A stultifyingly pedantic pedant writes: you mean "of which their bass is a dominant" -- the actual root of an augmented sixth chord is the raised 4th scale degree!

I honestly can't remember my thoughts on this, and will have to re-conjure them. Unfortunately, it's too early around here to faff around on the piano. However, I would also not suggest calling C-flat any kind of tonal center.

There are several layers of obfuscation in this piece, the final and most insulting one being the constant registral shifts from one chord to the next -- a stroke of genius in this song, but it makes talking about 'passing resolutions' and other linear phenomena characteristically difficult to talk about. D-flat in the voice doubles C# on the 3rd semiquaver of the piano, so the Cb and B are themselves resolutions of this suspension.

I don't feel like I'm answering your question... in my vocabulary, the term 'passing resolution' does not exist. Can you clarify what you mean by't?

The '-chen' in the voice does in a wrenching way give away what's about to happen, so I agree that that's a pivotal, if brief, compositional decision... certainly both hypothetical alternatives Bb and Gb are wrong for a number of reasons.

Just an invented term (hence scare quotes), I was referring to how the A# partially 'resolves' onto the C-flat, very important in terms of the harmonic implications, but on the other hand very transient, being on a weak beat and weak syllable, essentially subsumed into the broader line as part of a simple progression up to D-flat. Re secondary dominants: yes, clearly F#/G-flat is V of V of E major, but that's not absolutely clear until into bar 9, which could also, if the A# moved upwards to a B rather than downwards to an A natural, or that latter progression were otherwise avoided, have led to a proper resolution into B-C-flat (try changing the lower F# in the treble to an E, for example). That notwithstanding, I still regard a key serving as a dominant, which is reached via a secondary dominant, as some type of passing tonal centre - the secondary dominant is still a dominant of something. B/C-flat major is implied (in bar as much as E major is - only with the modifications in bar 9 is a resolution into B/C-flat displaced.

One other thought - supposing Schumann had spelled bars 8 and 9 differently in the piano, with the second dotted crotchet beat in bar 8 having a G-flat for the bass, and with B-flat/F-flat/D-flat/B-flat/G-flat for the arpeggio, and similarly with C-flat in the bass in bar 9, then he would have had to move from C-flat in bar 9 to C-natural in bar 10, which would be less than elegant on the page. Yet that is not necessary in the voice? But there I see this might be a case of what Schumann often did so brilliantly (as did Schubert before him and Brahms afterwards) - harmonising a melody distinctly from that harmony most obviously implied by the melody. The C-flat as a flattened supertonic of B-flat implies a modulation, but to what? To the flattened supertonic itself? Nothing in the voice alone to be implies a gravitation towards F-flat/E, only the piano part does that. But is that why it was necessary for him to spell vocal and piano parts differently when they appear simultaneously? The C-flat can be 'resolved' onto a C-natural (as the non-flattened supertonic of the re-established B-flat in the following bar), but that's possible because of the intervening B-flat in the vocal part, which doesn't exist in the bass of the piano. If he'd spelled the vocal line in accord with the piano (say shifting enharmonically to A# in the second half of bar , then he would have been left with an A# followed by a C-natural in bar 10, or a B# for the latter, but that would then have to resolve to a B-flat. I'm still not quite sure wholly what's going on in terms of necessary notation of particular harmonic features in these three bars, but can't really see how Schumann could have notated it other than how he did. So what is it about the melody/harmony that necessitates different spellings in voice and piano?

But how can something be both a dominant and a tonal center?

Can you give an example/s of this? I've a feeling I might know what you mean (although I'd interpret what's going on differently), but would be good to know which works you have in mind ...

I'll have a look - I'm thinking of various short piano pieces, including some from Mikrokosmos and For Children, but can't remember which ones (my piano teacher always used to bring them up in the context of never-ceasing arguments about tonality (and whether the presence of an over-arching tonal centre was a necessary condition for a good piece of music), from the point of view of 'the key of a piece is the key it ends in', which sort of accords with a Schenkerian view that I cannot wholly accept).

..and then potentially on to the tonic, hence Ian's use of 'implied' tonic? (Just trying to keep up here..)