Studying or practicing Ear Training?  If you're a musician, you should be.  You'll find my Palm software Ear Trainer invaluable.  Also, I teach guitar lessons in Orange County, CA.  (See OCGuitarLessons.com)

 
 
An extra string on your guitar can extend your harmonic capabilities.  Usually the extra string extends the pitch range on the bass side, tuned to a low B or A.  Unfortunately, this low of a pitch at the usual scale-length of a guitar begins to become "muddy"-sounding, owing to increased inharmonicity.  (Some luthiers, esp. Ralph Novak, employ a fanned-fret system in an attempt to remedy this.)

Some seven-string players, notably Lenny Breau, have added a higher string.  Unfortunately, to tune an added string a fourth above the existing high E is usually problematic.  For a nylon-string guitar, Breau discovered a particular brand of fishing line effective, but to avoid breakage on steel-string he either had to tune down or employ a custom-built instrument with a shorter-than-usual scale-length.

A less common employment of the extra string is within the usual range, to increase the richness of voicings, e.g. in Major Thirds Tuning on a seven-string the usual range of E2 to E4 is maintained but patterns are simpler, chords may be voiced more richly, and keyboardistic (or "pianistic") playing is easier.

Exploits and Customizations

Yamaha 12-String Acoustic Converted To a Scalloped-Fretted 19-String
(7 Playing Strings Plus 12 Sympathetic Strings)

DeArmond S67 7-String

Epiphone Les Paul 7-String

Fernandes Revolver 7-String with Sustainer, Reshaped

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