# 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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