Once one has the octave and the fifth, another interval comes into being, that which is called the perfect fourth. If we consider the progression from a note, to the one a fifth above it, and then to an octave above the original, we have a new interval between the fifth to the octave. This is NOT another fifth. Instead going from 3/2 of the original frequency to twice the original frequency means that you have gone up a ration of 4/3 in going from the fifth to the octave, a new ratio, and again a ratio of whole numbers. Again, two notes a fourth apart share harmonics- not as many as with a Pythagorean fifth, but now the every fourth harmonic of the lower note is shared with every third harmonic of the higher note. Again we would predict that two notes a fourth apart would again be harmonious, not as harmonious as the fifth (not as many harmonics are shared) but harmonious nevertheless. And again, within the music theory, the interval of the fourth is again considered a harmonious interval.