The very weird word counter INPUT: Any string consisting exclusively of lowercase letters via function argument, command line argument, STDIN, or similar.

OUTPUT: Print or return a number that will represent the sum of the distances of the letters according to the following metric:

You take the first and second letter and count the distance between them. The distance is defined by the QWERTY keyboard layout, where every adjacent letter in the same row has distance 1 and every adjacent letter in the same column has distance 2. To measure the distance between letters that aren't adjacent, you take the shortest path between the two.

Examples:

q-&gt;w is 1 distance apart
q-&gt;e is 2 distance
q-&gt;a is 2 distance
q-&gt;s is 3 distance (q-&gt;a-&gt;s or q-&gt;w-&gt;s)
q-&gt;m is 10 distance


Then you take the second and third letter, then the third and fourth, etc., until you reach the end of the input. The output is the sum of all those distances.

Example input and output:

INPUT: qwer
OUTPUT: 3

INPUT: qsx
OUTPUT: 5

INPUT: qmq
OUTPUT: 20

INPUT: tttt
OUTPUT: 0


Here is an image showing which letters are in the same column:



This is code golf, so the shortest code in bytes wins!
Python 2, 220 ... 124 119 Bytes

Huge thanks to Sp3000 for saving a lot of bytes.



f='qwertyuiopasdfghjkl zxcvbnm'.find
g=lambda i:sum(abs(f(x)%10-f(y)%10)+2*abs(f(x)/10-f(y)/10)for x,y in zip(i,i[1:]))


Usage:

g("tttt") -&gt; 0


Check it out here.

Slightly ungolfed + explanation:

f='qwertyuiopasdfghjkl zxcvbnm'.find  # Defining keyboard rows and aliasing find
g=lambda i:                           # Defining a function g which takes variable i
    sum(                              # Sum of 
        abs(f(x)%10-f(y)%10)          # horizontal distance, and
        + 2*abs(f(x)/10-f(y)/10)      # vertical distance,
        for x,y in zip(i,i[1:]))      # for each pair in the zipped list

# Example of zipping for those unaware:
# Let i = asksis, therefore i[1:] = sksis, and zip would make
# the list of pairs [(a,s),(s,k),(k,s),(s,i),(i,s)].