Our next task is to determine how a metrics can be established in a sensory continuum. Consider, for example, a continuous stretch of shades of grey passing from white to black. What do we mean exactly when we say that some definite shade is twice as dark as another? Obviously no definite meaning can be assigned to this statement until we have posited some convention permitting us to establish comparisons.
As another instance, take the case of a continuous stream of sounds varying in pitch. What do we mean by saying that some particular musical note is twice as high in pitch as some other, or that the interval between two notes is equal to the interval between two others? If we were to be guided solely by our ear we might assert that as certain musical notes, though differing in pitch, yet appear to present a certain undefinable similarity (the successive octaves), the intervals between these successive similar notes should be considered equal or congruent. We should thus define as equal the extension of notes subtending the successive octaves of a given musical note.
But if now we had learnt to measure the frequencies of vibration of the various musical sounds, a new type of measurement would immediately suggest itself. Starting from any musical note—say, the middle
of the piano, we should find that the octave of
was vibrating twice as fast, the following
three times as fast, and the second octave of our original