In music, the distance between any 2 notes is called an 'interval'.  There is a helpful way to memorize intervals, and here's an example.  If we are in the key of C, then C is 1,
C = 1, D = 2, E = 3, F = 4, G = 5, A = 6, B = 7, C = 8.
The interval between C and A is a 6th.
Now, let's invert-- what is the interval between A and C?
A = 1, B = 2, C = 3, D = 4, E = 5, F = 6, G = 7, A = 8
The interval between A and C is a 3rd.

The interesting trick here, is that these 2 intervals (C to A, A to C) are inverse of each other, and the sum of the intervals is 9. 
6 + 3 = 9

This works for all these intervals.  Here's another example:  What is the interval between C and F?
a 4th
What is the interval between F and C? 
Well, 4 + x = 9,
9 - 4 = 5
The interval between F and C is a 5th. 

I'll let you figure out the rest of the intervals-- in all keys, too!  Refer to the circle of Fifths to learn how to spell all the scales you need.

I had a profound realization recently, when I was studying chess moves.  (This should indicate the magnitude of my nerdiness, if nothing else does.)
In chess, the board looks like this:

Notice, going from the White king to the Black king, there are 8 rows (or ranks).  Just like in music--  The first row is 1, and the 8th row is the octave.  Like in music, when you inverse an octave, it's just like a mirror image-- high C to low C.  In chess, when you inverse the octave, and you make the Black king your own, and the 8th row becomes your back rank.

Look at Whites 2nd rank.  All his pawns are there.  Look at Blacks pawns-- where are they?  All on the 7th rank.  In chess, the inversion of the 2nd rank is the 7th-- White becomes black, but its weirdly the same. Notice=
2+7=9

Just like in music.  In chess, the center of the board is crucially important, and for black and white both, there is a center line.  When you cross this line, you enter the enemies turf.  For White, that is the 4th row.  For black, its the inverse-- the 5th.
4+5=9

The same principle applies going side to side.  The columns, or 'files' go from a to h-- think of 'h' as the octave of 'a'.  The inverse of file 'a' is 'h'.  The inverse of file 'd' is 'e'.

I love the symmetry here, amongst the numbers, and amongst two wildly different yet venerable pastimes.  Some people say that music is the emotional expression of math.  And others say that chess, despite its rigid rules and logic, is ultimately an artistic expression.  It's almost as if they are inverse of each other.

This lecture transcript has been floating around now for a while, I thank a friend for forwarding it, one of those 'inspiring email forwards', but for musicians, and actually inspring. 

Karl Paulnack is on faculty at the Boston Conservatory, and he gave a commencement speech to the new students of 2004.  He likens the role of a musician to more of a therapist and medical professional than entertainer.  Although he is speaking to future professional musicians, I think the message is also relevant to anyone who struggles to learn to play.  The act of playing your instrument, even alone, is a powerful means of experiencing music.  all the benefits of music can come through even the most basic of practice regimens (if you make sure to keep you practicing fun!)

Here's a link to the article where I read it:
http://greenroom.fromthetop.org/2009/03/11/karl-paulnack-to-the-boston-conservatory-freshman-class/