Often, in pronunciation guides for English speakers learning another language, they’ll instruct you to pronounce that language’s ‘a’ sound like the ‘a in father’. Now, the problem with pronunciation instructions like this is that in English especially, people have different dialects and will pronounce vowels differently. There are lots of examples I could give to show how this complicates things, but I’ll just focus on the case of ‘father’.
Sounds conventionally transcribed with ‘a’ in the Latin alphabet are usually open vowels, but may differ in their front or backness. The IPA provides three symbols for ‘a’ sounds: æ, a and ɑ (plus ɐ which is not completely open, so I’m not considering it in this discussion).
æ is prototypically the symbol for a front vowel, with some slight closing. The symbol reflects this, in that it is the closest a sound to an e. ɑ is the back counterpart, although it doesn’t imply any closing. It gets a bit more confusing with a, which doesn’t really have any specific prototype. It can be used to refer to any open vowel in the middle.
Within the range of sounds covered by a, there are two useful points. One is like æ, but without the slight closing. One is the most common vowel sound in any language: a central, open vowel. To disambiguate, the centralisation diacritic is sometimes used for the second sound, giving ä. But this level of precision is rarely used, and when you see that a language has a phoneme pronounced [a] you can’t be certain whether this is a front open vowel or a central open vowel.
Now, in English, we have the full range of different ‘a’ sounds. Quite universally in American and Australian English, and traditionally in British English, an ‘a’ as in ‘cat’ is pronounced [æ]. And this is the symbol used to transcribe the phoneme when talking about all English dialects.
Most English dialects have another ‘a’-like sound: the one found in ‘father’. This is always, as far as I know, further back than the ‘a’ in ‘cat’. In most North American English, this is merged with the ‘o’ sound of ‘cot’. Both are transcribed with [ɑ].
So, from a North American point of view, the instruction to pronounce ‘a’ in, say, Japanese (which has the usual central [ä]) as the ‘a’ in ‘father’ is pretty sensible, if not perfect–[ɑ] is closer to the centre than [æ] is. Especially since in many American dialects, these two phonemes are undergoing a shift where [æ] gets even closer and fronter to approach [e], and [ɑ] moves to the front and becomes pronounced more like [ä]–for these speakers ‘father’ is the perfect example.
But for non-North American dialects, it’s not perfect. For example, I, being from England, pronounce ‘father’ as [fɑːðə]–with a long [ɑ]. If someone told me to pronounce the ‘a’ in Japanese like I do in ‘father’, I might end up pronouncing ‘katakana’ as [kɑːtɑːkɑːnɑː]. Which would sound comically wrong, and take much longer to say than the actual pronunciation of [kätäkänä] (I assume so anyway, I don’t know the details of Japanese phonology).
For people like me who have an [ɑ] that’s always long, which includes most speakers from England, Australia and New Zealand, [æ] would probably be a closer approximation to the [ä] sound. Although at some point, we’d just have to accept that this is a sound with no proper equivalent in most varieties of English.
Note ‘most’. Because it actually gets worse–in Northern England, Wales, Ireland and Scotland*, the ‘a’ in cat is pronounced as [a], with varying degrees of backness–for all speakers it has none of the slight closing of [æ], and for some it’s a fully central [ä]. Plus, a full [æ] pronunciation in England is now either old-fashioned or vernacular–even in southern England, many people use the northern [a] sound. So for these speakers, there’s actually a really good approximation of [ä] in their native accent, but you’re telling them to use an different phoneme that’s much less like it!
* Although for many speakers in Scotland, the ‘a’ in ‘cat’ and ‘father’ are not distinguished, so telling them to use ‘father’ works well enough but you could just as well tell them to use the vowel in ‘cat’.
I guess the lesson to learn is: don’t rely an approximations for foreign phonemes on terms of your native language. Listen to the language’s speakers, and use the sound they use.