Multiplying the tonic by 2, 3 and 5 creates the octave, fifth and third respectively. The ear hears these intervals very well. We can easily sing them. Each one has a feel, a sort of harmonic flavor, that makes a fifth a fifth and a third a third.
It turns out that the ear can also easily hear compounds, that is, combinations of these low primes. Combining 2 with anything else simply puts it in another octave. But when you combine 3 and 5, or 3 and another 3, you get entirely new flavors. Here’s an example:
The final note is an octave plus a major second above the tonic — a major ninth. Its ratio is (3/2) x (3/2), or 9/4. It has a haunting sound, to me, a different beauty certainly. A new crayon in the box.
Next: The Major Seventh