# I think what you are saying about the perception of tempo is very
interesting.
# You want to integrate f(x) = (x*sqrt(1.5)/4)^2 right?
# This is not more involved than f(x)=x^2 because constants (the additional
scalars & denominators) does not affect the integration. I mean, if the
integral of f(x) is F(x) then the integral of c*f(x) is c*F(x). In your case

Integral[ x^2*(sqrt(1.5)/4)^2 ] = (sqrt(1.5)/4)*Integral[ x^2 ]
= 1.5/16 Integral [ x^2 ] = 3/32 * [ 1/3 * x^3] = x^3/32
# But I didn't understand where you get f(x) :-)
# See you!
-ugur guney-


On 9/7/07, Tim Mortimer <timmortimer@d2.net.au> wrote:
>
>
> http://www.nabble.com/file/p12552632/integratethis.jpg integratethis.jpg
>
> greets,
>
> The attached jpeg attempts to illustrate how i intend to use integration
> to
> calculate how much "absolute" time passes in a score when a tempo ramp of
> a
> certain duration becomes active for that period of the score.
>
> in the case of this example, the tempo decreases from 120BPM to 30BPM over
> a
> period of "4 beats"/ 4 "score seconds".
>
> My maths is pretty rusty (i did a year 12 bridging course a couple of
> years
> ago - at the time because i had just started using Max, & was desparately
> trying to find out what a Markov chain was, but anyway.. the Calculus &
> Trig
> have proven pretty useful along the way......)
>
> But I'm having trouble trying to integrate the formula in the attached
> jpeg.
>
> Can anyone assist? Once I have seen it done once, I should be able to
> apply
> the integrated function universally (more or less) to all my tempo ramping
> requirements.
>
> i do know the basics of integration (eg integrate x^2 = x^3/3 for
> example..)
>
> but im stumped by the additional scalars & denominators in my formula, &
> hence by what my desired "ramp time" formula looks like (but i'm pretty
> sure
> it's along the lines of what i'm proposing - see the diagram)
>
> I suppose what makes my approach "a bit fancy" (beyond the method, which
> some may feel excessive) is that i like to think that the perception of
> tempo - like that of pitch, is an exponential / power of 2 type
> relationship: i.e the perception of "speeding up" & "slowing down" is more
> meaningful in terms of "twice as fast" / "half as fast", & hence i want to
> use a power of 2 curve to impose this "perception" onto my tempo ramping -
> rather than simply a linear increase / decrease in BPM value...
>
> Ok, so i realise this may not be the  "normal" way to go about
> "accumulating" time vis a vis changes in score tempo, however doing it by
> the way I am attempting has some benefits (for me at least, in this
> instance) that i see...
>
> 1) it's ultimately more accurate
>
> 2) it will fit into my "python score paradigm" tempo system with a minimum
> of fuss (trust me - it has no accumulated score time - just a timeline
> "function" to which it makes reference to determine current absolute time
> for the scored event..)
>
> 3) u then only need to apply any tempo based calculations to significant
> score events, which is someways is surely computationally more efficient
> than accumulating ammassed time on each & every control rate clock - when
> on
> 99% of these "clock ticks" no event is actually taking place anyway
>
> so, err, any help from the maths brains out there on this one
> appreciated...
>
> kind regards
>
> Tim
> --
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