Pachelbel’s Canon

As you might expect, the demise of the IMSLP has put something of a damper on my grandiose plans of analyzing musical works on this blog. Today, however, we’re in luck, as Wikipedia provides all the source material I’ll need for this post.

The context for this is a question, of sorts, posed by a commenter named “funkhauser” (the reason I say “of sorts” will hopefully become clear):

It seems to me that a surprisingly large number of progressions of 8 chords found in say, your favorite piece, have a IV chord as the “middle” chord (i.e., if each chord lasts a quarter note and we are in 4/4 time then the IV chord is the first chord of the second measure). Two familiar examples are Pachel[bel]’s Canon and the beginning of Bach’s Air on the G string. The chord progressions are, roughly:
I – V – vi – iii – IV – I – IV – V
I – iii – vi – vi7 – IV – V7/V – V – V7

(…)However, I can’t think of an 8-chord progression I’ve heard in which the V chord is the middle chord…Imagine putting the V in the place of the IV in Pachelbel’s Canon or Bach’s Air. It would completely change the feel of the piece (and arguably, it would ruin it).

(…)What I’m wondering is: In Westergaard’s theory can we derive the important difference in function between IV and V? And can we derive the fact that the IV should occupy stronger beats and larger time spans than V?

Tisk tisk. It’s obvious that the questioner has not yet managed to throw off the Rameauvian shackles, and is still laboring under the impression that musical passages are constructed by juxtaposing “chords” in time. Well, funkhauser, you’ve come to the right place — disabusing innocent souls of this mistaken notion has become one of my missions in life.

The best way to start, I think, is to take a look at these passages and see what’s actually going on. Here’s how to construct the opening of Pachelbel’s canon:

1. The underlying basic structure is the usual \hat{3} - \hat{2} descent (with the \hat{2} on its way to \hat{1}, of course):

2. These structural lines will be realized as three textural lines, with span pitches assigned as follows:

(When I say “assigned” I technically mean borrowed, of course.)

3. The top two lines will both descend from the upper note to the lower:

4. The A is delayed by a lower neighbor, in familiar fashion:

5. We connect the F# to the A and the D to the F# by step motion. In fact, we’d like to have continuous quarter notes in these two voices, so on beat 3 of m.2 we’ll also elaborate the C# by a lower neighbor passing tone in the top voice (producing functional parallelism alignment with the bass) and borrow a G from the bass for the middle voice:

6. Actually, we’d like to have quarter notes in all three voices, so we elaborate the bass by means of borrowing :

(The A and the F# are of course borrowed from the span pitches of stage 2 above.)

7. Now, since this is supposed to be a canon, we’ll present the voices one by one.

8. Finally, this is how the texture is actually realized, in terms of which instruments play what.

Now, having analyzed the passage, let’s see if we can address funkhauser’s question. The first thing to note is that nowhere in the above derivation sequence is there any mention of “chords” at all. As a matter of fact, I didn’t even bother to check whether the progression claimed by funkhauser

I – V – vi – iii – IV – I – IV – V

is “accurate” or not — so that as I’m typing this, I literally don’t know what the “chords” of this passage are! I It’s important to emphasize this, because I just got through analyzing the passage in precise detail, attributing a specific function to every single note, and I have the passage itself, as well as my analysis of it, firmly entrenched in memory. Indeed, I can’t mentally replay the passage without instantly and simultaneously reconstructing my analysis. And yet — and yet — when it comes to selecting the appropriate Roman numeral for each of these quarter-note simultaneities, I am — at least at this immediate moment — about as clueless as a typical freshman theory student. (Though I do already know the first one will be I and the last one V.)

Having made that point, let me now pause to reflect on what the chords are…Okay, yes, funkhauser has got it “right”; though I suppose there is an ambiguity about beat 3 of m.1, since there are only two distinct pitch-classes in that simultaneity. Come to think of it, the same is true of both “IV” chords in m.2. Oh, and it’s also true of the very first chord!

(Notice how very different this type of thought is from the instinctive, intuitive reasoning that I used to construct the above analysis. Actually, “instinctive, intuitive” is not the correct description; what I meant to say was specifically musical. Whereas what I am doing here, in verifying funkhauser’s chord progression, is the totally abstract (if trivial) mathematical problem of verifying that two finite sets are equal to each other.)

Funkhauser asks about the difference in function of the IV and V chords. What I would like to point out is that there is no “IV chord” at all! The simultaneity on beat 1 of m.2 is just the coincidence of two passing tones, and that on beat 3 is just the coincidence of two neighbors a passing tone and a neighbor. To pick out these chords as fundamental objects in their own right (and as the same fundamental object, no less!), is to carve up musical reality in the wrong way, like putting dolphins in the same category as fish.

Strictly speaking, then, the answer to funkhauser’s question is “mu” — i.e., “your question depends on incorrect assumptions”. The “chords” of harmonic theory are simply not legitimate music-theoretical entities, any more than Earth, Air, Water, and Fire are chemical elements. Yes, these four things do exist, but they don’t play anything like the theoretical role that people once attributed to them. In fact, today we understand that not only are they not fundamental, but they’re not even the same kind of thing: “Earth” is a planet, “air” is a state of matter (gas), “water” is a chemical compound (H2O), and “fire” is a process (combustion).

So it is with “IV”, “V”, and all the rest. Yes, there are collections of notes in musical compositions to which you could give these labels, but to do so is to presuppose the wrong theory of music.

Like Aristotelian chemistry, harmonic theory may not seem obviously wrong until you’ve had considerable experience with the alternative. This explains why I invariably get reactions like “But…but…of course harmonic theory is correct (or useful) — look how ubiquitous progressions like I-IV-V-I are!”

Yes, and the “Four Elements” are also ubiquitous in the natural world.

For the moment, I will leave it as an exercise to come up with the correct analysis (or at least an analysis of the correct type) of the first two measures of the Air from Bach’s Third Orchestral Suite. Here’s a big hint:

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21 Responses to Pachelbel’s Canon

  1. SheetMusicFox has scores that used to be on IMSLP.

  2. funkhauser says:

    [komponisto],

    First of all, I would like to say thanks for devoting a whole page to answer my question. This is the first Schenkerian-style analysis I have found of Pachelbel’s canon, and I’m really glad to finally see one.

    Your derivation of the canon has already answered a number of questions for me. For instance, it seems that the extension of the tonic chord into the first two beats of m.2 (in step 1) is ultimately the source of the scale degree $4 – 1$ motion in the bass in m.2. Indeed, it seems to be the reason why the “IV chord” ends up being there, along with the I chord.

    However, this analysis has also brought with it a new question. Here is what I would like to ask you now: The analysis above was constructed with the canon in mind, that is, knowing what the end result should be. But now that I see this derivation, and agree with it, I would like to know if, in several of the steps you made above, there are reasons that certain operations you carried out are “particularly good”. That is, given the steps of the derivation, what reasons (if any) would Pachelbel have had to make such steps in the first place (okay, so he wasn’t explicitly using Westergaard’s theory but perhaps he intuitively understood elements of it)? To what degree might a composer be “naturally” inclined to compose Pachelbel’s canon specifically? Are there certain aspects of this piece that make it “better” (by which I mean more “natural”, if that makes sense) than others (even others very closely resembling it)? Maybe there aren’t, but that is what I would like to know.

    I can understand why, say, one would want the two top voices to both exhibit stepwise descent–functional parallelism. However, other things, like the initial rhythm of the 3 – 2 – 1 motion (see below), elude me in terms of a “natural” explanation. Do any of the steps ultimately come down to randomness or artistic choice, or can they be explained further?

    My specific questions are:

    1. In step 1, why might one be naturally inclined to let the tonic chord extend through beat 2 of m.2? Why not just extend to the end of m.1, and let scale degrees 2, 5, and 7 occur on beat 1 of m.2? Or some other rhythm?

    2. When you borrow notes for the bass in step 6, why might one be naturally inclined to make the choices you did? That is, why the A on m.1 beat 2, and the F# on m.1 beat 4? Why not the other way around? Why borrow the G on m.2 beat 1? Why not borrow the B? And why borrow the B on m.1 beat 3? Why not the D?

    Now, I will attempt to answer my own questions! I make no guarantee that these explanations are accurate or even plausible…

    1. Extending the tonic chord to m.2 beat 2 will result in a longer prolongation of the same original set of voices, thus increasing the simplicity/cohesion of the piece.

    2. The choice of borrowed notes in the bass provides variety. That is, we already have a D as the first bass note, so choose a B, rather than a D, as the third bass note. Then, since we already have a D and B, choose a G for the fifth note. As for the A and F#, perhaps we choose to have D – A – B – F# because D – A and B – F# are both fifth arpeggios (thus increasing similarity/cohesion).

    My “postulate” of course being that particular blends of similarity and variety are more natural and increase the appeal of a piece.

    I would very much like to hear your explanations (and critiques of mine). Also, I finally bought a copy of ITT, so feel free to direct me to any relevant pages/passages. Thanks again.

    By the way, I suppose my questions above are instances of a more general question:

    The Westergaardian operations seem so powerful and diverse in their production of musical material–I am very interested to know if we can, in general, understand certain subsets of these operations (or certain rhythmic choices, borrowing choices, etc. made when applying them) to be somehow more “natural” than others. Perhaps Westergaard addresses this–I still have a lot to read of the book…

  3. komponisto says:

    (Sorry for the delay in responding; my “day job” has been keeping me occupied recently.)

    Ponder:

    Thanks for the tip. While it doesn’t seem that SheetMusicFox has every score that IMSLP used to have, I was at least able to find the Pachelbel canon.

    Funkhauser:

    First of all, I would like to say thanks for devoting a whole page to answer my question. This is the first Schenkerian-style analysis I have found of Pachelbel’s canon, and I’m really glad to finally see one.

    Well likewise, thank you for the stimulating comments. I think this piece provides a particularly good illustration of what’s wrong with harmonic theory — as is perhaps not surprising, given that it was composed some 40 years before the invention of the latter.

    (This is not a trivial point. Advocates of harmony are obliged to explain why, if the concept of “root progression” is as integral as they claim it is to this music, no one bothered to mention this fact before 1722.)

    I would like to know if, in several of the steps you made above, there are reasons that certain operations you carried out are “particularly good”. That is, given the steps of the derivation, what reasons (if any) would Pachelbel have had to make such steps in the first place (okay, so he wasn’t explicitly using Westergaard’s theory but perhaps he intuitively understood elements of it)? To what degree might a composer be “naturally” inclined to compose Pachelbel’s canon specifically?

    Remember that the steps in the derivation do not necessarily correspond in any chronological sense to the composer’s creative process; what their order represents is the order of conceptual priority assigned to the events by the listener (ITT, p. 63).

    It’s quite plausible, for example, that the first aspect of this piece that Pachelbel thought of was the bass line. You’ll notice that the structure of this line is rather unambiguous (ITT p.64); it’s hard to imagine a substantially different way of deriving the notes from that of the analysis above. Consequently, the process of adding the upper lines may very well have amounted to a process of (re)discovering the structure of the bass line — composition via analysis, as it were.

    Had I composed this piece, my thought process would probably have been something along the following lines. First, I would have decided that I wanted to write a canon in D major for three violins and continuo, about 5 minutes in duration. Then, I would have decided to make it a passacaglia (in addition to being a canon), with the repeating bass line. Something in quadruple meter, moderate tempo, so that two measures would be an appropriate length for each iteration. Now, given that we want the whole thing to end with a \hat{2} - \hat{1} cadence (which in harmonic theory would be referred to with characteristic imprecision as “V-I”), and we want the penultimate measure of the piece to be the second measure of the final iteration, we’re pretty much forced into making each two-measure segment an elaboration of \hat{3}-\hat{2}. (A priori we might also consider \hat{5}-\cdots-\hat{2} or \hat{8}-\cdots-\hat{2}, but these structures require considerable effort to project, because the listener must be prevented from understanding a simpler underlying \hat{3}-\hat{2}; hence the result would be more rigid constraints on our various layers of elaboration, making a 5-minute canon/passacaglia virtually impossible to sustain.)

    It is at this point, then, that we confront your first question:

    1. In step 1, why might one be naturally inclined to let the tonic chord extend through beat 2 of m.2? Why not just extend to the end of m.1, and let scale degrees 2, 5, and 7 occur on beat 1 of m.2? Or some other rhythm?

    As a matter of fact, many other pieces do have different phrase rhythms — the Bach Air is one example. In the present case, I can think of at least two ways to end up with this particular rhythm:

    (1) If you’re starting with the bass line as given, it’s pretty much forced on you.
    (2) I happen to like the interval of a sixth, so I might have come up with the idea of an upper voice descending F#-A (or D-F#) in quarter notes; this requires six of them, of course.

    . When you borrow notes for the bass in step 6, why might one be naturally inclined to make the choices you did? That is, why the A on m.1 beat 2, and the F# on m.1 beat 4? Why not the other way around? Why borrow the G on m.2 beat 1? Why not borrow the B? And why borrow the B on m.1 beat 3? Why not the D?

    These alternatives would have made the line less interesting. See ITT, sec. 4.2 (in particular pp. 68-69).

  4. funkhauser says:

    Okay, thank you! I read the section on interest you mention–it answered several of my questions. And understanding that the steps above represent conceptual priority clarifies things quite a bit.

    By the way, what program did you use to generate the images of the staves and notes in your explanation above (or on any page of this blog, for that matter)?

  5. komponisto says:

    what program did you use to generate the images of the staves and notes in your explanation above (or on any page of this blog, for that matter)?

    GNU Lilypond.

  6. […] Bach: Air from Orchestral Suite No. 3, mm.1-2 Recall that in a previous post I challenged readers to analyze the first two measures of the Air from Bach’s Orchestral […]

  7. funkhauser says:

    [komponisto],

    I’ve been thinking more about your analysis of this passage and while I find it reasonable and elegant, I have come to a somewhat major point of disagreement. Namely, you derive scale degrees 4^ and 6^ on beat 1 of measure 2 as arising from passing tones in the arpeggiated descending interval of a sixth from 1^ to 3^ and from 3^ to 5^. However, to me beat 1 of measure 2 sounds like a point of arrival (albeit a temporary one), rather than like a collection of foreground passing tones. And furthermore, regardless of discrepancies of hearing, the fact is that these notes occupy a metrically very strong position. Why would Pachelbel put mere foreground passing tones on such a strong beat? In the treatments of counterpoint that I’m familiar with, passing tones are explained as being fleeting elaborations which should typically occupy weak beats. So if these notes come from the foreground elaboration you describe, then shouldn’t they occupy weak beats? Shouldn’t the goal tones 3^ and 5^ occupy the downbeat of measure 2, if anything?

    Given these considerations, I don’t think that what we have on beat 1 of measure 2 is a collection of foreground passing tones. I also feel the same way about beat 3 of measure 1. Now, I’m not suggesting that we must fall back on Rameauvian ideas to better explain the structure of this passage. Quite to the contrary. What I propose is that the introduction of the 4^ and 6^ simply occupies a position closer to the background. I would argue that the 4^ and 6^ arise as large scale neighbor tones to the 3^ and the 5^ in the original tonic triad (the tonic triad that spans the whole two measures).

    In particular, I propose that there is a neighbor structure 5^ – 6^ -5^, with the 6^ introduced on beat 3 of measure 1, and then resolved back to 5^ on beat 4 of measure 2. Similarly, I propose a neighbor structure 3^ – 4^ – 3^, with the 4^ introduced on beat 1 of measure 2, and resolved on beat 1 of measure 3. Of course this would leave a 4^ on beat 4 of measure 2, but I don’t see this as a problem since for example the last beat of 4^ can be replaced by a 5^ through borrowing.

    Now what about the notes of measure 1 beat 2, measure 1 beat 4, and measure 2 beat 2? I believe that in measure 1 the notes on beats 2 and 4 arise as passing tones, and the notes in measure 2 beat 2 arise as neighbor tones (to the larger scale neighbor tones). As for the notes of measure 2 beat 4, I would say that they arise possibly from the way you originally described, although I have a couple of different thoughts about them as well (which I won’t get into here for the sake of brevity).

    This analysis, I believe, more accurately depicts the relationships among the notes (at least as I hear them) and more fully respects metrical significance. I have come to similar conclusions about the first two measures of the Bach Air, as well as the first two measures of the Bach Prelude 21 of WTC I, which hopefully I can provide more detail on later. I regret that I am not able to graphically demonstrate my analysis, but I hope that my above explanation is intelligible. I welcome your thoughts.

  8. Great article. I never get tired of Pachelbel’s Canon chord progression… I – V – vi – iii – IV – I – IV – V

  9. Dacyn says:

    Sorry if these questions have obvious answers and I just did not understand, but:

    1. What exactly is the difference between the Schenkerian concept of “2-hat” and the harmonic-theory concept of “the V chord of the final cadence”? Specifically, does it have something to do with Schenkerians thinking that 2 is somehow the “most important note” of the chord? Because I haven’t seen any evidence for that claim.

    For example, it seems that harmonic theory predicts that the last chord will contain some of the notes 5,7,2,4, and this prediction seems to be borne out in practice. On the face of it, this seems like a success of the harmonic theory point of view. On the other hand, Schenkerian analysis seems to predict that the last chord will contain a 2, but doesn’t seem to directly make a prediction about any other notes. This seems wrong on three counts; first of all, I think the last chord does not always contain a 2, e.g. sometimes it may just be 5 and 7, secondly, even if there is a 2 it may not be the most important note (as seems true in this case), and thirdly, the hypothesis that the last chord contains a 2 would be consistent with the last chord being a (ii) chord, but this is rarely observed.

    So in what sense is the Schenkerian prediction superior to the harmonic-theory prediction here? Or am I misunderstanding, does Schenkerian analysis really give the same prediction as harmonic theory in this situation, and it is only in broader contexts where the predictions differ? If so, why is “2-hat” named after 2 rather than just being called (V)?

    Similarly, it is not clear to me what the difference between “3-hat” and “an important (I) chord before the final cadence” is.

    2. You say that your derivation doesn’t rely on the concept of chords, but I am not sure how “elaborating the bass by means of borrowing” is different from “putting in bass notes that create chords”. Can you give an example where these would be different?

    In conclusion, I’m not really sure what these “harmonic theory” concepts that you don’t find useful are. All the basic concepts like chords and (I)-(V)-(I) progressions seem to be useful and used in your analysis as well, albeit implicitly. Are there some more exotic predictions that harmonic theorists make that you don’t agree with? Or are you just saying that the concept of chords is not sufficient to completely understand music theory (and as far as I know, few people would claim that it is)?

    Anyway, I have just gotten here so if these questions are best answered by links to other posts, let me know.

    • Dacyn says:

      One question I forgot:

      3. It seems that when you start looking at the variants of the original theme that occur in Pachelbel’s canon, the main thing that stays invariant is the chords, while the two top lines you’ve emphasized here disappear and get replaced by other ways of realizing the same chords. How do you explain this without talking about chords as fundamental objects?

      • komponisto says:

        3. It seems that when you start looking at the variants of the original theme that occur in Pachelbel’s canon, the main thing that stays invariant is the chords, while the two top lines you’ve emphasized here disappear and get replaced by other ways of realizing the same chords. How do you explain this without talking about chords as fundamental objects?

        The textural lines change, but the structural lines (at least at a sufficiently high level of structure) remain the same. A common misunderstanding (which I think you’re evincing here) is that lines make sense only in pitch space, so that abstracting to pitch-class space requires invoking chords. In fact, however, a line in pitch-class space makes perfect sense as a concept (and is implicit in the notion of register transfer, ubiquitous in Schenkerian theory).

      • Dacyn says:

        So if I oversimplify what you are saying, it is basically that the thing that is staying the same is a set of pitch-classes, rather than a chord. Is that basically right? (Of course a chord is an example of a set of pitch-classes, but there are other examples as well so I’ll grant that there is a difference between these two ways of looking at things.)

        Or are you making the different claim that we should understand the voices as being determined by pitch-classes rather than by what instruments are actually playing them? So e.g. if we have a (I) chord with 5 in the middle voice and 3 in the upper voice, and then in a variation we have 5 in the upper voice and 3 in the middle voice, then we should think of the upper voice as being isomorphic with the middle voice of the old version, rather than the upper voice of the old version? I think this would run into problems if the mapping between voices wasn’t constant. To be clear, I am not trying to ask about empirical predictions, but I am asking: are these distinctions something that Schenkerian analysis would care about? And if so, why?

    • komponisto says:

      (N.B.: I’m the author; I’m in the process of an identity merge, changing my pseudonym on this blog to match the one I’ve used elsewhere.)

      What exactly is the difference between the Schenkerian concept of “2-hat” and the harmonic-theory concept of “the V chord of the final cadence”?

      “2^” denotes the second degree of the scale (the supertonic), as an individual tone. Schenker uses the circumflex notation only for tones of the fundamental line (Urlinie); today’s theorists use it in all contexts (so as to distinguish individual tones from the “chords” of harmonic theory, or the Stufen of Schenkerian theory).

      As an Urlinie-tone, 2^ must occur within a dominant (V) Stufe (this follows from the definition of the Ursatz). This is how the Schenkerian concept of 2^ (superficially) links up with the harmonic concept of “cadential V”. (Superficially, because the resemblance of Schenkerian Stufen to the “chords” of harmonic theory is only superficial.)

      So in what sense is the Schenkerian prediction superior to the harmonic-theory prediction here? Or am I misunderstanding

      You are indeed misunderstanding, and on a fundamental level. Schenkerian theory is not about making “predictions”, certainly not in the naïve sense you intend here. It’s not that kind of a “theory”. What it is is an interpretive paradigm with a significant normative/didactic component. Its “predictions” are roughly of the form “if you are the kind of listener that I (the Schenkerian theorist) think you ought to be, then you will interpret this musical data in this way…”. Thus it stands or falls on the kinds of grounds that aesthetic theories, not empirical theories, stand or fall. It would be more aptly compared to something like a computer programming language than to a scientific theory in the popular (Popperian) sense.

      Or are you just saying that the concept of chords is not sufficient to completely understand music theory?

      Pursuant to the above, what I’m saying is that the concept of chords is an ugly way to understand music theory. I don’t believe it’s impossible to understand music in terms of chords; but when I contemplate trying to do so, I get the same kind of feeling of revulsion as when I imagine doing arithmetic using Roman numerals instead of decimal digits, or programming a sophisticated web application in Cobol rather than Python. Only worse.

      My self of a decade ago had a harder time expressing this, and maybe I’m still not doing so perfectly; but hopefully this clarifies things somewhat.

      • Dacyn says:

        OK, so you seem to be agreeing that the “supertonic” part of 2^ is not necessarily the most important part. So I’m still confused as to why it is worth it to call it 2^, rather than V as is done in harmonic theory. Also, is there a canonical list of what kinds of “Stufe” are appropriate for higher parts of the Urlinie?

        I’m not sure I really understand the difference in practical terms between what you’re calling “aesthetic” and “empirical” theories… OK so replace my word “prediction” with “way of understanding”, in what way is the Schenkerian way of looking at things better than the harmonic-theory way of looking at things?

        To rephrase one of my other questions in terms of “aesthetics”, if the final V-I cadence did not have a supertonic in it would a Schenkerian analyst still call it 2^? I guess the real point is that I don’t see any motivation for distinguishing between V-I cadences that do and don’t have supertonics in them, so either a “yes” or a “no” answer seems like it creates problems for the Schenkerian theory.

      • komponisto says:

        OK, so you seem to be agreeing that the “supertonic” part of 2^ is not necessarily the most important part.

        Not exactly; 2^ specifically denotes the supertonic; it’s not another name for the “V chord” of harmonic theory. So “the ‘supertonic’ part of 2^” doesn’t strictly make sense.

        In this context, “2^” has an additional connotation beyond its denotation of the supertonic degree: it refers to the specific 2^ that occurs as the penultimate element of the Urlinie.

        For Schenker, the relevant effect of a (structurally) final cadence is not just the “progression” of a “V chord” to a “I chord”; it is specifically the resolution of the 2^ of the Urlinie to 1^, thus fulfilling what he called “the law of the passing tone”. In this sense, the 2^ is “most important”.

        However, this effect depends on the transition of Stufe from V to I, which in turn depends on the progression of the bass voice (from 5^ to 1^); this is, in some sense, “how we know” that 2^ of the upper voice has truly resolved to 1^. As Schenker says in §3 of Free Composition:

        “The combination of fundamental line and bass arpeggiation constitutes a unity…Neither the fundamental line nor the bass arpeggiation can stand alone. Only when acting together, when unified in a contrapuntal structure, do they produce art.”

        if the final V-I cadence did not have a supertonic in it would a Schenkerian analyst still call it 2^

        Schenkerian theory admits the concept of substitution:

        “…[A] tone which is not part of the fundamental line can substitute for a fundamental-line tone…Such a substitution…is easily recognizable as a substitution because the counterpointing bass arpeggiation clearly indicates the actual tone of the fundamental line, even though it is hidden.” (Free Composition, §145).

      • Dacyn says:

        So if I understand what you are saying, the Schenkerian concept of 2^ and the harmonic theory concept of “the V chord in the final cadence” are the same in the sense that they would apply in exactly the same situations, but are different in the sense that they are conceptualized differently, with the Schenkerian theory saying that the supertonic is the most important note of the chord, while the harmonic theory (to the degree that it says anything about a most important note) says that the root is the most important note. Both notes may be implied rather than directly present, so from that perspective I guess the theories are on an equal footing. I have to say the harmonic theory version seems more intuitive to me (maybe because I naturally listen for bass lines), but I guess at this point there’s not very much that one could say to objectively support either position.

        (N.B.: I’m the author; I’m in the process of an identity merge, changing my pseudonym on this blog to match the one I’ve used elsewhere.)

        Yeah, I got onto this website from one of your links from LessWrong. By the way, you never responded to my comments about the epistemology of mathematics there 😛

      • komponisto says:

        So if I understand what you are saying, the Schenkerian concept of 2^ and the harmonic theory concept of “the V chord in the final cadence” are the same in the sense that they would apply in exactly the same situations

        Well, no, they don’t apply in exactly the same situations, because they don’t denote the same effect. For one thing, harmonic theory doesn’t have the concept of structural levels (which is why we’re talking about “the final cadence” as a vague way of gesturing to the concept of “the highest-level structural cadence”, even though those aren’t by any means the same!).

        Schenkerian theory saying that the supertonic is the most important note of the chord

        The problem here is that asking about “the most important note of the chord” in the first place assumes harmonic theory, i.e. that “the chord” is the primary object of interest.

        Schenkerian theory does not consist of (or even involve) a mapping from chords to “most important” notes. It structures the data of music in a different way altogether. It views music as being made of lines. Lines, rather than chords, are the entities that contain notes. To the extent chords are of interest at all, they are viewed as the epiphenomenal results of the interaction of lines.

        (As I often like to point out, this is in fact the kind of model that is implicit in staff notation itself.)

        Schenkerian theory does involve claims about the hierarchical priority of lines. Most famously perhaps, the concept of the fundamental structure (Ursatz) subordinates inner voices to the (conceptual) outer voices, viz. the fundamental line (Urlinie) and the bass arpeggiation (Baßbrechung).

        I naturally listen for bass lines

        As you should, according to Schenker. Indeed in a footnote to the very passage I quoted earlier (from §3 of Free Composition), he writes:

        “Those new to the concept of musical background tend to turn their attention exclusively to the fundamental line, because it is the upper voice. They all too hastily accept any tone series as the fundamental line, without determining whether it rests securely upon the counterpoint of the lower voice. Thus they often meet with disappointment…”

      • Dacyn says:

        – “Well, no, they don’t apply in exactly the same situations, because they don’t denote the same effect.”

        Maybe we are meaning different things by “apply in exactly the same situations”, is there any situation in which there is a 2^ but not a V-I cadence (on the highest structural level) or vice-versa? I agree that if both are present, they are not necessarily the same object.

        After reading your comment, I think there are two different issues at stake here:

        A. Are the lines 2-1 and 5-1 more important, or are the simultaneities 5-7-2(-4) and 1-3-5 more important?
        B. Is the line 2-1 more important, or is the line 5-1 more important?

        An issue with the Schenkerian notation seems to be that it makes a bigger deal out of (B) than (A), but you seem to be saying (A) is more important and (B) doesn’t matter too much.

        In which case, I guess I can rephrase my initial question as follows: the Schenkerian picture 2-1, 5-1 doesn’t seem to say anything about the notes 7,4,3,5, which are both described by the harmonic theory and seem to actually appear in music. So does this mean the Schenkerian picture is missing something? I mean, I guess you can start talking about the lines 4-3 and 7-1, but then we run into issues like variations 7-5, 5-5, and 2-3…

      • komponisto says:

        (A) is a much bigger deal than (B). (B) is basically an esoteric point compared to (A).

        In which case, I guess I can rephrase my initial question as follows: the Schenkerian picture 2-1, 5-1 doesn’t seem to say anything about the notes 7,4,3,5, which are both described by the harmonic theory and seem to actually appear in music

        Here you seem to be confusing “the Schenkerian picture” with the Ursatz specifically. The latter technically involves only outer voices (and Stufen); but Schenkerian analysis is concerned with the entire complex of structural levels, from the Ursatz to the surface. And the texture (number and disposition of lines) may look different at different levels.

        As one expert has noted, “Schenker’s theory is not just a hierarchy of notes and rhythms, but a hierarchy of lines“.

      • Dacyn says:

        I think you are misunderstanding me, when I say “the Schenkerian picture 2-1, 5-1” I do not mean that Schenkerian analysis says that those are the only lines that exist, just that those are the only lines it is worth saying anything definite about across the class of all final/structural cadences regardless of what piece we are looking at. The other lines you can’t really say anything too definite about, because there are too many variations (as far as I understand). But if you look at simultaneities instead of lines, then it seems that you can say something more definite: first there will be 5-7-2(-4), then there will be 1-3-5. Of course there are variations on this as well, but there seem to be fewer of them…

      • komponisto says:

        You probably want to have a look at section 8.2 of Westergaard (on cadences), which contains a nice “derivation” of the “results” about verticalities that you’re talking about.

        More generally, I suspect you would find that a greater familiarity with species counterpoint (which, it should be recalled, is an integral component of Schenkerian theory, and the subject of chapters 4-6 of Westergaard) would address or dissolve the concerns motivating this line of questioning, so to speak.

        Indeed, I’m tempted to paraphrase you as asking “what part of Schenkerian theory deals with the vertical relationships between simultaneous lines?”, to which the answer is, of course, “species counterpoint”.

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