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Double Bass Double Time

Hey all my fellow bassists. I've been in a school jazz band for a couple months and have heard the term double time be thrown around. I have some idea what it is, but would like some clarification. For instance, during a walking bass line, would I play 8th's or half notes instead of quarter notes if the drummer switched to double time?
 
"Double-time" means that a slow song starts moving twice as fast. It's usually temporary. It's usually used on ballads, although it doesn't "have" to happen every time you play a ballad. Double-time can take two forms:

a) The pulse goes twice as fast but the bar lasts as long as it used to. That means your 4/4 becomes 8/8 and you play every eighth note. Eight counts in bar one with each count twice as fast as it used to; eight counts in bat two etc.

b) The pulse goes twice as fast and the bar goes twice as fast. Four counts in bar one with each count twice as fast as it used to; four counts in bat two etc.
 
I would add that in jazz we are usually working with a triplet continuum "feel" underlying per each quarter note (bababa, bababa, bababa, bababa). Making everything a coloration of those triplets yields some semblance of swing (hopefully :)).

In double-time, this triplet continuum, obviously, now goes by twice as fast. Don't just play faster quarter notes...play/feel the faster triplet continuum, regardless of how you color it.
 
The way I learned it is, this...
a) The pulse goes twice as fast but the bar lasts as long as it used to. That means your 4/4 becomes 8/8 and you play every eighth note. Eight counts in bar one with each count twice as fast as it used to; eight counts in bat two etc.
is called "double time feel", and this...
b) The pulse goes twice as fast and the bar goes twice as fast. Four counts in bar one with each count twice as fast as it used to; four counts in bat two etc.
is "double time".