How would you go about listing all of the integer ratios? I would like to have the integers listed in ascending order by the product of each integer's numerator and the denominator. This is because in music, more "complex" ratios sound more dissonant.
Every list would exclude ratios that can be simplified (2/4, 3/9).
List A1 would include those ratios with values less than or equal to 1, and List B1 would include those ratios greater than or equal to 1.
Lists A2 and B2 would exclude ratios of 1/n and n/1 where n is a composite number. List A3 would exclude ratios with values less than 1/2. List B3 would exclude ratios with values less than 2/1. Lists A4 and B4 would use both exclusions.
I think this is what I'm aiming for: List A1 = {1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 2/3, 1/7, ...} List A2 = {1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 3/2, ...} and so on.
Finally, is there anyone out there who understands how these lists relate to music theory? Thanks!
To list all the integer ratios as described, you can follow these steps:
1. First, create a list of all possible ratios by considering all pairs of positive integers (n, m) where n is the numerator and m is the denominator.
2. Next, eliminate simplified ratios by removing any ratios where both the numerator and denominator have a common factor greater than 1. For example, you would exclude ratios like 2/4, 3/9, etc.
3. Now, you need to sort the remaining ratios based on the product of each ratio's numerator and denominator. This will help you arrange the ratios in ascending order of complexity. The greater the product, the more "complex" the ratio.
4. Divide the sorted ratios into different lists based on the specific exclusions you mentioned. For example, List A1 would include ratios less than or equal to 1, List B1 would include ratios greater than or equal to 1, List A2 and B2 would exclude ratios with the simplified form of 1/n and n/1 where n is a composite number, and so on.
5. Once you have divided the ratios into different lists based on exclusions, you can further filter List A3 and B3 to exclude ratios less than 1/2 and 2/1, respectively.
6. Finally, you would end up with the desired lists, such as List A1, List B1, List A2, List B2, List A3, List B3, List A4, and List B4.
Regarding how these lists relate to music theory, these lists represent different intervals in music. In Western music theory, intervals are classified based on the ratio of frequencies between two pitches. For example, the ratio 2/1 represents an octave, 3/2 represents a perfect fifth, and so on. By listing these ratios, you are essentially exploring various musical intervals and their relative complexity or dissonance. The idea is that ratios with higher products (greater complexity) may sound more dissonant or less harmonious compared to simpler ratios.