recreational math, music
posted by Gray on .
How would you go about listing all of the integer ratios, preferably in a spreadsheet? I would like to have them listed in ascending order by the product of the numerator and the denominator, because in music, more "complex" ratios produce more dissonance. One list would include those ratios with values less than or equal to 1, and the complementary list would include those ratios greater than or equal to 1. The every list would exclude ratios that can be simplified (2/4, 3/9). A second set of two lists would exclude ratios of 1/n where n is a composite number, a third set would exclude ratios with values less than 1/2 (or greater than 2 in the complementary list), and a fourth set would use both exclusions. I think this is what I'm aiming for: 1/1, 1/2, [1/3], [[1/4]], [1/5], 2/3... along with the complementary list: 1/1, 2/1, [3/1], [[4/1]], [5/1], 3/2... where brackets indicate excluded ratios. Does anyone understand how the proposed exclusions relate to music theory? Thanks!