The Syracuse conjecture

Pick an integer between 1 and 10. If it is even, divide it by 2. If it is odd, multiply by 3 and add 1. For instance, a starting integer of 5 gives 16. Then repeat the procedure: 16 is even, so it gives 8, which gives 4, 2, 1 and then we go back to 4 and the resulting sequence remains 4,2,1,4…

The same happens with 8: 8,4,2,1,4… or 7: 7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1,4…

Mathematicians believe that the same is true for any positive integer. If we assign a note to each number, we obtain melodic motifs of different complexity all ending the same way. In a similar way, we could develop a harmonic structure based on these series.

Any iterated arithmetic procedure can be used similarly. For instance, the triangular numbers: 1, 3, 6, … n(n+1)/2:

Triangular numbers

Again, these ideas could be applied to any structural aspect, be it melodic, harmonic, rhythmic, etc. It is for the composer to decide how to assign a musical content to the numerical series described here.

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