In the realm of harmony...there can be no beginning and no conclusion, just as...
With such complex subject-matter, it doesn't help to have parts missing, however short, because they may contain vital clues or references, but I think the main points are clear.
The key to understanding this sentence is to recognise that it is a parallel description of Wagner's revolutionary harmonic scheme, the distinguishing feature of which is that the harmonies are never resolved (here: sich befriedigen) but continually evolve, change and develop without making use of conventional harmonic cadences or resolutions. Famously, the 'Tristan chord' with which the work opens remains unresolved throughout.
The reason why Wagner expresses the structure of his harmony in this rather elaborate language can be found in one word: Schopenhauer. As is well known, W was profoundly influenced by S, and to make sense of the German a (very brief and crude) summary of S on music is necessary.
For S, of course, the world has two aspects, will and representation, 'will' (Wille) being the underlying, grounding noumenal realm of true reality, while 'representation' (Vorstellung) denotes the physical, phenomenal world in which Wille is objectified and in which we ordinarily live. For S, the objectified world of representation in which we miserable humans are trapped is a bummer (striving, fighting, suffering, never-ending desire and torment) and redemption is to be found, as in Buddhism (hence the profoundly Buddhistic element in W's late operas), in renouncing desire and achieving release from the phenomenal world into the realm of pure noumenon.
Music is one of the principal ways by which this can be achieved (sex is another) because, for S, while the arts generally provide images (Abbilder) of the world of representation, music is a pure expression of the non-objectified realm of the noumenon.
W was utterly captivated by this notion which is of central importance to his work and this is why it is crucial that his harmony does not resolve itself by finding an object: by remaining unresolved, the music remains a noumenal Abbild of questing, unsatisfied desire, always turning back in on itself and renewing itself from within. Resolution is finally achieved only by means of the negation of desire through death.
This is why W uses the term 'Gegenstand' here, and other language echoing S (see first quote below), which must be retained in translation if the passage is not to become trivialised and meaningless. Maybe: 'In the realm of harmony...there can be no beginning and no conclusion, just as the blazing swirl of emotions, feeding on itself without an object and unaware of its cause, remains no more than itself, desiring, yearning, raging, pining, dying away, that is, dying without having resolved itself in the object, dying without really dying, and thus always returning back to itself.'
Not very intelligible English, but then W's German was pretty unintelligible too.
One final point: I don't know how deep you're going into this but, as Ken says, sexual imagery is far from inappropriate here, given S's emphasis on sex as a route to the noumenon and the successive waves of musical orgasm with which the work climaxes (sic), and is certainly an overtone in 'befriedigen'. If it were possible to use 'satisfy/satisfaction' in English instead of 'resolve/resolution' it would certainly not be wrong, though perhaps too narrow.
'alles Begehren, Streben, Wünschen, Verlangen, Sehnen, Hoffen, Lieben, Freuen, Jubeln und deren Gegenstücke als Äußerungen des Wollens’ („Die Welt als Wille und Vorstellung III, S.529f). '
[Der Mensch]...'ist Objektivation des Willens, d.h. zum Objekt gewordener Wille.
An dieser Stelle erklärt Schopenhauer, dass den verschiedenen Entwicklungsstufen der Erscheinung verschiedene Objektivationsstufen des Willens entsprechen...'
'music is distinguished from the other arts by the fact that it is not a copy [kpm: Abbild] of the phenomenon, or, more accurately, the adequate objectification of the will, but is the direct copy of the will itself'
'Der Wille wird objektiv, greifbar, in der Allgemeingültigkeit der Begriffe, zumal in dem, was Schopenhauer als Platonische Idee anspricht...Die als “Verdeutlichung der Sichtbarkeit des Willens” definierte Kunst hat in der Reinheit der Platonischen Idee ihren adäquaten Gegenstand. Die Ordnung der Künste ergibt sich aus dem Rang der jeweils repräsentierten Idee...In dieser Systematik wird die Stufenhierarchie der Willensobjektivation sichtbar...Aufgrund ihrer besonderen metaphysischen Wesenskennzeichnung wird die Musik außerhalb der Reihe dieser Künste angesiedelt.'
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