New answers tagged progressive-aspect
"Letters have been being sent" sounds fine, to me. At any rate, it is the pattern of auxiliaries described by Chomsky in Syntactic Structures: (Modal) (have+en) (be+ing) (be+en) where the affixes are affixed to the following verb form (either auxiliary or real verb) by Affix Hopping. Your example is from Letters have+en be+ing be+en send which ...
First, I agree that the example, as you stated it, does indeed sound a little strange. But I believe that it is the (over-)simplicity of the example -- a very short sentence, with no context around it -- that makes it sound strange. And I'm not saying anything about you; of course it is the case that these are precisely the kinds of examples that are used ...
The present perfect progressive in passive voice is extremely rare. BNC has only two examples. http://bnc.bl.uk/saraWeb.php?qy=has+been+being&mysubmit=Go COCA has 8 incidents. http://corpus.byu.edu/coca/ Search for "has been being". The link does not work as I thought.
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