Looking at the script, it seems that the INTERNATIONAL PHONETIC ALPHABET is what Henry Higgins would be most likely to use.  


It has been used since 1888, when it was devised by the International Phonetic Association as a standardized representation of the sounds of oral language.  

In addition to being used by linguists and Speech-Language Pathologists, it is also used by singers and actors.

The IPA is noted for representing only those qualities of speech that are distinctive in oral language: phonemes, intonation, and the separation of words and syllables.
It even includes qualities of speech like tooth-gnashing, lisping, and the sounds made with the cleft palate.
Letters and diacritics are the two basic types of symbols.  
Letters are divided into pulmonic consonants, non-pulmonic consonants, and vowels, while diacritics.  
Diacritics are small markings placed around letters to specify the pronunciation.
This is its official handbook, as released by the IPA, which I think is available in the Northwestern Library — if that would be of any interest:

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