diff --git a/doc/tempo.tex b/doc/tempo.tex new file mode 100644 index 0000000000..107c646479 --- /dev/null +++ b/doc/tempo.tex @@ -0,0 +1,301 @@ +% vim: tw=80 spl=en +% Copyright (C) 2018 Julien "_FrnchFrgg_" RIVAUD +% +% This program is free software: you can redistribute it and/or modify +% it under the terms of the GNU General Public License as published by +% the Free Software Foundation, either version 3 of the License, or +% (at your option) any later version. +% +% This program is distributed in the hope that it will be useful, +% but WITHOUT ANY WARRANTY; without even the implied warranty of +% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +% GNU General Public License for more details. +% +% You should have received a copy of the GNU General Public License +% along with this program. If not, see . +\documentclass[10pt]{article} + + +%\usepackage{fontspec} +%\usepackage{polyglossia} +%\setmainlanguage{english} +\usepackage[english]{babel} + +%\usepackage{lualatex-math} +\usepackage{amsmath} +%\usepackage[math-style=ISO,bold-style=ISO]{unicode-math} + +\usepackage{geometry} +\geometry{reset,margin=1.5cm,left=6cm,includefoot,marginparwidth=4.5cm} + +\usepackage{cellprops} + +\begin{document} + + + +\def\define#1{\textbf{#1}} +\def\d{\mathrm{d}} + +\section{A natural definition for tempo ramping} + +\subsection{Some definitions} + +The \define{time}, often denoted $t$, is the duration in some multiple unit of +seconds (samples, superclock, etc). + +We define $b$ to be the \define{beats function}: $b$~is the number of beats that +occurred since $t = 0$ until the current time. It is assumed to be continuous and +even derivable rather than being a staircase function: $b$~will thus value~$4.5$ +when exactly in the middle of beat~$4$, halfway\footnote{Not necessarily halfway +in time units, though.} between pulsations $4$~and~$5$. + +The \define{tempo function}, or simply tempo, is the number~$T$ of pulsations +per unit of time. Thus $T = \dfrac{\d b}{\d t}$. + +\def\DT{\Delta T} +\def\Dt{\Delta t} +\def\Db{\Delta b} + +A \define{tempo ramp} is a definition of $T$ for $0 \le t \le \Dt$ such that +$T_{t=0} = T_0$ and $T_{t=\Dt} = T_{\text{end}} = T_0 + \DT$ where $\Dt$, $T_0$ +and $T_{\text{end}}$~or~$\DT$ are given. Most of the times they are set by the +user, but $\Dt$ can be sometimes defined in beats, that is be the unique~$\Dt$ +such that $b_{|t=\Dt} = \Db$. + +\subsection{Linear tempo ramping} + +The simplest definition that comes to mind is a tempo ramp where the tempo +increases from $T_0$~to~$T_0 + \DT$ \emph{linearly} with time. + +Here $T = T_0 + \dfrac{\DT}{\Dt}\times t$, and thus +$b = \dfrac{\DT}{2\Dt}\times t^2 + T_0 \times t$, assuming $b_{|t=0} = 0$. + +If $\Dt$~is defined implicitly from a duration~$\Db$ in beats, then +\[ \Db = \dfrac{\DT}{2\Dt}\times \Dt^2 + T_0 \Dt + = \left( \frac{1}{2}\DT + T_0 \right) \times\Dt \] +thus +\[ + \Dt = \dfrac{2\Db}{\DT + 2T_0} +\] + +\subsection{Exponential tempo ramping} + +\def\e{\mathrm{e}} + +Humans are not very good at keeping track precisely of absolute time for long +durations, and are better at comparing short durations, that is maintain a +reasonably stable time span between successive pulsations, and have a precise +sense of elapsed time \emph{in terms of these pulsations}. + +In particular, when changing the tempo, it is less far-fetched to imagine a +human feeling the tempo increase as a function of elapsed \emph{beats} rather +than absolute time in seconds which is a pulsation that is very hard to maintain +constant and even sense when competing with the (increasing) musical pulsation. + +In other words, a definition more suitable than that of linear ramping would be +that the tempo increases \emph{linearly} with the value of~$b$. We thus want +\[ T = T_0 + \dfrac{\DT}{\Db}\times b + \text{, that is } +\dfrac{\d b}{\d t} - \dfrac{\DT}{\Db}\times b = T_0 \] + +The solutions of this order 1 linear differential equation are\footnote{The +exponential is the solution of the associated homogeneous equation and the +second term is the constant solution of the equation.} +$b = A \e^{\frac{\DT}{\Db}t} - \dfrac{\Db T_0}{\DT}$, and +since we want $b_{|t=0} = 0$, we get $A = \dfrac{\Db T_0}{\DT}$ that is +\[ b = \dfrac{\Db T_0}{\DT} (\e^{\frac{\DT}{\Db}t} - 1) \] + +Denoting $\omega = \dfrac{\DT}{\Db}$ we find +$b = \frac{T_0}{\omega} (\e^{\omega t} - 1)$. +(Note: $\omega$ is called $c$ in Nick's text, but I like $\omega$ better since +it is a frequency). + +The \emph{tempo} is $T = \dfrac{\d b}{\d t} += T_0 \e^{\frac{\DT}{\Db}t} = T_0 \e^{\omega t}$, and this is where the name +exponential tempo ramp comes from. + +Recall that $\Db$, $\DT$ and $T_0$ are parameters of the ramp, while $t$~is the +arbitrary time within the ramp at which we want to know $b$~and~$T$. + +Let us compute the reciprocal of $b$: +\[ + b = \dfrac{\Db T_0}{\DT} (\e^{\frac{\DT}{\Db}t} - 1) + \iff + \dfrac{\DT}{\Db T_0} b = \e^{\frac{\DT}{\Db}t} - 1 + \iff + 1 + \dfrac{\DT}{\Db T_0} b = \e^{\frac{\DT}{\Db}t} +\] +\[ + \iff + \log \left( 1 + \dfrac{\DT}{\Db T_0} b \right) = \frac{\DT}{\Db}t + \iff + t = \frac{\Db}{\DT} \log \left( 1 + \dfrac{\DT}{\Db T_0} b \right) +\] + +If $\Dt$~is defined directly instead of implicitly from a duration~$\Db$ in +beats, then we have to find out the value of~$\Db$. At $t=\Dt$, we have +$b = \Db$ thus +\[ + \Dt = \frac{\Db}{\DT} \log \left( 1 + \dfrac{\DT}{\Db T_0} \Db \right) + \iff + \Dt = \frac{\Db}{\DT} \log \left( 1 + \dfrac{\DT}{T_0} \right) + \iff + \Db = \frac{\DT\Dt}{\log \left( 1 + \dfrac{\DT}{T_0} \right) } +\] + +A more useful expression is +\[ + \omega = \frac{\DT}{\Db} + = \frac{1}{\Dt}\log\left(1+ \frac{\DT}{T_0}\right) +\] +which gives +\[ + b = \frac{\Dt T_0}{\log\left(1+ \frac{\DT}{T_0}\right)} + \e^{\frac{t}{\Dt}\log\left(1+ \frac{\DT}{T_0}\right)} + = \frac{\Dt T_0}{\log\left(1+ \frac{\DT}{T_0}\right)} + \left(1+ \frac{\DT}{T_0}\right)^{\frac{t}{\Dt}} +\] +and +\[ + T + = T_0 \e^{\frac{t}{\Dt}\log\left(1+ \frac{\DT}{T_0}\right)} + = T_0 \left(1+ \frac{\DT}{T_0}\right)^{\frac{t}{\Dt}} +\] + +We at last compute the reciprocals: +\[ + b = \frac{\Dt T_0}{\log\left(1+ \frac{\DT}{T_0}\right)} + \e^{\frac{t}{\Dt}\log\left(1+ \frac{\DT}{T_0}\right)} + \iff + \frac{\log\left(1+ \frac{\DT}{T_0}\right)}{\Dt T_0} b + = \e^{\frac{t}{\Dt}\log\left(1+ \frac{\DT}{T_0}\right)} +\] +\[ + \iff + \log\left( \frac{\log\left(1+ \frac{\DT}{T_0}\right)}{\Dt T_0} b \right) + = \frac{t}{\Dt}\log\left(1+ \frac{\DT}{T_0}\right) + \iff + t = \frac{ + \Dt + \log\left( \frac{\log\left(1+ \frac{\DT}{T_0}\right)}{\Dt T_0} b \right) + }{ + \log\left(1+ \frac{\DT}{T_0}\right) + } +\] + +\subsection{Summary with $T$ as main tempo representation} + +The following table gathers all results. For code factorization we might want +to only consider the red formulas, that is compute (and maybe cache) $\omega$ +beforehand. Note that in the formulas, $\omega$ is \emph{always} the inverse of +a time, never in per-beats. + +\[\everymath{\displaystyle} + \cellprops{ td:nth-child(1) { math-mode: text; } } + \begin{array}{lccc} + \hline + When you know & \Dt & \Db & \omega \\ + \hline + Compute $\omega$ & + \color{red} + \frac{1}{\Dt}\log\left(1+ \frac{\DT}{T_0}\right) + & + \color{red} + \frac{\DT}{\Db} + & + \\ + Compute $b$ from $t$ & + \frac{\Dt T_0}{\log\left(1+ \frac{\DT}{T_0}\right)} + \left(1+ \frac{\DT}{T_0}\right)^{\frac{t}{\Dt}} + & + \frac{\Db T_0}{\DT} (\e^{\frac{\DT}{\Db}t} - 1) + & + \color{red} + \frac{T_0}{\omega} (\e^{\omega t} - 1) + \\ + Compute $T$ from $t$ & + T_0 \left(1+ \frac{\DT}{T_0}\right)^{\frac{t}{\Dt}} + & + T_0 \e^{\frac{\DT}{\Db}t} + & + \color{red} + T_0 \e^{\omega t} + \\ + Compute $t$ from $b$ & + \frac{ + \Dt + \log\left( \frac{\log\left(1+ \frac{\DT}{T_0}\right)} + {\Dt T_0} b \right) + }{ + \log\left(1+ \frac{\DT}{T_0}\right) + } + & + \frac{\Db}{\DT} \log \left( 1 + \frac{\DT}{\Db T_0} b \right) + & + \color{red} + \frac{1}{\omega} \log \left( 1 + \frac{\omega b}{T_0} \right) + \\ + Compute $T$ from $b$ & + T_0 + \frac{1}{\Dt}\log\left(1+ \frac{\DT}{T_0}\right) \times b + & + T_0 + \frac{\DT}{\Db}\times b + & + \color{red} + T_0 + \omega b + \\ + \hline +\end{array}\] + +Note that there are often library functions to compute $\log(1+x)$ directly +from~$x$ more precise than computing $1+x$ then its logarithm. The main reason +is that when adding 1to some very small number you loose a lot of precision +because the exponent is now tailored to the representation of~$1$, and that +the Taylor-McLaurin series for $\log(1+x)$ is a very efficient mean to compute +an approximate value for it. + +\subsection{Using $S = \frac{1}{T}$ to represent tempo} + +\def\DS{\Delta S} + +\everymath{\displaystyle} + +Since the base time unit in ardour is not seconds but far smaller than +that\footnote{It was samples in previous versions of Ardour, and is even more +precise in later versions.}, any value of $T$~or~$\DT$ will be very small. A +better value to store, that can easily be rounded to integer without a huge +loss of precision, is $S = 1/T$. + +Instead of having $T_0$~and~$T_1 = T_0 + \DT$, we thus assume that when +computing the formulas we have access to $S_0$~and~$S_1$. + +Then from $\Dt$, +$\omega = \frac{1}{\Dt}\log\left(1 + \frac{\DT}{T_0}\right) + = \frac{1}{\Dt}\log\left(\frac{T_1}{T_0}\right) + = \frac{1}{\Dt}\log\left(\frac{S_0}{S_1}\right)$. + +From beats, $\omega = \frac{\DT}{\Db} = \frac{S_0 - S_1}{S_0S_1\Db}$ (or just +use $\omega = \left( \frac{1}{S_1} - \frac{1}{S_0} \right) / \Db$). + +To compute $b$~from~$t$ we can use +$b = \frac{\e^{\omega t}-1}{S_0 \omega}$. + +And for $S$~from~$t$ we have +$S = \frac{S_0}{\e^{\omega t}} = S_0\e^{-\omega t}$. + +As for $t$~from~$b$ the formula becomes +$t = \frac{1}{\omega} \log \left( 1 + S_0 \omega b \right)$. + +Lastly, for $S$~from~$b$ we can see: +\[ S = 1/T = \frac{1}{1/S_0 + \omega b} = \frac{S_0}{1+S_0\omega b}\] + +Maybe we can also to compute and cache +$\omega' = S_0 \omega = \frac{S_0}{\Dt}\log\left(\frac{S_0}{S_1}\right) + = \frac{S_0 - S_1}{S_1\Db}$ +Which is in « per-beats » and occurs often in the formulas (but not +exclusively). Note that the fact that $\omega$~or~$\omega'$ is relevant is +\emph{not} depending on whether we compute from time or from beats, or even if +the ramp length is defined in time or in beats. + +\end{document} +