Ancient Greek accents

Given that I was foolhardy enough to put a page on my site about how to transcribe Ancient Greek in IPA, I could hardly complain that an email arrived from “J.J.” saying

I'd like to ask you a question about using IPA alphabet for transcribing Ancient Greek. The Wikipedia has a recommended set of symbols for use on its pages, but I do not get one thing — how to correctly mark the tone and stress.
Does /àá/ for a diphthong or long vowel, e.g. η or ηυ, mean that if accented it needs to be transcribed as ɛ̀ɛ́ and ɛ̀ːú̯? The Wiktionary has a different system… Ἀχιλλεύς is akʰilːe͜ʊ́s, but I keep thinking that the tone and stress marks should be on both vowels of a diphthong, i.e. akʰilːè͜ʊ́s. How does one correctly notate the tone and stress in Ancient Greek words?

(Typographical note: the acute and grave accents are meant to display in proper alignment above the vowel symbols. Your browser may not manage that.)
I answered

I think it depends on how explicit you want to be. In languages that have only two tones, high vs low, as Ancient Greek did at the level of the mora, it is sufficient to show high tones and leave low tones unmarked.

As I understand it, a diphthong or long vowel written in classical Greek with an acute accent had a low tone on the first mora, a high on the second. Transcribed fully this would require a low tone mark on the first vowel symbol, a high tone mark on the second. However it would be unambiguous to write it just with a high tone mark on the second symbol.

A diphthong or long vowel written with a circumflex (perispomenon) had a high tone on the first mora and a low on the second, producing a compound tone. Transcribed fully this would require a high tone mark on the first vowel symbol, a low tone mark on the second. But it could also be written unambiguously just with a high tone mark on the first.

Short vowels are straightforward: if written in classical Greek with an acute accent they were high, otherwise low.

According to Allen, the Greeks appreciated the fact that there is no need to mark most low tones.

In one early system of marking, every low tone was indicated by the grave accent-mark — e.g. Θὲόδὼρός; but such a practice was clearly uneconomical and inelegant, and was later replaced by the current (Byzantine) system whereby only the high and compound tones are indicated (by the acute and circumflex symbols). The grave symbol was, however, then substituted for an acute where this occurred on a final mora (‘oxytone’ words), except in the case of interrogatives (e.g. τίς) or when followed by an enclitic or pause — thus ἀγαθός ἐστιν, ἔστιν ἀγαθός·, but ἀγαθὸς ταμίας.

In modern Greek it’s straightforward: the ancient accents have become simple word stress. Correspondingly in modern spelling the three marks of ‘polytonic’ notation, ὰ ά ᾶ, have been collapsed into a single ‘monotonic’ mark, ά.