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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
, 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
and its harmonics.

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Shahrokh Yadegari
2001-03-01