four times as fast. It would then appear plausible to define equal intervals between musical notes by the differences in their rates of vibration, and we should infer that the distance between
and its first octave was equal to the distance between this last note and the following
, and equal again to the distance between this
and the following superoctave
.
We should thus have obtained a definition of equal stretches of sounds which was at variance with our original definition, in which all octave intervals were regarded as equal or congruent. In view of these conflicting results, we could not well escape the conclusion that a sensory continuum of itself offers us no precise means of defining equal stretches, and that whatever definition we might finally select would be a mere matter of choice, an arbitrarily posited convention.