other (unfortunately, harmonic analysis does have a tendency to be binary). In my view, the former is the better reading as it continues the sequence of descending fifths initiated earlier (part of which you can see in this example); this is despite the fact that the presence of the third below these roots would seem to argue for the latter reading. In addition, because the peaks of the upper voice are so prominent (F5 and E5, respectively), one would have to incorporate them into the harmonic analysis, but they are 'unresolved' sevenths in the latter reading - we would either have to mentally prolong the F over the next bar, where it becomes a ninth, or we would have to take an implicit resolution to E for granted (even if the analogue to this doesn't work when this E becomes the next 'seventh'). Of course, in the former reading, these thirds below the root in the lower part become part of an extended neighbour figuration to the tonic (i.e. G-A leads to B-flat in the first case). But then again, it's possible to capture the ambiguity of this harmonic progression by saying that the latter reading is shadowed within the former.
   
   As for the second part of your question, it depends on whom you ask. Schoenberg, I believe, would always see diminished chords on the leading tone as dominant chords without the root, even if the fifth (scale degree 4) or indeed the diminished seventh (flat scale degree 6) is present. Indeed, the two chords perform such similar functions anyway (very much unlike I6 or i6 and iii or III), so this collapse in distinction doesn't matter too much in the larger picture. But I wouldn't conflate the two, and neither would I read F-A, in the key of C major, as anything other than IV, unless it is absolutely important to get the point across that a D is strongly implied. So yes, I6 or i6 and iii or III would be the exception.
2. E5 against Bb3, third crotchet.
3. Again, we need to return to the idea of there being a hierarchy of harmonies in tonal music. It would be extremely tiring, not to mention unilluminating, if we were to treat and understand every single change in harmony as being equal to each other. I will explain this by means of analogy, although I admit that there are pitfalls. The go-to example would be linguistics. I take it that you are a native speaker of English? Take a look at this opening sentence:
   
   'It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season  of  Light,  it  was  the  season  of  Darkness,  it  was  the  spring  of  hope,  it  was  the  winter  of  despair,  we  had  everything before us, we had nothing before us, we were all  going  direct  to  Heaven,  we  were  all  going  direct  the  other way—in short, the period was so far like the present period,  that  some  of  its  noisiest  authorities  insisted  on  its  being  received,  for  good  or  for  evil,  in  the  superlative degree of comparison only.' [Apologies, I have no idea how to use html and so I can't indent this without indenting the entire bullet point.]
   
   A bit of a mouthful, isn't it? Clearly a dépassé style of writing! Of course, if you started off by identifying the parts of speech of each clause and subclause à la:
   
   It (noun) was (copula) the best (superlative) of (preposition) times (noun) [...]
   
   you would get nowhere in reading this sentence, let alone the entire book! Obviously, a larger sense of the picture is needed here: the entire half of the sentence before the em-dash enumerates parallel pairs that illustrate the crux of Dickens' argument:
   
   The noisest authorities of this period (subject noun phrase) insisted on (predicate) this period's being received in superlative degrees of good or evil (object noun phrase).
   
   Makes much more sense now, doesn't it? It is the same with tonal music: although the constituent local parts may become very complex, they tend to be able to be subsumed within a broader framework that allows us to make broader sense of the music:
   
   
   
   
   [correction: V7 at end of bracket should really read V13, even if V7 was used to emphasise the dominant prolongation]
   
   Another example to illustrate this can be taken from the visual arts. Of course, the many pitfalls of comparing a temporal art such as music, to a spatial art such as, well, the visual arts, have been well-recognised in recent years; see Jonathan Sterne's introduction to The Sound Studies Reader for a good overview, if it interests you. But the point is well-depicted nonetheless in this piece of fractal art:
   
   
   
   Clearly, although we can distinguish (even from this level of overview) a common denominator to the entire picture, our eyes are still able to trace the larger outlines of this artwork. Yes, it is more difficult in a temporal art such as music because our working memory has its limits, but we are able to intuitively sense these larger paint-strokes nonetheless, and to articulate them given the proper theoretical training.
   
   On that note, it is interesting to regard tonal music as being fractal in its structure. Although you wouldn't get anything so calculated (and at the same time so unsophisticated) as you may sometimes find in Bartok (such as the recursion of the golden ratio in the division of the first movement of the Music for Strings, Percussion and Celesta) and Ligeti, you could argue, as Schenker did, that the background harmonic–voice-leading structures are often reflected in the middleground and the foreground, although there is the caveat that Schenker's theories are designed to project just such images of unity onto most any tonal music ...
4. But let us return to your question of just what larger harmonic progressions are typically expressed (which was what I said), which ties into this question and number 7. It's not so much to 'establish the key' as it is about prolonging certain harmonies (most often the tonic) and creating something coherent from the larger motions. Think about a car trip. If you keep veering off the main road, it becomes less a journey to your initially-conceived destination than it does a journey that happens to include said destination in the itinerary. In your case, we get from the tonic to the 'dominant', but we then spiral off to the supertonic. It's almost as if you were taking the alternate route at each possibility!
   
   So yes, if your aim is to prolong the tonic, then I-V-I (and its multitudinous variants and elaborations) is your go-to middle-scale progression, whereas I-IV-I (or iv or ii6/5 or vi or Ger10/iv [diminished tenth=inversion of augmented sixth with scale degree #4 in the bass] or some other pre-dominant substitute) is usually reserved for when the piece is about to end. Analogously, for dominant prolongations, you get V-I-V and V-V/V-V (and substitutes); V-ii-V is spotted with some frequency from Chopin onwards.
   
   I should clarify that when I invoked closure, I meant that the opening phrase(s) expounding the motive/theme should end satisfyingly on either the tonic (this is by far the more common one amongst the inventions) or the dominant (see the B minor invention), if only because this is so customary in the style. I suppose you could argue why in the light of rhetoric, although I'm ignorant of just how this would have been applied in the early 18th century (Mattheson is the figure to look at here), but let's take my layman take on it: if my sentences were all run-on, how much harder would this become to read? Or, if this is easier: preferably, you should establish where you are in a story before the narrative can unfold, and closing off the introductory phrase(s) allows you to announce, 'let the action begin'; or, you could compare it to good essay-writing: if you move on to developing your thesis without rounding off your introductory ideas, your readers would be rightly confused! Is this helping? It's difficult to explain in words, so perhaps it'd be best for you to look at the openings of all of the inventions.
   
   Let's return to the C major invention, then. By bar 3, I meant the start of it, not the end (which is why you were confused):
   
   
   
   
   
   Does the reduction above help you visualise and, hopefully, translate what I've been illustrating with words and pictures into aural understanding?
5. There is only one way to prepare dissonances such as sevenths, fourths and seconds in traditional species counterpoint, and that is they need to be suspended from a consonance thus:
   
   
   
   If the dissonance is approached by step, it is best that is a descending passing note or an upper (complete) neighbour note:
   
   
   
   In later styles (or even in Scarlatti, already), you can find sevenths in this double-neighbour configuration:
   
   
   
   Sevenths (with the exception of leading notes) are usually resolved downwards by step, especially dominant sevenths, but even these are exceptionally seen to be resolved upwards with an intervening chromatic scale degree #4:
   
   
   
   Unprepared sevenths, especially the diminished seventh, can be used to good effect. But sticking them any old how is not good counterpoint.
6. See 4.
7. Yes and no; the right hand has the same figuration four times in a row, sure, but the left hand breaks from it in the second minim of bar 4, entering into a larger gesture of a bar's length (unlike the minim-length sequential iteration). If you look at the sequence beginning on bar 15, you will hear that the sequence is broken into iterations of two bars' length rather than one bar or of half-a-bar; also, Bach breaks it where you might expect the third iteration to begin.
   
   This is a good example of an over-long sequence:
   
   But sometimes even Quantz' rule-of-thumb is of no help:
8. The choice of cadence depends on how strong a sense of closure you feel is appropriate given the point in the structure in which you are. It takes some trial-and-error, and sometimes the most logical or predictable choice is not necessarily always the best. Experience and familiarity with tonal music are your best friends here.

Feel free to ask more questions, but I must warn you that I'm not prepared to give you lengthy answers from now on; I see no reason why I should substitute for your H&C and analysis lecturers without being paid for it.

Best wishes,

John

P.S.: I've noticed a few errors and a few points that were quite poorly expressed, and so I have fixed them. If anything remains to be clarified, do ask.

Thanks John,

It was extremely generous of you to write such detailed explanations; I really appreciate you sharing your knowledge! You clearly know a lot! Time to begin reading Mr Aldwell and co. If you were in Australia, I would definitely book a composition lesson with you!

Thanks again mate!