In the last of his 15 Sinfonias (aka Three-Part Inventions), Bach uses quite a daring approach to the tonic chord. Occurring in a sequential progression by downward thirds, it can be described most plainly as a V7-iii9-I cadence.

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Analyzing a similar example from the WTC, Walter Piston, or rather his editor DeVoto, interprets the A as an anticipation, resolving with change of octave to the A in the alto (Harmony, 4th ed., p.128) which seems rather far-fetched.

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Have more recent theorists got anything of sense to say about the idiom? And are there precedents from the 17th century, before the congealment of functional harmony?

  • Thanks for all the insightful answers! I somehow didn't see them when I joined Music.SE and posted this question four years ago. – Mirlan Jun 20 at 3:58
  • And by the way, if anyone's interested in the history, I think this is a peculiarly "German" kind of figuration: I've found comparable examples in Hassler's toccatas, but none in French or Italian Baroque music. – Mirlan Jun 20 at 3:59

It's easy to see a figure consisting of a stack of thirds, and think that this represents the chord. But this isn't necessarily the case. The figures here could more plainly be seen as merely ornamenting a central pitch with upper and lower auxiliary tones, a third above and below. It may seem odd to think of notes a third away as neighbors, since that term is technically used only for stepwise motion, but the key is that this is a rapid figuration, and it is always the first note of each group which is repeated (thrice), and is always on the beat. The other notes act primarily as unaccented decorations or ornaments, and would certainly be seen as non-functional if they were only a step away, instead of a third.

I'm trying to be careful about calling them "non-functional" rather than "non-harmonic" because they notes may indeed be harmonic (and usually are, though not always), but they are less important in determining the harmony. IOW, the harmony does not depend on these ornaments.

Thus I would progressively simplify the figure as follows (again, I'm using N and P -- Neighbor and Passing tones, in a very loose, and technically wrong manner): enter image description here

Based on this reading, I've prepared a series of four reductions (separated by double bars) which demonstrates my analysis. Note that I've also included the measure preceding the three that you posted, as it helps to indicate the linear nature of this progression. I've also converted the meter to a more straightforward 3/4 for no particular reason. You'll probably have to right-click -> View Image in order to read the thing at full size. enter image description here

  1. For the most part, all I've done here is replace the elaborate figure with a simpler dotted figure that emphasizes the functional accented note, and the falling thirds motif. You'll note the clash in the penultimate beat between the stepwise A-G-F♯ in the R.H. and the skipwise (A-F♯-D) in the L.H.

  2. In the R.H., I've removed the repeated-octave notes in the third beat of each bar (incorporating them into the previous note an octave lower), and I've lengthened the accented notes where they bleed into the next chord. With those actions done, it was natural to view the R.H. as consisting of two voices: the second imitates the first a beat later and a fifth lower, and the remaining sixteenth notes act as upward resolutions for implied suspended sevenths. In the L.H. I've merely removed the unaccented tones, as if they were passing tones. This reveals the passage progresses in fifths, instead of thirds.

  3. In the R.H. I've removed the unaccented sixteenth notes (which were doubling the bass notes), and I've made the previously implied 7-6 suspensions explicit. In the L.H., I've also gotten rid of the repeated octaves. This represents what I see as the pure 3-voice counterpoint that underlies this progression.

  4. I've further removed the alto voice, and the third beat of the bass (which is always the fifth of the following note) in order to emphasize the linear nature of the progression.

My harmonic analysis comes from levels 2 and 3, and allocates one chord per beat. I'll leave off the implied sevenths from the chord symbols, since they aren't actually written (except for the final A7).

G Em/G C♯dim | F♯m D/F♯ Bm | Em C♯dim/E A7 | D

IV (ii6) vii | iii (I6) vi | ii (vii6) V7 | I

You'll notice that within each bar, the chords are dropping by a third each beat, then raising a fourth across the bar lines. If you were to remove the intermediate chords on beat two (in parenthesis) -- which are all first inversion chords that retain the same bass note as the previous chord -- then you're left with a simple circle of fifths progression. If you further remove the third beat chords, you're left with a simple descending scale, as in reduction 4:

IV - iii - ii - I

From the harmonic analysis, we can also see why the confusion arises: most of the time, when the figure occurs, it is ornamenting the third of the current chord. For example, in the first measure, in the first beat, it occurs on the B, which is the third of the G chord. On the second beat it occurs (in the bass) on the third of the E minor chord (which happens to be in first inversion, so the ornamenting notes a third below end up conveniently playing the root. On the third beat, the figure occurs on the E in the R.H. (which is the third of the C♯ diminished). But in that third beat, it also occurs on the chord root in the L.H., so the decorating notes end up being a third beneath the root. For the most part, this note a third beneath the root doesn't interrupt the flow of the harmony too much (it actually helps to stabilize the diminished chord) but in the penultimate measure, it does slightly weaken the dominant harmony there, by having that F♯ that clashes with the rest of the A7 chord.

  • To clarify for other readers, you're suggesting that the tertian figurations are mostly not actually functional chord tones, but ornamentations that prolong chord tones? In your 2nd level analysis, you incorporate the V7 but drop it in level 4. Are you arguing that the passage in question resolves in a Plagal cadence through passing-tone motion? – jjmusicnotes Jan 5 '15 at 5:31
  • @jjmusicnotes A sort-of-yes to the first question, with the caveat that, while the chords don't necessarily depend on those notes, in most cases, the thirds do end up belonging to the chords -- except for the lower thirds in the last beat of the bass. As to the 2nd question, I hadn't thought of it as a plagal cadence. The overall piece is in Bm, but in the previous 4 bars, there's a Bm-E-A-D Co5's progression, which this section extends and accelerates, modulating to D. At a very high level, I see Co5 prog's as just a decorated form of a scale: Bm-(E)-A-(D)-G-(C#dim)-F#m-(Bm)-Em-(A7)-D. – Caleb Hines Jan 5 '15 at 6:28

If I may offer a different analysis of the excerpt in question, it appears to me that the progression moves as thus:

I6/3 - ii - V6/5 - I

Intermediary arpeggiations are upper-tertian embellishment that essentially just serves as smooth harmonic leading between chords. This is a very common progression for the period and a very common progression for JS Bach. A iii9 is such a weak chord that, as others have mentioned, its use would be negligible, and it certainly wouldn't stand up in a background Schenkerian Analysis.

Having played through all of JS Bach's 371 Harmonized Chorales, I can tell you that I cannot recall a time where he ever used a iii chord in a cadence, or even as part of an anticipation. Anticipation suggests non-chord tones and/or dissonance. A iii chord is merely an upper-tertian extension of a I chord, and therefore is glossed over aurally. I think it would be very difficult to distinguish a iii as a plausible anticipation aurally.

Did JS Bach use iii chords at all? Absolutely. Sometimes to prolong I, sometimes to serve different connective functions, but I would say never at a cadence; not unless it was re-purposed as a secondary dominant/leading tone chord

As you might already know, mediant and chromatic mediant relationships did not become popularly in music until at least Beethoven (moving from I to bVI in developments! Of course, you can reason that he went to the relative major of the subdominant minor - two ways of getting to the same place.) If someone has an earlier example than Beethoven, feel free to share!

I mention this because it is important to remember the time period. During the 17th century, composers were not in the habit of thinking vertically. Yes, they were conscious of vertical intervals (you must be to resolve your counterpoint correctly), however, composers did not think in vertical blocks like so many composers do today. By and large, composers thought linearly, and creating their harmonic progressions as embellishments of a 1st-Species framework (Hint: this is one reason why Schenkerian analysis works in this period!). Apart from the larger, intended harmonic motion (note the first beat of each measure in the example), the motion in between facilitates the prolongation of chord tones through upper-tertian harmony, and as such, should be seen moreover as an extension of those larger, horizontal harmonic shapes.

If you took this thought literally, the passage is reduced to a progression of I6/3-ii-I which, in Schenkerian analysis, the ii would be seen as a passing tone between F# and D, with the whole passage resulting in a prolongation of I.

As this is only an excerpt and we are not analyzing the entire piece or movement, it is difficult to determine exactly how important any of the chord outlines are in the excerpt. Because of this reasoning and the given excerpt, I am not being strict in my thoughts concerning harmonic downbeats, and, as I mentioned above, inserted the V6/5 in my analysis above.


I'm inclined to agree with @dwn. The relative major here isn't particularly tonicised: it is just emphasised a bit by melodic motion through the upper auxiliary, and a slowdown of harmonic rhythm. It is otherwise just a natural conclusion to the preceding sequence. It's the next 2 bars that start to look a bit more like a conventional cadence, only Bach reinterprets D in the second half of the 2nd bar as the seventh of a dominant 7th on E and moves to A, and thence to E, using the figuration of the last bar of your example.

The nature of the piece is very much "toccata": Bach isn't going to make any cadences before the end that aren't undercut in some fashion, and the easiest way to end a sequence without stopping is just to change the figuration (and harmonic rhythm in this case), and start a new sequence.

As for the WTC example, I think it is proper to say that the entire bass figuration after the tie is an anticipation. Look at the outline it takes up to and including the low F: E-A-C-F. Then take a look at how he delays the resolution of the suspension over the F to emphasise the IV7 sonority.

The A in specific is an anticipation of a note that isn't there: if he had continued the melodic sequence (up a 3rd, down a 5th) in the bass, the note immediately following F would have been A. That, however, wouldn't have confirmed F as the root as well as C does, and wouldn't have allowed the ascending scale segment to set up ii6 (which is anticipated on the 4th beat of the bar). The setup in the second half of the bar, as he has it, can be read as IV7, vi6, ii, vi 6-4 (and thence ii6), and also more simply as an elaboration with passing notes of IV7 - the ambiguity, I think, is deliberate, but would have been somewhat broken if he used A in the bass after the low F. There's even a little ambiguity as to whether the D [semiquaver] in the descant [in the figuration] on the 4th* [3rd] beat is a neighbouring* [passing] tone or a resolution of the suspension, which throws the real weight of resolution onto the start of the next bar* [beat].

Now here's the interesting notion: what happens immediately after the low F is suggesting a harmonic continuation of the "up a 3rd, down a fifth" movement with its suggestion of vi and ii. Notice too that there is a pattern working through the entire bar, that of anticipating the following harmony by a variety of means. I haven't analysed the entire prelude, but I would not be surprised to see that pattern showing up a lot.

The A in question is thus something of an anticipation of a "pensato", and yeah, I think Bach was sneaky enough to do something like that. I don't think there's any question that the harmony in the first half of the measure is the dominant 7th on C: the A in question doesn't have enough weight (unprepared, on the 4th [semi]quaver of the bar* [2nd beat]) to become a root.

The main point to take away from all this, I think, is that Baroque composers, including Bach, didn't necessarily see all the voice leading as a figuring of the bass: a lot of notes (even in the bass) would be considered non-harmonic tones, and those could be used both freely and motivically, as was done here. The A in question can be seen as a kind of nota cambiata that helps to trace a broken chord on F and anticipates the following harmony through melodic means.

Edit: The last sentence of the fourth paragraph (2nd after the break) should read "There's even a little ambiguity as to whether the D semiquaver in the descant in the figuration on the 3rd beat is a passing tone or a resolution of the suspension, which throws the real weight of resolution onto the start of the next beat."

The last sentence of the 6th paragraph should read "I don't think there's any question that the harmony in the first half of the measure is the dominant 7th on C: the A in question doesn't have enough weight (unprepared, on the 4th semiquaver of the 2nd beat) to become a root."

This is the problem of doing an analysis of this sort in the wee hours of the morning: I become innumerate. <wry grin>


I'm not well versed, but to me it just looks like a sort of not-yet-rooted tonic, with the G (last treble note in the 2nd measure) being a neighbor tone. I'd agree that the effect of mediant as anticipation seems negligible.

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