"Take Five" (Dave Brubeck Quartet)

"Take Five" is one of those curious songs that seems to cross every boundary.  Even people who claim to not like jazz will embrace "Take Five" as one of their perennial favorite songs.  Personally, I think it has as much to do with its eternal cool factor as it does with its indelible saxophone melody.  Saying that you're a fan of "Take Five" is like casually flashing your membership card to the avant-garde.
Brubeck and his band had come back from a 1958 U.S. Department of State-sponsored world tour, where they had acted as "jazz ambassadors," sharing the uniquely American art form with developing nations and countries that had been cut off from the West by the Iron Curtain.  Along the way, they jammed with local musicians in places like India, Pakistan, and Turkey, where they soaked up the intricate folk rhythms, which often were in time signatures like 7/8 or 9/8—time signatures that were uncommon to jazz in 1958.
All of those experiences were distilled into the album Time Out (1959), which presented a set of seven original compositions, all in time signatures other than waltz (3/4) or common (4/4) time.
During one of the sessions for Time Out at Columbia's famed 30th Street Studio in New York, drummer Joe Morello was playing around with a rhythm in 5/4 time, and Brubeck asked saxophonist Paul Desmond if he could come up with a melody that played off Morello's beat.  Desmond doodled around for a bit and came up with two passages but struggled with how to unite them into a single song.  So Brubeck stepped in and added the staccato piano vamp, creating a foundation for the verse, and then figured out the release heading into the bridge section.  And "Take Five" (named for its 5/4 time signature) was born.
Curiously enough, the song is credited only to Desmond, even though it is the tune most associated with Brubeck.

"Unsquare Dance" (Dave Brubeck Quartet)

In 1959, the Dave Brubeck Quartet released the hugely successful Time Out—an album that proved it was feasible to play jazz in time signatures other than 3/4 and 4/4 and still retain the genre's heart and soul.  
In 1961, the band released the follow up, Time Further Out, which again explored tricky time signatures.  Like 7/4—the time signature of the track "Unsquare Dance."  
The title is an obvious play on "square dance," which if you attended a physical education class in the past 50 years, you no doubt were forced to perform under duress by a whistle-toting Johnny Unitas look-alike.
Just like the traditional music played at a square dance (or hoedown, if you will), "Unsquare Dance" has a folksy, two-step feel that seems readymade for do-si-do'ing and promenading.  Until you start to count it out, that is.  The twist is that extra beat at the end of each measure: 1-2, 1-2, 1-2-3.
What starts out kind of hokey sounding ("square" in beatnik-speak), reveals itself to be a wildly inventive, funky exploration of rhythm, with bassist Eugene Wright playing a skeletal bassline over handclaps, and Joe Morello playing syncopated percussion on the rim of his snare drum.  
Even when Brubeck returns at the end of the song to do his little "shave-and-a-haircut" ending, it's with a knowing wink.

"Blue Rondo à la Turk" (The Dave Brubeck Quartet)

Anytime anyone tells me they don't like jazz, I point them to either Miles Davis's Kind of Blue or The Dave Brubeck Quartet's Time Out, both from 1959.  
In the case of Time Out, I find that the uniqueness of the odd time signatures--which is the whole reason why the album is called Time Out--keeps anti-jazz curmudgeons engaged long enough that they actually start realizing how inventive and great jazz can be.
Basically, the album came about after Brubeck and his band went on a State Department-sponsored tour of Europe and Asia in the late 50s.  Along the way, Brubeck and co. kept hearing these intricate, non-Western folk rhythms that just blew them away.  In particular, the band had made a stop in Turkey, where Brubeck heard some street musicians playing this funky folk rhythm in 9/8 time: 
1-2, 1-2, 1-2, 1-2-3 / 1-2, 1-2, 1-2, 1-2-3 / 1-2, 1-2, 1-2, 1-2-3 / 1-2-3, 1-2-3, 1-2-3.
And that cinched it: their next album was going to avoid typical 4/4 and 3/4 time signatures as much as possible.
That Turkish folk rhythm, as you might have guessed, was the basis for "Blue Rondo à la Turk."  In fact, the name of the song spells out literally what's happening: the song is written in the rondo form, where the main theme (built on the Turkish rhythm in 9/8) cycles with alternating episodes (in swinging 4/4), which are based on a typical blues chord progression of 1-4-5.
Forgetting about the unique time signature for sec, it's just a great showpiece for Brubeck's chunky block chords and alto sax virtuoso Paul Desmond's light as a feather, fluid as quicksilver improvisations.  I also can't neglect Eugene Wright's bass playing or drummer Joe Morello's unfaltering time-keeping abilities.  In short, everything works on this track--easily a contender for the best-ever album-opening song.