Wave Convergents

Here is the simplest way I can think of putting it.

1. For a given interval a/b form the convergents [1/1, p/q,...a/b].

2. take f(t) = e^iat + e^ibt.

3. a = pg + R
b = qg - R,

where g = (a + b)/(p + q) and R = (aq - pb)/(p + q).

4. substituting 3. into 2. we get

f(t) = (e^iRt)*(e^pgt) + (e^-iRt)*(e^qgt).

Conclusion: Since p and q are *necessarily* whole numbers (irrespective of a and b which might be irrational), and since g is their GCD, then the original wave 1. can always be analysed into simpler whole-numbered ratios belonging to the harmonic series with fundamental g but with a modulated amplitude (iRt) in each component. The p/q might be a JI interval, for example.

The 3-limit 81/64 = a/b becomes a 5-limit 5/4 = p/q gives one example.

The deduction of Step 3. above I've given in previous posts.

Rick

And everything I said in my last email still applies. This will work even if
p and q are not convergents of a/b.

-Mike

On Sun, May 23, 2010 at 11:31 PM, rick <rick_ballan@...> wrote:

>
>
> Here is the simplest way I can think of putting it.
>
> 1. For a given interval a/b form the convergents [1/1, p/q,...a/b].
>
> 2. take f(t) = e^iat + e^ibt.
>
> 3. a = pg + R
> b = qg - R,
>
> where g = (a + b)/(p + q) and R = (aq - pb)/(p + q).
>
> 4. substituting 3. into 2. we get
>
> f(t) = (e^iRt)*(e^pgt) + (e^-iRt)*(e^qgt).
>
> Conclusion: Since p and q are *necessarily* whole numbers (irrespective of
> a and b which might be irrational), and since g is their GCD, then the
> original wave 1. can always be analysed into simpler whole-numbered ratios
> belonging to the harmonic series with fundamental g but with a modulated
> amplitude (iRt) in each component. The p/q might be a JI interval, for
> example.
>
> The 3-limit 81/64 = a/b becomes a 5-limit 5/4 = p/q gives one example.
>
> The deduction of Step 3. above I've given in previous posts.
>
> Rick
>
>
>