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From what I know, stress can only be assigned at the level of the word (as in English) or the level of the sentence (as in French). Can any natural language assign it syntactically, e.g., "the first syllable of every verb is stressed"? It doesn't seem like this should be impossible but I didn't manage to find anything with some preliminary googling. I found an article from 2013 titled "Some evidence for syntactic stress in Hittite", but it's not publicly available (nor am I sure if it's even talking about the same thing that I am).

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First, word stress can be computed with reference to syntactic properties, so "the first syllable of a verb is stressed" (as opposed to perhaps the penult in nouns) is perfectly possible. This may be outside the scope of what you're looking for, since stress assignment would not refer to properties of other words, it would refer to morphological information about a word that relates ultimately to syntax. I suggest as a starting point that "sentence stress" has to crucially involve properties of two words. And then you have to sort out the problem of clitics, which are usually treated as separate syntactic words which are phonologically dependent on a host word (typically meaning that they don't get their own stress; they may or may not affect the position of stress within the host word), or be stressable themselves when attached to a word. Given the typical analysis where a clitic "attaches" to a host, the clitic may or may not be visible when word stress is assigned.

There is one well-established sense of "sentence stress", name intonational phenomena, where citation stress patterns change under phrasal concatenation. For example, "thirteen" in American English usually has final stress, but in phrases like "thirteen men", stress is on the first syllable. The classical phonological analysis of this is that there are degrees of stress, thirteen has secondary stress on the first syllable and primary stress on the final. When combining thirteen+men, stress is shifted to the first syllable. But another analysis is that stress is erased from the final, leaving the initial stress the most prominent. There are many theories for dealing with intonation, one of them being the class of "tonal" accounts where intonemes like H and L are "aligned" to stressed syllables, and other elements of a sentence. In such a theory, what might seem to be movement of stress at the phrasal level actually reduces to variable positioning of these intonational elements. In that sense, the phenomena can be analyzed without invoking the notion "sentence stress". (As you can see, this is becoming a problem of the type "what do you mean by 'syntactic stress'?").

There are languages which have what is best termed phrase-positional prominence, for example, in many Bantu languages (Mwiini and Chewa are most widely discussed), the penult of the phrase (not the word) is "prominent". This paper by Hyman surveys the topic, noting that the typical exponent of stress in these languages is lengthening. The first step in accounting for these languages is syntactically determining what word-combinations constitute a phrase: typically, in "verb plus noun", that combination constitutes one phrase; in V N N, that is usually parsed as (V N) (N). Then, the rule is, "the penult of the phrase is stressed (prominent, lengthened, or whatever you want to say).

I suspect that Durnford's analysis reduces to an example of this latter type, that is, phrasal grouping. Claims about "stress" in Hittite are even more conjectural than such claims about Shona. The premise that stress is relevant is argues in the prior paper by McNeil, assuming that metric systems can be divided into quantitative vs. stress-based, and since (it is argued) it's not quantity, it must be stress.

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The most widely known analysis of English stress is that proposed in Chomsky and Halle's The Sound Pattern of English. There, stress is assigned both to words and other constituents. For discussions and proposals, google "nuclear stress rule". I am skeptical of the supposed opposition you mention between English word stress and French sentence stress.

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