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!
To list all the integer ratios in a spreadsheet, you can follow these steps:
1. Open a new spreadsheet document, such as in Microsoft Excel or Google Sheets.
2. Create two columns: one for the numerator and one for the denominator.
3. Start by listing the ratios with a numerator of 1 and a denominator greater than or equal to 1. Example: 1/1, 1/2, 1/3, and so on.
4. Expand the denominator column to include higher values, if needed.
5. Continue by listing ratios with a numerator greater than 1. Example: 2/1, 3/1, 2/3, and so on.
6. You can use formulas to automatically calculate the product of the numerator and denominator in a separate column.
7. Sort the spreadsheet based on the product column in ascending order.
8. Exclude ratios that can be simplified, such as 2/4 or 3/9.
9. For the second set, exclude ratios of 1/n where n is a composite number (a number with more divisors than 1 and itself).
10. For the third set, exclude ratios with values less than 1/2 or greater than 2 (the reciprocal values of your proposed limits).
11. For the fourth set, combine both exclusions and exclude the ratios accordingly.
Regarding the relationship to music theory, the exclusions you mentioned are related to the concept of consonance and dissonance in music. Consonant intervals or ratios are typically simpler and more stable, while dissonant intervals or ratios are more complex and create tension. By excluding certain ratios, you are focusing on the more complex and potentially dissonant ones, which can be used for specific musical effects or harmonic explorations.