A friend approached me with him not completely understanding the technique of how Tristan Murail explained his approach of distorting the overtones of a certain basenote. In this blogpost I'm going to explain the way I approached the calculations.

The source [Desintegration of Tristan Murail -- An analysis of harmonic generation and use of spectra - Kenneth Lampl] described a curve in the shape of 

$$ f(x) = ax^b + c. $$

With $a=f_1$, the base frequency, $b=1$ and $c=0$ one obtains the conventional overtones with $x$ values being whole numbers. For example, if $a=f_1=100$, then $f(x=1)=100 \cdot 1 = 100$, $f(x=2)=100 \cdot 2 = 200$ and so on.

The way Murail distorts the overtone series is with following these steps.

1. Choose a base note, for example $f_1=100$.
2. Choose two partials, $p_1$ and $p_2$. For example, let's make them the 3rd and the 7th overtones resulting in $p_1=3$ and $p_2 = 7$, and their corresponding frequencies $f_{p_1}=f_3 = 300$ Hz and $f_{p_2} = f_7 = 700$ Hz.
3. Nudge them a little bit, more concretely detune them with a certain amount. For example make the 3rd partial 10 Hz lower, $f_{p_1}' = f_{3}' = f_3 - 10 =290$ Hz. Make the other partial 50 cent higher resulting in $f_{p_2}' =f_7' = f_7 + 50$ cent $=926.37$ Hz.
4. Find the parameters $a,b,c$ for the curve $f(x)= ax^b + c$ such that evaluating them at $f(1)=f_1$, $f(p_1) = f_{p_1}'$ and $f(p_2) = f_{p_2}'$. In this case we want to find the parameters for which the curve results in $f(1) = 100$, $f(3)=290$ and $f(7) = 926.37$.