Thanks to earlier questions (and some reading), I understand the basics of the phonological hierarchy
Features < Root < Skeletal Slot < (mora / onset-rhyme-nucleus-coda) < Syllable < Foot < Word < Phrase
This all seems coherent and probably has real scientific value. But is this hierarchy capable of accounting for prosodic phenomena like stress, tone, intonation and so on?
I doubt that there is such a hierarchy, with higher level units displaying special behavior which does not emerge from the lower level things they are made up of. Take syllables, for instance.
I have a theory adapted from Saussure's idea that syllables are the paradigmatic counterpart to a syntagmatic difference between individual sounds that are explosive (with increasing aperture) and those that are implosive (with decreasing aperture). The consonants at the beginning of a syllable are explosive and those at the end are implosive. There are not two different things involved between the feature difference (as we would call it these days) explosive versus implosive and the grouping of sounds into syllables -- these are different perspectives on the same phenomenon.
I don't think Saussure's theory quite works, as he proposed it, because aperture doesn't always increase in the onset of a syllable and doesn't always decrease in syllable offset. For instance, the "s" of "stop" has greater aperture than the "t" that follows. (This is so obvious, one wonders what Saussure was thinking, when he discussed this in Cours de linguistique general.)
It is a little recognized fact about the SPE feature system that consonants can differ in stress, as well as vowels. In my version of Saussure's theory, I use the stress feature instead of Saussure's explosive/implosive. Syllables begin with stressed consonants and end with unstressed ones. Then when consonants change syllable membership, this can be alternatively described as a change in the stress of the consonants.
The problem in this respect with the prosodic hierarchy (PH) is determining what it would mean for the hierarchy to account for prosodic phenomena like stress, tone, intonation and so on. There are two parts to the problem: identifying "prosodic phenomena", and saying what you mean by the hierarchy "accounting for" those phenomena. To make life simpler, let's limit prosodic phenomena to sub-phonemic pitch and duration variation subsumed under the rubric "intonation" (because intonationologists make extensive use of the PH in their analyses). Something in the theory of the PH seems to be useful for such analysis. However, there is an awful lot about intonation that isn't accounted for by the PH. For example, the PH is pretty much useless in determining whether pitch goes up, down or stays the same; it doesn't tell you how much it goes up when it does; it doesn't tell you why it goes up when you're contrasting a choice of nouns (why doesn't it go down). If you move to tone, things get even worse because there are so many things that the PH doesn't explain. The reason is, the PH doesn't purport to explain those things. PH is a theory of grouping: all of its claims are about the parts of phonological systems that say "when these two (or more) things are in a particular relationship". That's all it does.
The utility of a theory (whose scope is now properly delimited) can't be judged in a vacuum, it only makes sense to evaluate the theory relative to an alternative. Fortunately, this has been done. Recall that the core concepts of the PH were promulgated in the mid 80's, and there were two competing theories of relations – direct reference, and indirect reference. Kaisse and Odden advocated the direct reference approach, Nespor, Vogel and Selkirk advocated the indirect reference approach. The central issue distinguishing these approaches is whether phonological rules can directly refer to properties of grammatical structure (e.g. can phonology know that two words are "in the same NP"). Or is there a pre-phonological interface component which groups words hierarchically into parts of the PH, subject to the standard conditions on hierarchical tree-building (Pi immediately dominates all and only Pi-1; every Pi-1 is immediately dominated by exactly one Pi; the inventory of elements is (U,I,P,W)). Neither side in this debate had much of a stake in specific claims about the kinds of non-phonological structural relations that could in fact enter directly into rules (in the DR account) or construction of prosodic units (for IR accounts), but the most popular understanding was that it was usually about "c-command", or "is the head of".
My evaluation of the matter is that the debate fizzled out, mostly because of the difficulty of identifying anything empirical that could distinguish the accounts. 80% of "the facts" boil down to a simple observation: languages can stipulate that x and y interact when x has a certain relationship to y, and not otherwise, and that relationship is exemplified by pairs like "noun,adjective", "noun,determiner" but not "noun,adverb" (it is more complicated that that, though). The mass of facts that the debate is about reduces to the concept that the head of a phrase 'groups together' with its modifiers, in a way that two words not in that relationship don't do (like, subject and object in a V-final language). Both theories had machinery for getting that fact. Other facts exist that may cause problems, but they are part of the arcana of the topic where "something else must be going on".
There is, additionally, a question as to whether the PH is a single coherent thing, and most often people distinguish the "postlexical" hierarchy from the word-internal hierarchy (with Pwd at the top). The strongest claims of strict dominance relationship have been mostly rescinded (what survives is that Pi immediately dominates Pi-(0..n), that is, a node can dominate itself or anything lower, but not anything higher). There are many specific questions about particular levels of the hierarchy, for example what principles explain why feet come in three main varieties: one-or-two syllable feet, two-syllable feet, and three-syllable feet, and there isn't a general analogous question for the entire PH (though there has been a related claim about the mora-syllable relation).
Once you fold in features, there is nothing concrete and testable to the concept of "hierarchy", apart from the non-linguistic fact that immediately perceptible facts have to be subsumed into higher abstractions, in order to retain the perceptible in memory.
