The Well-tempered Scale

Imagine that we have recorded a melody on tape. If we play the tape twice as fast as it was recorded, the melody is transposed up an octave, and if we play the tape one and a half times faster, the melody is transposed a fifth above. In almost any tonal scale other than the well-tempered scale, not all the new notes resulting from transposition by time scaling would fall exactly over the scale values. In other words, the melodies in the well-tempered scale are invariant against time scaling with a similarity factor of $\sqrt[12]{2}$, meaning that if we transpose any melody according to any of the frequency factors of the scale, we come up with a melody whose notes are all in scale. Schroeder[43, page 99] explains the different power laws which govern this property of the well-tempered scale, and he also explains that if we had all the notes of a piano (which was tuned exactly according to the well-tempered scale), sounded simultaneously, we would hear a self-similar Weierstrass function with $\beta = \sqrt[12]{2}$ and its harmonics.