Poets rarely use more notation than prose provides. Hopkins used stress-accents and Dickinson's dashes may have lengths that correspond to the intended length of pauses, but apart from these, line-breaks are the only non-prose way to indicate stress and pauses.
More advanced notations exist, used by linguists transcribing soundtracks, and by poetry theorists. Poetry theorists less often work from performance, marking up stress using purely the text, and breaking lines into "feet" - all of which makes their work rather subjective, and easy to manipulate to support their theoretical viewpoint.
Of diacritical marks the most commonly used have been
In rhythmic analysis, diacritical marking of normal text is less satisfactory for most purposes than some kind of graphic transcription, extracting essential features
Less often, numbers have been used to indicate stress or pitch. Notations to support other features exist - e.g where long/short syllables (classical Greek and Latin poetry) or pitch (Chinese poetry) matter.
Serious music tends to have been written before played, written in measures even if one can't hear them. "traditional" music often gets written down after it's been played. The choice of notation matters to the composer and the archivist of "traditional" music. It also matters to experimentalists and theorists. Quoting from Musical notation - according to Philip Tagg and Richard Middleton
Before music notation became standardised, several methods were tried. After standardisation the number of performance cues increased, but the notation remains restrictive. Alternatives have been developed. Graphic notation refers to the contemporary use of non-traditional symbols and text to convey information about the performance of a piece of music. It is used for experimental music, which in many cases is difficult to transcribe in standard notation.
Conventional musical notation has often been adapted for rhythmic analysis of verse. For English, occasional and partial use of musical notation begins with C. Gildon (Complete Art of Poetry, 1718), and recurs frequently in the later 18th and 19th century; since S. Lanier (Science of English Verse, 1880) full musical notation has often been used by writers whose analysis of verse is musical or exclusively temporal. Poe, for example, used it. At the very least, duration of pause may be indicated by conventional musical notation for rests.
Music scores are for performers rather than receivers. In poetry, receivers "perform" the poetry when they read it. Music notation tends to restrict composers. Extra poetry notation would in contrast restrict readers/theorists. It's unclear to me why so many poets with widely varying views on poetry allow such liberal interpretations of their work. Given the increasing specialisation of of the poetry audience, the paucity of auxiliary notation is surprising.
Given the resistance to other notations, it's not so surprising that the line-break is so popular - a single tool with many uses. However, it performs none of them very precisely.
An interpretion needn't follow the author/composer's intentions. Sometimes (e.g. when actors in Macbeth dress as Nazis) it's done deliberately. Elsewhen (e.g when old music isn't played on period instruments) the decision may not have been consciously made. Some of these choices are controversial. Poetry readers have at least as much freedom as performers in other Arts. At poetry recitals audiences don't seem to mind an English male performing an American woman's poetry as long as the subject matter isn't too incongruous - sounds can be changed ("tomato" and "Z" pronounced differently) as long as words aren't.
Private readers may well make many unconscious (and uninformed) decisions. Perhaps it's assumed that such misreadings are of minor consequence compared to conceptual misunderstandings, and that it's not worth cluttering the visual impact of the text to reduce the errors. The "look" of the text matters, especially when expressive line-breaks are used.
Nowadays it's nearly as easy to have sound online as text. Even movies aren't too hard, with optional subtitles, like a DVD. Perhaps the age of "The Master Text" is coming to a close.
Even restricting oneself to text, I don't think one need tie oneself down to a minimalist notation, especially now that color and graphics are cheaply reproduced.
This article has benefitted in particular from the first 3 sources (for which much thanks)