neat JI math trick

My apologies if this has been posted before, I haven't read all 91,000 messages :)
If you're looking for a ratio in a certain range, you can take two nearby ratios and use one of the three pythagorean means. For example neutral thirds between 6/5 and 5/4 include 49/40, 60/49 and square root (3/2). But to find a small-number ratio, use this trick:
(a/b, c/d) --> (a+c)/(b+d)
The neutral 3rd between 6/5 and 5/4 is 11/9. Neutral sixth between 8/5 and 5/3 is 13/8. Tritone between 7/5 and 10/7 is 17/12. This new ratio always lies between the other two, and is usually the smallest-numbered ratio that does. But not always. The neutral 2nd between 16/15 and 10/9 is 13/12, the smallest-number neutral 2nd is actually 11/10.
So does this kind of mean have a name?

some people call them 'freshman sums'. It is used here quite a bit and generates what is called the Farey series

--- In tuning@yahoogroups.com, "kiteg" <kiteg@...> wrote:
>
> My apologies if this has been posted before, I haven't read all 91,000 messages :)
> If you're looking for a ratio in a certain range, you can take two nearby ratios and use one of the three pythagorean means. For example neutral thirds between 6/5 and 5/4 include 49/40, 60/49 and square root (3/2). But to find a small-number ratio, use this trick:
> (a/b, c/d) --> (a+c)/(b+d)
> The neutral 3rd between 6/5 and 5/4 is 11/9. Neutral sixth between 8/5 and 5/3 is 13/8. Tritone between 7/5 and 10/7 is 17/12. This new ratio always lies between the other two, and is usually the smallest-numbered ratio that does. But not always. The neutral 2nd between 16/15 and 10/9 is 13/12, the smallest-number neutral 2nd is actually 11/10.
> So does this kind of mean have a name?
>

--- In tuning@yahoogroups.com, "banaphshu" <kraiggrady@...> wrote:
>
> some people call them 'freshman sums'. It is used here quite a bit and generates what is called the Farey series

Mathematicians call it the mediant:

http://en.wikipedia.org/wiki/Mediant_%28mathematics%29

Thanks for the info & links, guys!
Kite

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "banaphshu" <kraiggrady@> wrote:
> >
> > some people call them 'freshman sums'. It is used here quite a bit and generates what is called the Farey series
>
> Mathematicians call it the mediant:
>
> http://en.wikipedia.org/wiki/Mediant_%28mathematics%29
>