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Is there any pronunciation difference between both? Shouldn't IPA use one symbol per phonem/allophone? Curiously, this happens with the schwa, but not with "true" vowels, eg the A in car [kɑɹ].

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Most Americans do not pronounce [əɹ] with two different, consecutive sounds (phones), but with one phone. You say "Shouldn't IPA use one symbol for phoneme?" The problem is that [əɹ] is two symbols for one phoneme. So [ɚ] is used instead.

For the other r-colored vowels, many Americans use single phones for [ɑ˞] and [ɔ˞], but many also use two, so [aɹ] and [ɔɹ] are considered to be adequate IPA representations of these. Wikipedia says:

In words such as start, many speakers have r-coloring only in the coda of the vowel, rather than as a simultaneous articulation modifying the whole duration. This can be represented in IPA by using a succession of two symbols such as [ɑɚ] or [ɑɹ], rather than the unitary symbol [ɑ˞].

For the other three r-colored vowels, [ɛɹ], [ɪɹ], and [ʊɹ], even fewer use a single phone than for [ɑɹ] and [ɔr]. (In fact, I don't know whether any Americans use a single phone for [ɛɹ] and [ɪɹ].)

If you try to represent American and British speech with the same IPA symbols, you run into big problems. What vowel do you use for palm, box, cloth, caught? Those are two vowels in American and three vowels in British speech, and there's not any easy way to deduce the American pronunciation from the British, and vice versa. So why not use [ɚ]?

And the American pronunciation of wanderer, /ˈwɑn.dɚ.ɚ/, is indeed different from the British pronunciation, /ˈwɒn.dər.ə/, in the der syllable.

Peter Shor
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  • Three vowels in AmE as well if you don’t have the cot–caught merger. (Also there’s no reason to assume that /ər/ is one phoneme. It may be pronounced as a single sound, a rhotacised schwa, but that doesn’t mean it’s not phonemically a sequence of two phonemes.) – Janus Bahs Jacquet Jul 01 '19 at 22:34
  • You mean palm and box are different vowels? Or cloth and caught are different vowels? They're not for me. – Peter Shor Jul 01 '19 at 22:40
  • Yes, palm /ɑ/, box and cloth /ɔ/, and caught /ɒ/ (to give phonemes – the /ɒ/ phoneme in particular varies a lot in actual pronunciation). Also to some AmE speakers, I’m pretty sure all four words would have /ɑ/. Can’t think off the top of my head right now which merger that would be, but one of ‘em. – Janus Bahs Jacquet Jul 01 '19 at 22:44
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    The dictionaries say that palm and box have /ɑ/ and cloth and caught have /ɔ/. Which matches my dialect. If you merge cot and caught, they all get the same phoneme. I don't know what dialect you're speaking, but are you sure that some British English didn't get mixed in? – Peter Shor Jul 01 '19 at 22:50
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    @PeterShor The cot/caught merger is not binary. Palm has /ɔ/ on a lot of the East Coast. But I have heard it pronounced with /ɑ/ by people from Upstate New York who pronounce caught with /ɔ/. When I try to put a /ɑ/ between "p" and "l" I wind up with a coughing fit. Or it comes out as /æ/. I don't know how they do it. – Spencer Jul 01 '19 at 23:11
  • I really cannot perceive a difference between my pronunciation of after ( [æftər] ) and the Wiktionary/Cambridge dictionary audios of [æftɚ]. When going directly from [t] to some kind of [r] , my mouth unavoidably quickly passes through the uncolored schwa first. I'd teach the pronunciation of this word syllabically as [æ] - [ftə] - [r]. From what you're saying, this is not how Americans speak. Same applies for first ([fɜrst] and [fɝst]). – Alan Evangelista Jul 02 '19 at 11:37
  • @AlanEvangelista: Can you hear the difference between the two pronunciations of anorak in Cambridge dictionary? Let me add that [fɜrst] and [fɝst] are allophones in American English, so people would tend not to notice the difference, and for being understood, it doesn't matter which one you use. – Peter Shor Jul 02 '19 at 12:30
  • @JanusBahsJacquet: Like Peter Shor, I think you may have mixed up different dialects. Most American English speakers, regardless of whether they have the cot-caught merger, have lost /ɒ/ as a phoneme distinct from "thought" and /ɑ/. I can't even think of any American accent with three phonemes in this area. Accents with two phonemes do differ in which words they assign to which category, but within an accent of American English there's typically not a three-way distinction between /ɑ/, /ɔ/, and /ɒ/. – herisson Jul 02 '19 at 12:43
  • @sumelic: I do agree with Spencer that the vowel in palm (and balm, calm, etc.) varies a lot on the East Coast. I probably should have used spa for the example word in my comment. And I am positive that some people in New York City have three different vowels in horror, aura, and ford, but that's not a typical American English accent (and the majority of them probably only have two). – Peter Shor Jul 02 '19 at 12:55
  • @sumelic New Yorkers, as Peter mentions, were what I had in mind for the /ɒ/-like phoneme. I have definitely heard New Yorkers pronoun the vowel in box and cloth ([ɔ]) differently from the one in caught (something like [oə] or even [uə]). I can’t say for sure whether these people pronounced palm with [ɑ] or [ɔ] (which would make it only two vowels again), but if we use spa or father instead, there’s less variation. I’ll admit my “if you don’t have the cot–caught merger” conditioning wasn’t very accurate, though. – Janus Bahs Jacquet Jul 02 '19 at 14:18