According to a book on American English pronunciation ('Lesson 12: Central and Back Vowels', page 136. I don't know the name of the book, though), /a/ is the vowel of father, box or calm. However, almost every dictionary out there establishes a difference between /a:/ and /o/, which would be the equivalento to what the book calls /a/. I find it a conflciting message. Or are the sounds of /a:/ and /o/ alike after all as the book indicates?

This way, /a:/ is found in arm, father and /o/ is found in hot, rock.

Note I chose to write the phonetics symbols in the ASCII code.
I'm not quite sure I understand your symbols. For one thing, North American English doesn't really distinguish vowel length, vowel phonemes are realized as long vowel allophones before voiced consonant phonemes in the coda of a syllable. Secondly, [ o ] is used in words such as "hope" or "vote", in dialects that have monophthongized [ oU ]. [ a ] is used in NCVS shifted dialects.

In the Western US, and Western and Central Canada, arm, father, hot, cot, rock, dawn, and caught are all pronounced with [ A ]. People with the Canadian shift or the California shift, pronounce them all with [ O ] .

There are some people that lack the cot-caught merger, and thus pronounce "cot" as [ kAt ] and "caught" as [ kOt ]
Some people from the Northern Midwest pronounce "cot" as [ kat ] and "caught" as [ kAt ] .

The bother-father merger is almost universal in North American English, except in Northeastern New England, such as the Boston accent, and New York and New Jersey English.

If you're aiming for a Western accent, you can simply pronounce all of them with [ A ] . (The Western accent is considered one of the least accented varieties of North American English.)

[ O ] is the open-mid back rounded vowel
[ A ] is the open back unrounded vowel
[ a ] is the open front unrounded vowel
This is where I took the symbols from:

/o/ is located in the ninth line, for example.
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Ok. Here's the thing. Some North American dialects do not possess the father-bother merger: such as Boston, New York, and New Jersey. All three of those dialects are considered non-standard, and I don't recommend that you learn them. Virtually all other dialects use exactly the same vowel in words such as father and bother. Many Southern dialects use a different vowel in the words cot and caught. I also do not recommed that you learn a Southern accent. Most of the Northern Midwest makes a distinction between words like "cot" and "caught", but a new shift is causing "caught" to be pronounced as "cot"--although "cot" shifts as well, so they remain distinct. Because of this new shift, the closest dialect to "General American", (the dialect that is considered neutral and unaccent to most people) is currently the dialect that is spoken in the Western US, and Western and Central Canada. In this dialect, father, bother, cot, and caught all are pronounced with the same* vowel. The vowel that is used for those words is dependent upon whether the speaker possesses the California or Canadian vowel shift. Even if other speakers make a distinction between "cot" and "caught", or even "father" or "bother", speakers from this dialect will perceive them as being simply variations on the *same sound.

Therefore, some speakers use different vowels, other speakers do not. I would recommend that you use the [url=" "]Open back unrounded vowel[/url] for all of those words.
First off, the New York and New Jersey accents are father-bother merged, as are the rest of the Mid-Atlantic accents (Baltimore, Philadelphia, Wilmington). Only New England is unmerged.

Secondly, considering that roughly half the country is cot-caught merged, while the other half is not, doesn't it make more sense for the OP to adopt the speech patterns of wherever he plans on living? Whether you merge or not, you will still sound accented to at least half the country.
Just in case you didn't notice it, you are responding to a post that is more than four years old.

That's OK, but I'm just saying that I don't think you'll get a response.

Students: We have free audio pronunciation exercises.