I'm looking for the definition of a drum beat where the main beat is slow, about 70 bpm, but the sub beat is about 3 times faster. What is the official name/style of that playing? for example andalusia by joe satriani.
Not sure about 3 times, but sometimes if the beat is constant and somebody is playing something that seems twice as fast we may refer to that person playing double time. An example would be somebody switching from quarter notes to eighth notes.
On the flip side, if someone is playing half-time it has the opposite feel. Changed quarter notes into half notes. Here is an example of a drummer showing half-time:youtube video
As the comments suggest, you might benefit by reading up on some basics of musical notation. THe "Beats per minute" term, while often used, is normally referring to the fundamental meter of the piece. For example, in 3/4 time, there are three beats per measure, quarter note getting a beat.
But (duh), any musical instrument can play multiple notes per beat. If your putative drummer is playing sixteenths, there'd be four 'hits' per beat. If he's whacking the bass drum on the first quarter note of the measure ("downbeat") and hitting something else on every quarter note, you'd be hearing a 3-to-1 rhythm. And so on.