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 half-note separation = 1.0595^4 = 1.260 (approximately 5/4)
With five half-note separation:
f2/f1 = 1.0595^5 = 1.335 (approximately 4/3)
With seven half-note 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.
To list all the integer ratios as you described, you can follow these steps:
1. Determine the maximum value for the numerator and the denominator. This will help you define the range for generating the ratios.
2. Generate a list of all possible ratios, looping through all possible numerator and denominator values within the defined range.
3. Filter out the ratios that can be simplified. To do this, you can use the greatest common divisor (GCD) algorithm to check if the numerator and denominator share any common factors. If they do, then the ratio can be simplified and should be excluded from the list.
4. Sort the remaining ratios based on the product of each integer's numerator and denominator. You can calculate this product and use it as a key for sorting the ratios in ascending order.
As for the specific lists you mentioned, you would need to apply the exclusions you described:
- List A1: Generate all ratios with values less than or equal to 1. Exclude any ratios that can be simplified. Sort the remaining ratios by the product of the numerator and denominator.
- List B1: Generate all ratios with values greater than or equal to 1. Exclude any ratios that can be simplified. Sort the remaining ratios by the product of the numerator and denominator.
- List A2: Similar to List A1, but also exclude ratios of 1/n and n/1 where n is a composite number (not prime).
- List B2: Similar to List B1, but also exclude ratios of 1/n and n/1 where n is a composite number (not prime).
- List A3: Similar to List A2, but also exclude ratios with values less than 1/2.
- List B3: Similar to List B2, but also exclude ratios with values less than 2/1.
- List A4 and B4: Apply the exclusions from both List A2 and List A3 to List A1 and List A2 respectively.
In terms of how these lists relate to music theory, they are closely tied to the concept of just intonation. Just intonation is a tuning system that uses pure integer ratios to create harmonious sounds. These ratios represent the relationship between different pitches and can be found in various musical contexts, such as scales, chords, and intervals. The lists you described provide a systematic way of organizing these ratios based on their complexity, as indicated by the product of the numerator and denominator.