Wikipedia:Reference desk/Archives/Mathematics/2022 October 29

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October 29

Requesting an specific list made of 13 numbers that follow some rules. This is needed to create a musical tuning (no need to know music theory to answer the question).

What is the list of 13 numbers that best fit those rules? This will create a musical tunning (way to decide what frequency each, key, hole,..... of the instrument will play)

The rules:
1-It's a list of 13 rational numbers
2-Each number is between 1 and 2
3-The number 1/1 and the number 2/1 must be 2 of the numbers at this list.
4-Each number must be the reduced term version of the number (no 10/6 as some example, if such number is at the list it must be 5/3)
5-No two numbers are the same.
6-Take the ratio of each pair of numbers (remember 1/1 and 2/1 are two of the 13 numbers of the list):
6.1-Look at the numerators and denominators of lowest term version of those ratios
6.2-Pick out the largest number among these numerators and denominators
6.2.1-The goal is to minimize the largest number. (if tied, minimize the second largest number and if still tied, the third largest....)
179.134.99.224 (talk) 02:09, 29 October 2022 (UTC)

All the "versions" of a rational number are considered to be the same number, so 10/6 is the same as 5/3; it's not a different "version". I'm pretty sure that means condition 4 is unnecessary. If the list is 1/1, 13/12, 7/6, 4/3, ... , 2/1 then the most "complex" ratio is 24/23 and the highest value is 24. One might then ask if it's possible to do better than 24. You can simplify the problem a bit by clearing fractions; in that version you want a sequence of 13 integers, n₁ < n₂ < ... < n₁₃ = 2n₁, and when you look at the ratios n_i:n_j in lowest terms the largest number involved is as small as possible. In the example, this translates to the numbers 12, 13, ... , 24. --RDBury (talk) 04:13, 29 October 2022 (UTC)

PS. I created a spreadsheet to at least compute value you're trying to minimize. If the sequence is n₁ < n₂ < ... < n₁₃ as above then the value is:

$\max _{i,j}{\frac {n_{i}}{\gcd(n_{i},n_{j})}}$

I'm having no luck getting a better value than 24, but it's not the only sequence where the value is 24; another is 5354228880/24, 5354228880/23, 5354228880/22, ... 5354228880/12. You can assume the n_i's have no common factors. With this assumption I haven't found a third sequence with value 24, so afaik there may be only two. --RDBury (talk) 05:35, 29 October 2022 (UTC)

How is the sequence 5354228880/24, 5354228880/23,......, a sequence where the value is 24?

1: 5354228880/12 (that is 446185740/1) is there, so the list would have the biggest number that is a numerator or denominator being 446185740 (ratio between 446185740/1 and 1/1), remember the numerator counts too when looking for a value that will try to minimize the largest numerator or denominator.

2: 1/1 and 2/1 are two of the 13 numbers and you must also calculate the ratio with them too.

3: 446185740/1 is not a number between 1 and 2.....

4: Remember you must minimize the biggest number that is a numerator or denominator of one of the ratios. But if a list is tied, then you should try to minimize the second largerst numerator or denominator. And if still tied, the third largest numerator or denominator...... and this goes on and on.

179.134.99.224 (talk) 18:32, 29 October 2022 (UTC)

I was proposing it as a solution to the integer version of the problem. in the original it would be 1, 24/23, 12/11, 8/7, 6/5, 24/19, ... 24/13, 2. It's easier to set up on a spreadsheet if you clear fractions. Sorry for the confusion. --RDBury (talk) 20:26, 29 October 2022 (UTC)

If I understand the problem correctly, the answer should be obvious: 1, 24/23, 22/21, 20/19, 18/17, 16/15, 14/13, 12/11, 10/9, 8/7, 6/5, 4/3, and 2. All fractions are reduced, all 24 integers and all 13 rational numbers are different, and the highest integer is 24. Dhrm77 (talk) 19:49, 30 October 2022 (UTC)

The lowest-term form of the ratio between 24/23 and 22/21 is 252/253. The maximum of 252 and 253 is 253. For the two solutions by RDB, we get instead a maximum of merely 24.  --Lambiam 23:19, 30 October 2022 (UTC)

Putting the problem in math jargon, and generalizing it a little bit: Given $1=x_{0}<x_{1}<\cdots <x_{n-1}<x_{n}=2$, where each $x_{i}$ is rational, let M be the greatest denominator among the fractions $x_{i}/x_{j}$ where $0\leq i<j\leq n$. (The numerator is less, so we can ignore it for simplicity.) How small can M be? — RDBury's two solutions with $M=2n$ are the overtone series and the undertone series. I'll be very surprised if these can be beaten or matched. —Tamfang (talk) 17:03, 2 November 2022 (UTC)

I neglected the tiebreaker part of the question. So: if $X$ is a sequence of $x_{i}$ as above, let $S_{X}$ be the list, in decreasing order, of numerators and denominators. Over the family of possible $X$, which $S_{X}$ comes first in lexicographic order? —Tamfang (talk) 17:43, 2 November 2022 (UTC)

The two RDB solutions produce exactly the same multiset, since there is a one-to-one correspondence between the two: for every ratio $p:q$ in one, we have the ratio $q:p$ in the other.  --Lambiam 19:30, 2 November 2022 (UTC)

yep. —Tamfang (talk) 19:42, 2 November 2022 (UTC)