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Tempo ramps - demystify comments a bit.

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nick_m 2016-02-26 13:46:06 +11:00
parent 895bbbba11
commit d8549c6ff5
1 changed files with 23 additions and 12 deletions
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libs/ardour/tempo.cc
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@ -261,6 +261,7 @@ TempoSection::frame_at_beat (double beat, framecnt_t frame_rate) const
{ {
return frame_at_tick (beat * BBT_Time::ticks_per_beat, frame_rate); return frame_at_tick (beat * BBT_Time::ticks_per_beat, frame_rate);
} }
/* /*
Ramp Overview Ramp Overview
@ -273,7 +274,7 @@ Ta----|--------------|* |
| * | | | * | |
T0----|* | | T0----|* | |
* | | | * | | |
_______________|___|_____ _______________|___|____
time a t (next tempo) time a t (next tempo)
[ c ] defines c [ c ] defines c
@ -281,32 +282,42 @@ Duration in beats at time a is the integral of some Tempo function.
In our case, the Tempo function (Tempo at time t) is In our case, the Tempo function (Tempo at time t) is
T(t) = T0(e^(ct)) T(t) = T0(e^(ct))
where c is the function constant: where c is the function constant
c = log(Ta/T0)/a c = log(Ta/T0)/a
and so
a = log(Ta/T0)/c a = log(Ta/T0)/c
Given the function constant, the integral of the Tempo function (the beat function, which is the duration in beats at some time t) is: The integral over t of our Tempo function (the beat function, which is the duration in beats at some time t) is:
b(t) = T0(e^(ct) - 1) / c b(t) = T0(e^(ct) - 1) / c
To find the time t at any beat b, we use the inverse function of the beat function (the time function) which can be shown to be : To find the time t at beat duration b, we use the inverse function of the beat function (the time function) which can be shown to be:
t(b) = log((cb / T0) + 1) / c t(b) = log((cb / T0) + 1) / c
We define c for a tempo ramp by placing a new tempo at some distance t away from our existing one. The time t at which Tempo T occurs is a as above:
The problem is that we usually don't know t. t(T) = log(T / T0) / c
We do know the duration in beats where the next tempo section lies.
Where t = a (when we define c), we can solve a in terms of beat duration b and the two relevant tempos using the beat function:
a = b log (Ta / T0) / (T0 (e^(log (Ta / T0)) - 1))
We then use a to set the function constant c (above). We define c for this tempo ramp by placing a new tempo section at some time t after this one.
Solving a and setting c in one operation allows us to further reduce the problem to: Our problem is that we usually don't know t.
We usually do know the duration in beats between this and the next tempo section.
Where t = a (i.e. when a is equal to the time of the next tempo section), we can solve t in terms of
beat duration and our two tempos.
A bit of scribbling with the beat function gives us:
t = b log (Ta / T0) / (T0 (e^(log (Ta / T0)) - 1))
By substituting our expanded t as a in the c function above, we see that our problem is reduced to:
c = T0 (e^(log (Ta / T0)) - 1) / b c = T0 (e^(log (Ta / T0)) - 1) / b
We can now evaluate and store c for use in beat, time and tempo calculations until the following tempo section
(the one that defines c in conjunction with this one) is changed or moved.
Most of this stuff is taken from this paper: Most of this stuff is taken from this paper:
WHERE’S THE BEAT? WHERE’S THE BEAT?
TOOLS FOR DYNAMIC TEMPO CALCULATIONS TOOLS FOR DYNAMIC TEMPO CALCULATIONS
Jan C. Schacher Jan C. Schacher
Martin Neukom Martin Neukom
Zurich University of Arts
Institute for Computer Music and Sound Technology
https://www.zhdk.ch/fileadmin/data_subsites/data_icst/Downloads/Timegrid/ICST_Tempopolyphony_ICMC07.pdf https://www.zhdk.ch/fileadmin/data_subsites/data_icst/Downloads/Timegrid/ICST_Tempopolyphony_ICMC07.pdf