I understand how it works for Ohayō the o here is omitting the u from う, so without the macron it would be "ohayou". Even Tōkyō becomes Toukyou. But what about other vowels? such as ī ē ā ū what are these characters shortening or omitting?

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It depends on the romanization system which your writing or the words you encounter are based on. In nihon shiki and kunrei shiki, for example, the macron is not used and in its place a circumflex appears above the vowel letter. Among the major romanization systems of the Japanese language, you only see macrons used in Hepburn.

The Wikipedia page on Hepburn is pretty clear and detailed and hard to improve upon, so I will just borrow the information from the relevant section:

In Hepburn, vowel combinations that form a long sound are usually indicated with a macron ( ¯ ). Other adjacent vowels, such as those separated by a morpheme boundary, are written separately:

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There are many variations on the Hepburn system for indicating long vowels with a macron. For example, 東京 (とうきょう) is properly romanized as Tōkyō, but can also be written as:

  • Tokyo – not indicated at all. Common for Japanese words that have been adopted into English, and the de facto convention for Hepburn used in signs and other English-language information around Japan.
  • Tôkyô – indicated with circumflex accents, as in the alternative Nihon-shiki and Kunrei-shiki romanizations. They are often used when macrons are unavailable or difficult to input, due to their visual similarity.
  • Tohkyoh – indicated with an h (only applies after o). This is sometimes known as "passport Hepburn", as the Japanese Foreign Ministry has authorized (but not required) it in passports.
  • Toukyou – written using kana spelling: ō as ou or oo (depending on the kana). This is also known as wāpuro style, as it reflects how text is entered into a Japanese word processor by using a keyboard with Roman characters. Wāpuro more accurately represents the way that ō is written in kana by differentiating between おう (as in とうきょう (東京), Toukyou in wāpuro) and おお (as in とおい (遠い), tooi in wāpuro); however, it fails to differentiate between long vowels and vowels separated by a morpheme boundary.
  • Tookyoo – written by doubling the long vowels. Some dictionaries such as the Pocket Kenkyusha Japanese Dictionary and Basic English Writers' Japanese-English Wordbook follow this style, and it is also used in the JSL form of romanization.
  • Ah so in hepburn if I'm understanding correctly, a word like 魂 tamashī, it's really just omitting the 2nd i. So without the macron it would be tamashii?
    – aeonic
    Aug 23, 2021 at 22:42
  • @aeonic are you using the words "omitting" and "shortening" as expressing some kind of difference in representing the words?
    – Leebo
    Aug 24, 2021 at 0:27
  • @aeonic, in 魂【たましい】 romanized as tamashī, the macron + i → ī is both "i"s. So nothing is omitted. Aug 25, 2021 at 0:03
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    @Eddie, I noticed an error in the table above in the section for romanization of double-"i" -- in the word おいしい, the final two "i"s are not part of a single morpheme. I just updated the page on Wikipedia to use the word おにいさん instead as an example. FYI. 😃 Aug 25, 2021 at 0:05
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    @Eiríkr Útlendi as I told leebo, I meant shortening but rather than the semantics of omit versus shorten, I just needed to know that ī = ii. mystery solved.
    – aeonic
    Aug 26, 2021 at 9:55

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