Intro 

The baroque guitar had nine or ten strings in five courses, a slightly rounded back intermediate in construction between a lute and a modern classical guitar, and frets fastened around the neck similarly to the lute. 

Popular in the baroque period of music (say 1600-1750), it is the earliest likely ancestor of the modern guitar generally given the name guitar (but see Historic guitar tunings for earlier instruments and particularly earlier four course guitars) and the direct ancestor of the six-string romantic guitar which introduced the E-A-d-g-b-e'  tuning and was itself the ancestor of the modern classical guitar and from there all subsequent guitars. Its most likely ancestor is the Spanish vihuela of the 15th and 16th centuries (and not to be confused with the later South American vihuela), but there is much speculation on this point and also on the origin of the name.  

Modern instruments are generally used for performance of baroque music rather than for newer repertoire.

Scale length and pitch

About 27.1" (688mm) which is considerably longer than a modern guitar, and requires either lighter strings, tuning to a pitch significantly below the modern standard of a'=440Hz, or both.

Some authorities recommend tuning one semitone flat, which would give a'=415.30Hz, others recommend tuning a' in the range 416-430Hz.

Some modern (and not very authentic) reproductions use a shorter scale length and fixed wire frets, to accomodate modern guitarists and allow them them to play easily with other instruments. These can use commercially available guitar strings and a'=440Hz tuning. 

Strings and gauges

The baroque guitar predates wire-wound bass strings, which may cut into the traditional gut frets. Plain gut for all strings is probably most authentic, but is rejected by many modern players for the same reasons it was abandoned when economical wire wound strings became available.

Tunings 

Baroque guitar tunings are generally A-D-G-B-E, but wthin this pattern there is some variation:

• The first course may be single (nine string tunings) or double (ten string tunings).
• There may be a reentry between the the G and D courses, or between the D and A courses, or the tuning may be non-reentrant. 
• The A and D courses may be octave tuned, or just the A course may be octave tuned, or all courses may be unison.

Not all possibilities within this pattern are attested. Some that are attested are listed below.

Nine string tunings

As described by modern musicologist James Tyler:

• a a - d d - g g - b b - e' (nine-string, reentrant between 4th and 5th courses, all double courses unison)

• a' a - d d - g g - b b - e' (nine-string, reentrant between 4th and 5th courses, octave A)

• a' a - d' d - g g - b b - e' (nine-string, reentrant between 4th and 5th courses, octave D and A)

Ten string tunings  

Gaspar Sanz  (4 April 1640 – 1710), Spain

• a a - d' d' - g g - b b - e' e'  (ten-string, reentrant between 3rd (G) and 4th (D) courses)

Robert de Visée (ca. 1650  – 1725), France

• a a - d' d - g g - b b - e' e'  (ten-string, reentrant between 4th (D) and 5th (A) courses, octave D)

Girolamo Montesardo ((fl. 1606–c.1620), Italy

• a A - d' d - g g - b b - e' e'  (ten-string, non-reentrant, octave D and A)

External links

http://en.wikipedia.org/wiki/Baroque_guitar contains diagrams of the various tunings

http://www.classicalguitar.ws/baroque.shtml more explicit diagrams and some descriptions

http://www.cs.dartmouth.edu/~lsa/aboutLute/BaroqueGuitar.html Lute Society page gives the Tyler tunings

http://www.cincinnatiearlymusic.com/baroque_guitar.html Building a Baroque Guitar

http://www.earlyromanticguitar.com/erg/evolution.htm fascinating new theories on the evolution of the guitar