Formula for the size of the nth tonality diamond

If dia(n) is the size of the nth tonality diamond for any odd integer n,
then a recursive formula for dia(n) is

dia(n) = dia(n-2) + phi(n)

where phi(n) is Euler's totient function; that is, phi(n) is the
number of integers less than n relatively prime to n.

http://en.wikipedia.org/wiki/Totient_function

Using this you can find dia(101) is 2107, dia(1001) is 203381 and
dia(10001) is too many levels of recursion for Maple to handle, but it
probably would be possible to get an order of growth formula.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> If dia(n) is the size of the nth tonality diamond for any odd integer n,
> then a recursive formula for dia(n) is
>
> dia(n) = dia(n-2) + phi(n)

Simpler is just to say

dia(n) = sum_{odd integers <= n} phi(n)

Then dia(101)=2107, dia(1001)=203381 and dia(10001)=20273125.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> dia(n) = sum_{odd integers <= n} phi(n)
>
> Then dia(101)=2107, dia(1001)=203381 and dia(10001)=20273125.

An asymptotic formula for dia(n) is

dia(n) ~ (2/pi^2) n^2

I think the error term is O(n log n). I apolgize for the irrelevance
of this thread to actual music, but it is interesting to see that we
can consider the size of the tonality diamond as a number-theoretic
function and make sense of it. That the tonality diamond grows as the
square of the odd limit tells you why higher limit stuff can get
pretty impractical fairly quickly; dia(11) is 29 and was already
giving Partch some difficulty. Dia(13) is 41, which may be another
good reason to stop at 13.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> An asymptotic formula for dia(n) is
>
> dia(n) ~ (2/pi^2) n^2
>
> I think the error term is O(n log n).

I found this web page for a closely related function to dia(n):

http://mathworld.wolfram.com/TotientSummatoryFunction.html

From that you can see that it has a very similar asymptotic formula and
error term.

With all this talk of diamonds, the following may be of
interest...

http://lumma.org/music/theory/diamonds.txt

-Carl

At 08:36 PM 6/20/2005, you wrote:
>If dia(n) is the size of the nth tonality diamond for any odd integer n,
>then a recursive formula for dia(n) is
>
>dia(n) = dia(n-2) + phi(n)
>
>where phi(n) is Euler's totient function; that is, phi(n) is the
>number of integers less than n relatively prime to n.
>
>http://en.wikipedia.org/wiki/Totient_function
>
>Using this you can find dia(101) is 2107, dia(1001) is 203381 and
>dia(10001) is too many levels of recursion for Maple to handle, but it
>probably would be possible to get an order of growth formula.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> With all this talk of diamonds, the following may be of
> interest...
>
> http://lumma.org/music/theory/diamonds.txt

Why don't you take the pages you've written and set up an actual
theory page which gives access?