JLPT 2 has this word 短~ (たん)
EDICT lists it as
fault; defect; weak point; or
minor (music), however there isn't any examples at all on the usage of the word.
Does anyone have some example usages (applications) of the word ?
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There are two set phrases that come to mind:
Using 短 in this way is extremely kanbun-y and, apart from set phrases like the above, as Dave M G says it would be unlikely to occur in regular speech.
In the musical sense of "minor", 短 is used as a prefix, like this:
Note names and sharp/flat terminology are a whole different question (hint), but basically 短調 = "minor key", 短音階 = "minor scale", 短三和音 = "minor triad", etc. You can replace 短 with 長 to get the major version of any of these.
Using 短 in the context of music to mean "minor" is completely normal -- in fact it is the standard way to express the concept of "minor" found in classical Western harmony.
I note that you cite the word as "短~"; could it be that JLPT 2 is referring to the use of 短 as a prefix? (I ask because of the squiggly line after it which often indicates "insert something here" in Japanese language learning texts.) As Dave says, 短~ appears this way in 短所 "weak point, fault", 短気 "short-tempered" etc. with the meaning "insufficient, faulty" but it also very commonly carries the meaning of literal shortness: 短期 "short term", 短距離 "short distance", 短銃 "pistol (= short gun)", etc. You wouldn't learn this searching for 短 on its own, though, as AFAIK 短 as the standalone Sino-Japanese word たん only has the figurative meaning of "fault" discussed above, and not the literal meaning of "shortness".
That definition is very uncommon. A word like
短所【たんしょ】(weak point) uses it in that sense, but in the vast majority of cases, the kanji
短 means "short" and changes words in a way that has that meaning in mind.
Take a look at this list of words beginning with
短 to see how it's used.