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/j/ is the semivocalic equivalent of /i/, /w/ of /u/, /ɥ/ of /y/, /ɰ/ of /ɯ/, and so forth, and I've also seen /ɹ/ described as the semivocalic equivalent of /ɚ/. Considering all of this, it seems that approximants can have vocalic equivalents in certain contexts. I'm trying to work out what would be the vocalic equivalent of the bilabial approximant /β̞/. Is there a vocalic equivalent to this sound, and if so what would it be?

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First off, a quick note about the use of / / vs. the use of [ ]. Usually the former is used for phonemes and the latter for phones. Since you are really talking about phones here, I'm going to shift to using [ ].

This is sort of a tricky question. In the examples you give, the state of both the tongue and the lips is expressed by the IPA symbol:

[i]/[j] - spread lips, high front tongue
[y]/[ɥ] - rounded lips, high front tongue
[ɯ]/[ɰ] - spread lips, high back tongue
[u]/[w] - rounded lips, high back tongue

With vowels in general, it's crucial to know what the tongue is doing to be able to distinguish them from one another. In most of the symbol pairs you listed, the approximant symbol was derived from its vowel cousin's symbol, so the state of the tongue during the articulation of the approximant is assumed to be the same as for its vowel cousin.

The bilabial approximant symbol, on the other hand, has been derived from its fricative cousin. As such, it puts emphasis on the state of the lips--they are compressed but unrounded. So we know we are looking for a vowel that involves compressed lips but all bets are off as far as the placement of the tongue is concerned.

Just thinking in terms of sounds from real languages, one candidate that seems to fit the bill is the [ɯᵝ] sound in Japanese, usually represented phonemically as just /u/, since it doesn't contrast with any other high back vowel. Actually, if we're nitpicking, even [ɯᵝ] is probably not exactly right for this sound, since the tongue placement is usually slightly more centralized than the symbol implies. At any rate, the superscript β is used to indicate lip compression. The approximant cousin of this vowel is usually represented (as you might expect) as [ɰᵝ] rather than as [β̞], since the former gives information about tongue placement.

In essence, you could think of [ɰᵝ] as a narrow version of [β̞]--that is, [ɰᵝ] specifies a subset of the sounds specified by [β̞]. And so [ɯᵝ] specifies a subset of the vowel sounds that are cousins to the approximant sounds represented by [β̞]. One might imagine a vowel that is the fronted counterpart to [ɯᵝ], perhaps represented as [iᵝ], that would be just as valid as the vowel cousin of [β̞]. I don't know if that sound is documented in an actual language, but it is logically possible.

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