[PD] exponential tempo control

[Chuckk Hubbard]

Just a note, I came up with a pretty obvious proof for:
 SUM (k=0 to N-1) i^k = (i^N - 1) / (i - 1)
  (i - 1) (i^0 + i^1 + i^2 + .... + i^(N-1))
=
-i^0 + i^1 - i^1 + i^2 - i^2 ... + i^(N-1) - i^(N-1) + i^N
= i^N - 1
All the terms cancel except the lowest and highest powers of i.

 On 10/26/05, Chuckk Hubbard <badmuthahubbard at gmail.com> wrote:
>
> My problem: having tempo change exponentially, but controlling when it
> would reach the new tempo. So I know the old BPM (a), the new BPM (b), and
> the number of beats to get from one to the other (B). I set metro to 100 ms.
> What I don't know is the amount of time those however many beats will take
> while tempo is changing, how many increments of change there will be (n),
> and what factor to multiply by tempo with each increment (i). The idea is
> that, whatever that factor is, it will multiply 10 times a second.
> So:
> i^n=b/a
>  Also, the sum of all of the tempos passed through:
> a*i^0 + a*i^1 + a*i^2 + a*i^3 + .... + a*i^(n-1)
>  So the average BPM for the whole time is:
> (a*i^0 + a*i^1 + a*i^2 + a*i^3 + .... + a*i^(n-1))/n = v (average)
>  Seconds per beat is the reciprocal of v times 60, and increments-per-beat
> is 10 times that, and n is then B times that, so:
> n = 600B/v = 600Bn/SUMk=0->(n-1)(a*i^k)
>  if:
> n = 600Bn/SUM""""
> then:
> 600B/SUM"""" = 1, or:
> 600B = (a*i^0 + a*i^1 + a*i^2 ..... + a*i^(n-1))
>  dividing by a:
> 600B/a = i^0 + i^1 + i^2 .... i^(n-1)
>   I did some experimenting with constants and found that SUM(k=0->(n-1)
> i^k is always equal to:
> ((i^n)-1)/(i-1)
>
> For example, 1+3+9+27+81+243=364, which is (729-1)/2
>  But since i^n=b/a, this sum is also:
> ((b/a)-1)/(i-1)
>  So:
> 600B/a = ((b/a)-1)/(i-1)
>  Yay. Now I know i.
>  i = ((b/a - 1)a/600B)+1
>  And n is simply the log base i of b/a.
>  I round n to an integer, multiply by 100, and that is the delay before
> looking up the next tempostructure. Outside of this abstraction, there is
> simply a metro that bangs a float multiplying tempo by i at 100 ms until
> stopped by the next tempo. Outputting n at all proved to be unnecessary.
>   -Chuckk
>
> --
> "It is not when truth is dirty, but when it is shallow, that the lover of
> knowledge is reluctant to step into its waters."
> -Friedrich Nietzsche, "Thus Spoke Zarathustra"
>
>


--
"It is not when truth is dirty, but when it is shallow, that the lover of
knowledge is reluctant to step into its waters."
-Friedrich Nietzsche, "Thus Spoke Zarathustra"
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