highlight that doubling the fre-
quencies of consecutive sounds gives a especially nice eect when they overlap.
To build and tune instruments with xed sounds (harpsichord, piano, . . . ) the
tempered scale is used. Moreover, this scale oers the possibility of modular
compositions, i.e. has the feature to move from a tonality to another.
In order to construct musical scales, that are nice to hear, it is necessary
to follow certain rules: if a scale has a frequency f then it should have also the
frequency 2f and then the frequency f=2. The main issue is that the sound
sequences give a melody portable on our scale, in other words the melody
should be able to be reproduced by starting from a sound of any frequency.
The melody is characterized by the ratios of the frequencies of consecutive
sounds for psychoacoustic reasons. Since we wish to play a melody from dif-
ferent frequencies, these ratios should be assigned constant. Let