How to publish a paper that does not seem to be within the scope of any journal? I have written a paper on applying a mathematical theory to a linguistic problem. I privately submitted it to several renowned professors (some of them very famous indeed) in the field of linguistics, semantics and so on. They made some suggestions and expressed some criticism on some details, but for the most part they praised the quality of the paper.

Following their suggestion, I submitted it to a leading journal in the field. The answer I got from the journal’s preliminary assessment was that the paper had huge potential, but was too mathematical for them. They advised me to resort to their sister publication, where much more focus on natural language processing and mathematics is to be found. The answer was the mirror image: nice paper, definitely scienficially worthy, but too soft, that is, too linguistic or informal for advanced mathematicians or computer scientists.

Now I am in no man’s land. The paper is indeed interdisciplinary (as corresponds to a linguist using some mathematics as he needs them), but I find it very unfortunate that due to a false sense of the own domain, noone seems to feel responsible for its contents, which I deem valuable. Before trying any further submission, I was entertaining the option of submitting it to the ArXiv, but I do not know whether that will make any sense.

How can I proceed to find a journal that considers my paper to fit within its scope?

I posted this question on Math Stack Exchange before and was recommended to repost it here.
Interdisciplinary papers have this problem a lot. A - now - very influential paper in my field, I was told by the first author, was in limbo for 4 years before it was published, despite the senior author being world-famous. 

Persistence is the key. Try to find someone influential who is knowledgeable and you can convince to help you expanding/adapting/explaining the work better and you could then co-submit with. Or else, you could consider aiming for a general-purpose journal such as PLoS ONE, and whose reviewers are advised to select by novelty and correctness, rather than (subjectively judged) relevance. 

Learning to write for the specific audience of a journal can also help.