Superparticular decomposition

For J. Gill

I don't have special interest for superparticular ratios. I want only to signal a math property, maybe already well-known.

Let S(N) = (N+1)/N a superparticular ratio.
The only superparticular ratios dividing S(N) are S(N+K) where K is a factor of (N+1)N.

For instance, with S(4)= 5/4, the K are the factors of 4*5 = <1 2 4 5 10 20> and then the factors are

S(5), S(6), S(8), S(9), S(14), S(24)

or

6/5, 7/6, 9/8, 10/9, 15/14, 25/24

Pierre

--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:
> For J. Gill
>
> I don't have special interest for superparticular ratios. I want only to signal a math property, maybe already well-known.
>
> Let S(N) = (N+1)/N a superparticular ratio.
> The only superparticular ratios dividing S(N) are S(N+K) where K is a factor of (N+1)N.
>
> For instance, with S(4)= 5/4, the K are the factors of 4*5 = <1 2 4 5 10 20> and then the factors are
>
> S(5), S(6), S(8), S(9), S(14), S(24)
>
> or
>
> 6/5, 7/6, 9/8, 10/9, 15/14, 25/24
>
> Pierre

J Gill: Thanks for this interesting message, Pierre!
While being simple, it is thought-provoking...

__________________________________________________________

I noticed, at: http://www.cut-the-knot.com/blue/Stern.html
that your perception has resulted in the following discovery,
noted by Alexander Bogomolny (a brief excerpt quoted below):

Pierre Lamothe from Canada informed me recently of a property of the Stern-Brocot tree I was unaware of. Pierre discovered that property in his research on music and harmony.

Let's associate with any irreducible fraction p/q the number w(p/q) = 1/pq - its simplicity. The property discovered by Pierre states that the sum of simplicities of all fractions in any row of the Stern-Brocot tree equals 1!

Best Regards, J Gill