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I know that the diacritic ̈ is for centralization and ̞ for lowered. So when [e] is centralized, it becomes [ɘ] and when lowered it becomes [ə]. See the diagram:

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If someone gives me the transcription [ë̞] (centralized and lowered [e]), how should I know which sound it is? I saw this in a textbook. The book said "anyone who is well-versed in IPA will know what sound [ë̞] is" I read the whole chapter but they didn't tell what sound it was.

When someone gives [ë̞], how do linguists pronounce it? Is it [ə]? I know that mixing diacritics can get you another sound for which IPA already has another symbol.

Also

  • is [ï] another symbol for [ɨ]? and
  • [ɛ̈] for [ɜ]?
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  • I am not asking multiple questions, but the main point is one: when one vowel gets centralized, it becomes the other? – Sphinx Jan 24 at 10:01
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Typically linguists use such diacritics when the sound they're describing is in between the sounds associated with unmodified base glyphs of the IPA. So I would not expect the author to write [ë̞] to represent the same sound as [ə], but rather a sound intermediate between [e] and [ə]

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I agree with @drammock and would add this. IPA is really a phonetically-based system of phonological symbolization. When a person write a certain sound of a language as [ɪ] or [e], that is in part a statement about what the vowel sounds like, where [ɪ] or [e] represent approximate targets. There is substantial variation in the formants of [ɪ] or [e] across languages that have those values, so that [ë] is a statement that in a token, [e] is centralized relative to a standard value. That might be "Expert IPA productions", or it might be and more often is "relative to canonical value in this language". Therefore [ë̞] might be a sensible thing to write for a contextual variant of /e/ especially when /ə/ is itself a bit off the IPA-expert standard.

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Phonetic transcription is an example of cooperative communication. As such, it follows from the maxim of quantity, that if someone is using [ë̞] there must not have been a simpler transcription that adequately describes the phone in question

Therefore the choice of this particular transcription tells us that [e] is likely the closest cardinal vowel to the phone, but that the phone is more central and lower than cardinal [e]

In a loose or phonemic transcription, it'd be possible that [e] isn't the closest cardinal vowel, and was chosen to show some other information (possibly this vowel patterns with front vowels rather than central ones, or maybe it results from a recent split with genuine [e]), but as this is in square brackets I am assuming we are speaking purely in phonetic terms here

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