I don't quite understand the difference between /a/ and /æ/. Google gives the transcription for 'add' as /ad/, while Wiktionary returns /æd/.
Are these sounds actually distinct or is this just two different ways of writing the same one?
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There are many different ways of writing English phonetically. Linguists and people interested in language (such as you often find on this site) often use the International Phonetic Alphabet (IPA) system.
According to that system, the vowel sound in add is written /æ/.
Google's transcription isn't incorrect, it just doesn't use IPA symbols.
Google, of course, isn't a dictionary. It outsources those tasks, and then, for some reason, integrates the information into its search results in such a way that it appears to be Google's information.
Beneath the "Dictionary" box that they display, there's a link that says, "From Oxford", which takes you to the Lexico website. This site appears to use the New Oxford American Dictionary transcription method, or "NOAD" in this list: Traditional respelling systems
If you look at that handy chart, you'll see that IPA's "æ" corresponds to NOAD's "a"
Unfortunately, I managed already to lose access to the account from which I posted this question. Since new accounts can't comment, I can only reply to @Juhasz by adding an answer. I understand that this is not what answers are for, but, unfortunately, this is currently my only option.
I don't think Google uses the NOAD system, since it doesn't, say, use /ē/ or /ä/ in its transcriptions. According to this page, Lexico doesn't use /æ/ for its UK dictionary at all. All other symbols, however, appear to be the same as they are in the IPA. I find this rather weird. Searching for 'aid' gives /eɪd/, by the way.
The US dictionary entry gives "/ad/ /æd/" as transcriptions, but the UK dictionary entry only gives "/ad/". This probably means that google uses the transcription from the UK dictionary, despite linking to the US counterpart.
It seems tempting to assume that /æ/ and /a/ are two symbols representing the same sound, but according to the IPA Vowel chart that LPH linked to, this isn't the case. Could it perhaps be that /æ/ isn't used in RP?
I'm still rather confused which sound best fits words such as 'add' or 'cat' pronounced in RP, but /a/ seems more like how I pronounce it. Even though Cambridge gives /æd/ as the transcription, the audio sounds more like /ad/ to me.
The best is to refer to IPA pronunciation, which you can check here. /æ/ is the typical pronunciation of "a" in English, the so called short pronunciation (as in "hat", "man", "bat"). It is the symbol retained for short "a" in Longman Pronunciation Dictionary (Cambridge, successor of Jones's dictionary), which dictionary reproduces faithfully RP pronunciation (including as well GenAm and widely used in France and all over the world). /a/ is not found in English; it is a vowel found in French, for instance; it is a so called cardinal vowel, that is, a vowel that phoneticians have chosen for reference (ref.).
Complement suggested by the comments (user john Doe)
Does this mean that Google's transcription is incorrect? I am mostly interested in RP, by the way.
I wouldn't say "incorrect"; it means rather that the people recording the pronunciation wanted to take into account a variant, which diverged from main stream pronunciation or from what use to be main stream and has been more or less replaced. I listened to google's pronunciation (add) and I must conclude that the sound is faithful to the transcription: this /a/ is an allophone of the French /a/ (meaning "very similar" to it). One problem is the lack of reference to any accent (Is that international English? I don't know.). I must say too that this corroborates my finding elsewhere of both pronunciations side by side; this means that there is a tendency, that is more or less important, of a shift towards the cardinal vowel. Nevertheless, most dictionaries do not record this pronunciation and mention only /æ/, the traditional or RP sound (which is still pronounced normally). Moreover they do take into account the accent, as in this source (Cambridge) where you can see that the vowel quality is the same in RP and in AmE, although the length has not been taken into account as the transcription for AmE should have been /æ:/ (the sound is even a little too long, other sources do have a shorter sound).
The vowels [a] and [æ] are close to each other.
Some phoneticians consider that the vowel of add or shack in modern British English has changed from [æ] to [a], and so some (not all) British dictionaries now represent it by /a/. See Lexico. The vowel has not changed in American English, so /æ/ is the vowel in add or shack in American English. See Lexico's U.S. pronunciation.
See this blog entry for some sound clips demonstrating how this vowel has changed.