math and music

posted by .

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!

  • math and music -

    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.

  • math and music -

    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.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. math

    Something times something equals 97 Are you supposed to factor 97 as the product of integers?
  2. algebra

    w10y12z _______ w6z5 Are those exponents like [(w^10)*(y^12)*z]/[(w^6)*(z^5)]?
  3. music

    hi um could you help me i have to do a music assignment and i to talk to my class about a piece of music from my fave artist like what instruments are used the structure of the music and those kinds of things but im not good at music …
  4. Math

    how do you reduce a fraction Dividde both the numerator and denominator by the largest common integer possible. Then cancel them out. For example, in the fraction 15/25, the largest integer that is divisible evenly into both numerator …
  5. math

    there are 8 consecutive integers that add up to 31. Only two of the integers are equal. what is the greatest number for any one integer?
  6. recreational math, music

    How would you go about listing all of the integer ratios, preferably in a spreadsheet?
  7. math and music

    How would you go about listing all of the integer ratios?
  8. Math

    Is i^2=-1 for how many integers N is (N+i)^4 an integer?
  9. Algebra

    the denominator of a fractio is an integer that is one greater than the numerator. A second fraction is the reciprocal of the integer that is one less than the numerator of the first fraction. If the product of the two fraction is …
  10. Math

    Paulo withdraws the same amount from his bank account each week to pay for lunch. Over the past four weeks, he withdrew one hundred twenty dollars. Which rule best applies to determine the change in his account each week?

More Similar Questions