12  = 
   12 = 11+1 = 10+2 = 9+3 = 9+2+1 = 8+4 = 8+3+1 = 7+5 = 7+4+1 = 7+3+2 = 6+5+1 = 6+4+2 = 6+3+2+1 = 5+4+3 = 5+4+2+1
Here each sum accumulates mutually different magnitudes; these 15 exhaust all the possibilities (when the summand order doesn't count). 

12  = 
  11+1 = 9+3 = 9+1+1+1 = 7+5 = 7+3+1+1 = 7+1+1+1+1+1 = 5+5+1 = 5+3+3+1 = 5+3+1+1+1+1 = 5+1+1+1+1+1+1+1
   = 3+3+3+3 = 3+3+3+1+1+1 = 3+3+1+1+1+1+1+1 = 3+1+1+1+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1+1
Now each sum accumulates odd magnitudes, and likewise these 15 exhaust all possibilities. 

15 possibilities both times; this is not a coincidence nor is it a special property of 12.  Start with any whole number and it turns out that the two tasks always lead to the same numbers of solutions.  

Now a claim like that requires proof; just testing finitely many cases does not suffice.  One sort of proof involves inventing some way of seeing an instance of one sort sum as a secret code for an instance of the other sort of sum (and vice-versa).  Then the implicit codebook matches up the two sets of solutions.  

Suprisingly, mathematicians know more than one recipe here for doing the coding and decoding.

And starting specifically with 12, coding by one recipe and decoding by the other happens to walk you back and forth between the all the solutions of both problems.  The structure of this piece is that process, with all the intermediate steps. The rhythms alone sonify the sums, but the pitches will guide you through the execution of the coding.

Different and Odd, for alto[aka tenor] horn and piano

Composer, mathematician, visual artist. 
Seeks collaboration with performers and choreographers.

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Different and Odd, for alto[aka tenor] horn and piano

David Victor Feldman