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I understand that I, IV, and V chords in both major and minor keys are common chords that make up a good progression. I understand that those 3 chords cover all the notes diatonic to the key that we need to make up a melody:

 -  scale degrees    
I:  1, 3, 5
IV: 4, 6, 1
V:  5, 7, 2

At what stage do we need to use other chords, like ii, iii, vi, or viii?

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Woody Guthrie said, "If play more than three chords, you're just showing off." We use the ii, iii, vi and VIIdim pretty much to show off. – Dave Jacoby Aug 29 '12 at 1:39
These are all fine questions, but it would be best if you would split them up into three separate questions. – American Luke Aug 29 '12 at 1:58
I'm going to answer the middle question--the other two are much more disjunct from the question title. – NReilingh Aug 30 '12 at 19:17
Specified the question, and added some information to clarify what the asker (assumedly) already understands. – NReilingh Aug 30 '12 at 19:33
There are scholarly answers to this vote-worthy question, but my simple answer is "You use more than three chords when you learn enough about music to know more than three chords." – user1044 Aug 31 '12 at 14:21

At what stage do we need to use other chords, like ii, iii, vi, or viii?

To cut through the academia: we need to use other chords when they make the sound we want to hear.

Let's say you're singing in the key of C major.

Sing a G over a C chord. It sounds a certain way.

Sing a G over a G chord. It sounds different.

Sing a G over an Em chord. It sounds different again.

Sing a G over an Am7 chord. It sounds different again.

Those are all chords in the key, which contain a G.

Which of those is correct at a given moment in the melody? That's entirely subjective. You can change how a melody is perceived by changing the chords below it.

Of course, familiar tunes usually have a familiar chord progression, and if you deviate from that, it will jar to the audience.

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Given the set of all possible diatonic melodies, you could theoretically harmonize them using only three chords, but in doing so you give up control of the harmonic rhythm. You would end up changing chords at odd times in order to catch melody notes where they fall.

Consider a tune in C major, and in one bar you have the notes E, C, and A--in that order. To harmonize this (albeit in a very rudimentary and amusical way; though it serves to illustrate the point), you could use a I chord for the first two notes and a IV chord for the last one.

However, the rhythm of the notes could be anything, and could dictate that you change chords at a strange time. Also, the harmonic rhythm of the music may only be asking for one chord change for the entire bar! To solve this problem, I would use a vi chord--it covers all of the melody notes in that bar, and does not require a change in the middle.

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Luckily, not all melodies use only the note material from a diatonic scale. Most songs use leading-tones to move the melody from one mode to another (which is called tonal modulation).

Furthermore, a lot of modern Western music use a circle progression, in the harmonization of a melody, as each chord has its own place and feel in a harmonic progression. In a circle progression, you'll often see chords getting "dominatized" (making a minor chord to a major), to create suspense and a feeling that the chord pulls towards another chord.

Theoretically you could harmonize a melody (which only used notes from the diatonic modes!), with only I, IV and V, but much more often you'll want to break this pattern up with different cadences, such as ii - V - I, the interrupted cadence V - VI to create a more livid and exciting arrangement.

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