Interesting piece of writing - I was curious as to the writer, so I looked it up, and was surprised to find it was (a translation of) Kandinsky's writing, from On the Spiritual in Art, which I must re-read. Thank you!
In answer to your question, Kandinsky is making an analogy between the dull nonresonant sound an old, long-buried, cracked vase would make when it is dug up and the reaction he sees the soul, or the spiritual part of our being, would make in response to its being acted upon by a materialistic impulse.
Our soul rings cracked when we seem to play upon it,
What's the subject? Our soul.
What about it? rings. Rings is the verb.
How does it ring? cracked. Cracked here is either used an adverb which modifies 'rings' or as an adjective which modifies 'soul'.
When does it ring? when we seem to play upon it. That is an adverbial clause which modifies 'rings'.
as does a costly vase, long buried in the earth, which is found to have a flaw when it is dug up once more.
How does it ring? as does a costly vase
What kind of vase? long buried / which is found to have a flaw in it - adjectival phrase/clause modifying 'vase'.
Buried where? in the earth. Adverbial phrase modifying 'buried'.
When does it ring? when it is dug up. Adverbial clause modifying 'ring'.
Dug up when? once more. Adverbial phrase modifying 'dug up'.
There are several things here that would make me want to go back to the original and check how accurate the translation is. Does 'once more' imply it has already been dug up once before? Is there distinction between soul and spirit, that Anna Wierzbicka has written eloquently about that has been 'lost in translation'? Is the choice of 'cracked' accurate? And what exactly did he mean by 'primitives'? Not to mention the fact that I would imagine the vase 'ringing' after it was dug out, not when. Alright, it doesn't say 'when it's being dug up', but it still seems ambiguous.
Ref to AW's paper is: Soul and Mind: Linguistic Evidence for Ethnopsychology and Cultural History - Anna Wierzbicka - American Anthropologist - New Series, Vol. 91, No. 1 (Mar., 1989), pp. 41-58.