Mean 'tones'

I was playing with ratios recently, and was looking into
Meantone type tunings, based on the 'mean' of two
fractions:

where the mean of two fractions

b/a and d/c is (b+d) / (a+c)

for example

mean of 5/4 and 6/5 is 11/9

So the 'mean tone' is the mean of
10/9 and 9/8 = 19/17

Taking this in cents = 192.56
Which leads to a 5th of 696.28 cents
And a Major third of 385.12 cents

Moving to the mean of the pythagorean and
just major thirds

81/64 and 5/4 (=80/64)
we get 161/128 = 397.1 cents

adding 2 octaves and dividing by 4 to obtain a fifth of
699.28 cents

So although this seems an odd way to end up pretty close to
12EDO, I find it interesting that such a technique of 'meanthird'
temperament leads to 12EDO.

Another interesting feature of this type of mean is convergence on the
tritone or half octave:

for any ratio b/a, there is the complement 2a/b

The mean of this is (2a+b)/(a+b)

This can then be complemented, to arrive at a recursive convergent
sequence of ratios to 2^0.5

What I find interesting is that for two different ratios, say 5/4 and
6/5, they both converge, but they never equal each other at some point
in the sequence. Is there a name for this?

5/4 8/5 13/9 18/13 31/22 44/31 etc
6/5 5/3 11/8 16/11 27/19 38/27 etc

Mark

--- In tuning@yahoogroups.com, "Mark Gould" <mark.gould@a...> wrote:

> I was playing with ratios recently, and was looking into
> Meantone type tunings, based on the 'mean' of two
> fractions:
>
> where the mean of two fractions
>
> b/a and d/c is (b+d) / (a+c)

This is, assuming b/a and d/c are reduced, something called
the "mediant"; and should not be called the "mean". It shows up,
among other places, in the theory of Farey sequencs and Stern-Brocot
trees.

> What I find interesting is that for two different ratios, say 5/4
and
> 6/5, they both converge, but they never equal each other at some
point
> in the sequence. Is there a name for this?

Is there a name for what, exactly? You've come up with a way to get
sequences to converge to sqrt(2), for one thing. If instead of a
mediant, you had taken an arithmetic mean, you would have Newton's
method, which would converge much faster, though in this context that
may be a bad thing, depending on what you want such sequences for.
Note also that the "mediant" is a mediant in terms of indeterminates,
and might not be the mediant when we use actual numbers.

>Is there a name for what, exactly? You've come up with a way to get
>sequences to converge to sqrt(2), for one thing. If instead of a
>mediant, you had taken an arithmetic mean, you would have Newton's
>method, which would converge much faster,

I wonder what an 'newton's basin' type plot of this would look
like... might be cool if it converges more slowly...

-Carl

--- In tuning@yahoogroups.com, "Mark Gould" <mark.gould@a...> wrote:
> I was playing with ratios recently, and was looking into
> Meantone type tunings, based on the 'mean' of two
> fractions:
>
> where the mean of two fractions
>
> b/a and d/c is (b+d) / (a+c)
...

Mark,

You might find this interesting.
http://dkeenan.com/music/noblemediant.txt