I have seen in some music theory books triads are called common triads or sometimes I see common chords. What does common exactly point to? Does common chord mean a complete chord? in that case do we have incomplete triads? if not what does "common" exactly mean for a triad?
Your book may use 'common' to simply mean 'basic'. Maybe as a synonym for 'Primary triads', I IV and V, the 'three chord trick'. But 'common' is also used to describe chords that appear in two different keys. F major chord is 'common' to C major and Bb major. It can therefore be used as a stepping-stone when modulating from C major to Bb major.
There are three possible meanings. One is more colloquial (the one mentioned by Tim) and another is more technical (pointed out by Laurence).
The third meaning--which I think is most likely--hasn't been mentioned yet, so I'll put them all together in one place. In general, a "common chord" could mean:
- a commonly-used chord (given that the context is a music theory book, this usage seems unlikely)
- a chord that is shared between two different keys (Laurence gives a great example)
- a chord with a defined chord quality (major, minor, etc.)
To understand #3, we must look at the word chord itself. In one sense, a chord is any two notes played simultaneously. But in another sense, a chord is three notes with a defined chord quality (major, minor, etc.). This second term is also called a definite chord, a common chord, or even triad. The modification common triad is a bit redundant, since any triad will have a defined chord quality. However, the author is likely simply calling attention to the fact that any triad has a defined chord quality.
I don't know of a term for the "opposite" of this, but we might call it an indefinite chord: for example, a chord with only two notes does not have a defined quality (major, minor, etc.).
Common here means most used. A triad consists of three notes (tri) and the common ones use root, third and fifth. In C, it's C E G; in Cm it's C Eb G. Another way to look at it is a pile of thirds - C>E is a third, and E>G is too.
There are other three note chords, such as sus chords, as in C sus 4, where the 3rd is replaced by a 4th. Not so common, and not 'stacked thirds'.
Can't imagine what an 'incomplete triad' might be! Possibly what guitarists call a power chord (5th)..?!
EDIT: Checking in my 'bible', the following is stated: triad= 3 note chord with root, third and fifth. Common chord = triad with perfect fifth. Major common chord = common chord with maj.3rd. Minor common chord = common chord with min.3rd. Aug.triad = 1, M3,#5, Dim. triad = 1, M3, b5.Those last two don't particularly apply to the question, but are still triads. No mention of 'common triads'...