My Pyraminx Solution
There are many ways to approach almost any puzzle you find.  After a lot of thought and practice, this is the pyraminx solution that I find most productive.  It doesn't take much to learn.  Most of the time is lost in recognition, until you get used to the puzzle, then the solves become more fluid.  I have broken 9 seconds for an average, and maintained an average number of turns of 20 (somewhat consistently).  Since there is no 'official' notation for they pyraminx, I made my own. (You can find it >>here<<)
Step 1- Pick a tip, and correctly orient it.  This is so trivial that I hesitated counting it as a step....
Step 2- After correcting a tip, place 2 adjacent edges to that tip.  Notice you are leaving an empty edge slot (A)  after doing this.
Step 3- Using the empty 'A' slot, you are going to correctly orient centers.  (i.e. adjust center, and then move on to next center by making (D) turn. Adjust that center, and then move on to the next one by making a (D) turn.  And so on.  Sometimes the centers will already be oriented correctly.
Step3b- As a side note, I solve the other 3 tips during Step3.  While making the (R) or (R') turn to orient the center, I also adjust the tip.  It's a little awkward at first, but I've found this is the fastest way to solve the remaining three tips.
Step 4- Now it's time to place the edge that belonged in the 'A' slot.  It is important not to effect center orientation in this step...
<---Sequence:  L' R' L R
(((((Mirror)))):
R L R' L'
Step 5- Orient and Permute remaining 3 edges.  All the possible sequences are below:
R' L R L' U L' U' L
U L' U' L
U' R U R'
L R' L' R' U' R' U
R' L R L U L U'
>>Main Page<<
Summary:

Step 1 + Step 3b (tip solving):  2.66 turns avg.
Step 2:                                  ~4.50 turns avg.
Step 3:                                   4.00 turns avg.
Step 4:                                   4.66 turns. avg.
Step 5:                                   5.83 turns avg.