math and music
posted by Gray on .
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!

There are an infinite number of integer ratios that cannot be reduced to ratios of smaller integers. Only a few of these play a roll in music composition.
Furthermore, on a piano at least, the actual ratio of frequencies played is usually a multiple of the 12th root of 2 (1.0595), to keep the ratio between adjacent notes the same. This is the basis of the chromatic scale. It is called "equal tempering"
It is rather a coincidence that so many of the frequency ratios available are very close to integer ratios like 3/2 4/3 and 5/4.
f2/f1 with four halfnote separation = 1.0595^4 = 1.260 (approximately 5/4)
With five halfnote separation:
f2/f1 = 1.0595^5 = 1.335 (approximately 4/3)
With seven halfnote separation:
f2/f1 = (1.0595)^7 = 1.499 (approximately 3/2)
The irregular placement of black keys on a piano facilitates the playing of chords with nearly integer frequency ratios. 
Thanks. I already know all of what you said, except for that last sentence (Wouldn't that only apply in certain keys? C seems much easier for 3rds, 4ths, and 5ths than C#). I just want to know how to generate that list in a spreadsheet.