Baroque music (at least in my admittedly limited understanding) is stately, ornamental and cerebral.  This is music that aims at your heart, but intends to get there through your head.  The elements that characterize baroque music to me are 1) basso continuo, or a distinct, foundational bass line; 2) counterpoint, which is the weaving together of unique yet complimentary musical parts that can each stand alone, but that combine to compose a greater whole; and 3) "featured instrumentation," or the tendency for a particular instrument to be the "star" of a composition or section.  Pachelbel's Canon in D (frequently heard at weddings these days) is a good example of these features.  I find much of baroque music to be rather challenging to listen to with my modern ears.  That's not to say it isn't good, or even great, it just generally takes more effort on my part to really appreciate it.
My Favorites: Concerts Royaux

My Thoughts: Couperin is quintessential baroque music.  Very frilly and regal, this is the stuff I picture lords and ladies minueting to (is that a verb?).  I'm not crazy about all begins to sound alike to me after a while (at least what I've heard of it), but  it's still very nice, quite pretty, and great as gentle background music.  Like I said of baroque music in general, I find that if I concentrate on Couperin and look for the counterpoint the music does start to stand out a bit more.  Couperin is also known among keyboardists (e.g., organists, harpsichordists, pianists) as a pioneer of keyboard composition, and was influential on later composers like Strauss and Brahms.
My Favorites: Violin Concertos in E, G Minor, F and F Minor (i.e., "The Four Seasons," especially Spring); Gloria in D; Mandolin Concerto in C

My Thoughts: It's taken me a while to get up the nerve to write about Vivaldi.  I remember listening to his music for the first time in my English class in high school, and being completely amazed by his ability to evoke the imagery of the seasons in his concertos.  Vivaldi is one of those rare artists who are undeniably products of their times, but also utterly transcendent of them.  Vivaldi's use of counterpoint is masterful, and his reliance on the virtuosity of the featured instrumentalist is definitely Baroque, but listening to Vivaldi is not as intellectually demanding as most Baroque music is (at least to me).  You really feel this stuff when you hear it, and it's elegant and ornamental, but not over-the-top.  In addition to his secular work, Vivaldi's Gloria in D is absolutely sublime, in a league with Palestrina's work.  This is beautiful music, and Vivaldi is one of my favorites.
My Favorites: Concerto in B-Flat for 3 Oboes; Concerto in D for Trumpet

My Thoughts: I realize I've already used "quintessential baroque music"  to describe Couperin, but the term really does apply to Telemann as well.  His work is like an elegant watch, intricately constructed and perfectly balanced.  This is cerebral music, and while it's technically close to perfection as its form is concerned, it just doesn't move me in the way that Vivaldi or Bach does.  This is not to say that Telemann's work isn't beautiful or pleasant to listen to; on the contrary, like the music of Couperin, Telemann makes great background music.  It just doesn't make me sit up and take notice in the way some of the true baroque masters' work does.
My Favorites: Menuet 1 & 2; La Laborde-Rondement; Les Indes Gallantes; Les Boreades: Overture

My Thoughts: If you combine my comments on Couperin and Telemann you'll have a good idea how I feel about Rameau.  To be fair, I had a hard time getting my hands on his music; the list above reflects the extent of my listening.  Also, it's worth noting that Rameau's biggest contributions to music were related to music theory, opera and the harpsichord, which are all pretty much beyond my direct interest.  Don't get me wrong; it's nice, pretty, tinkly harpsichord-centered baroque music, but given that I'm not especially enamored with the harpsichord Rameau's work doesn't particulary excite me.
My Favorites: coming soon!

My Thoughts: coming soon!
My Favorites: Messiah Oratorio (HWV 56); Music for the Royal Fireworks (HWV 351), especially the Overture & La Réjouissance

My Thoughts: Everybody's heard of Handel, whether they realize it or not.  I mean, in this day and age, who hasn't heard at least a snippet of the Hallelujah Chorus (even if it was as a ringtone, or as a clichéd sitcom sound byte)?  Handel is also fairly unique among composers of vocal classical music in that he wrote in English, and there's something to be said for hearing sublime music sung in your native tongue.  And the Messiah Oratorio certainly is sublime, stirring music utterly transcendant of its time and place, which is probably why it is still phenomenally popular.  Handel's other compositions are outstanding examples of Baroque Music, although for me they mostly fall short of the Messiah, overplayed and trite though parts of it have become to our jaded modern ears.
My Favorites: Orfeo ed Euridice (particularly the overture and the early chorus pieces)

My Thoughts: Music people largely recognize Gluck for his contributions to the development of opera, and thus his music is not really on my short list of absolute favorites (see Opera Disclaimer).  That said, I do respect the nature of Gluck's contribution to opera, insofar as he insisted on the fundamental importance of musical drama (as opposed to vocal virtuosity, for example, which was in vogue at the time).  Gluck tried to balance the amount of diva-oriented arias in his operas with more recitative and choral pieces, and it is these latter that I most enjoy.  Unfortunately, Gluck recordings are rather difficult to find, so Orfeo ed Euridice is really all I've been able to listen to, so perhaps my opinion will change a bit when I've heard more.

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