Regularity of the tonality diamond

The smallest element of n-limit diamond greater than 1 is (n+1)/n;
this leads to the largest interval between elements of the diamond,
again (n+1)/n. The n-diamond comma, the smallest such interval, seems
to be greater than or equal to (n^2+1)/n^2, but never very much greater;
cents(n-diamond comma)/cents((n^2+1)/n) approaches 1, and gets close
right away. If so, the largest over the smallest ratio for the diamond
is close to n; hence, it goes to infinity with increasing n. However,
the asymptotic formula for dia(n) tells us that on average, an
interval between elements of the diamond is (pi^2/2) 1/n^2 octave,
whereas the n-diamond comma is about (1/ln(2)) 1/n^2 octave. Taking
the ratio, an average interval over the smallest interval is about
ln(2)pi^2/2, or
3.42. This is irregular but not terrifically irregular. It might be
interesting to define a systematic way of filling in the gaps between
1 and (n+1)/n and 2n/(n+1) and 2, leading to a scale for each odd n
where the largest over the smallest ratio was bounded.

> Taking the ratio, an average interval over the smallest interval
> is about ln(2)pi^2/2, or 3.42. This is irregular but not
> terrifically irregular.

Is the average the right thing here, because aren't the majority
of the intervals almost as small as the smallest?

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@...> wrote:
>
> > Taking the ratio, an average interval over the smallest interval
> > is about ln(2)pi^2/2, or 3.42. This is irregular but not
> > terrifically irregular.
>
> Is the average the right thing here, because aren't the majority
> of the intervals almost as small as the smallest?

You want a median?

>> > Taking the ratio, an average interval over the smallest interval
>> > is about ln(2)pi^2/2, or 3.42. This is irregular but not
>> > terrifically irregular.
>>
>> Is the average the right thing here, because aren't the majority
>> of the intervals almost as small as the smallest?
>
>You want a median?

Scala reports biggest / smallest, which might pertain to propriety.

-Carl