In this way the equation -
r₁³ / r₂³ = t₁² / t₂²
assumes the form-
r₁³ / r₂³ = (t₁ / t₂) / (t₂ / t₁)
The right-hand side of the equation is now constituted by the double ratio of the linear values of the periods of two planets, and this is something with which we can connect a quite concrete idea.

To see this, let us choose the periods of two definite planets - say, Earth and Jupiter. For these the equation assumes the following form ('J' and 'E' indicating 'Jupiter' and 'Earth' respectively):
r_J³ / r_E³ = (t_J / t_E) / (t_E / t_J)
Let us now see what meaning we can attach to the two expressions
t_J / t_E and t_E / t_J.

During one rotation of Jupiter round the sun the earth circles 12 times round it. This we are wont to express by saying that Jupiter needs 12 earth-years for one rotation; in symbols:
t_J / t_E = 12 / 1
To find the analogous expression for the reciprocal ratio:
t_E / t_J = 1 / 12
we must obviously form the concept 'Jupiter-year', which covers one rotation of Jupiter, just as the concept 'earth-year' covers one rotation of the earth (always round the sun). Measured in this time-scale, the earth needs for one of her rotations 1 / 12 of a Jupiter-year.

With the help of these concepts we are now able to express the double ratio of the planetary periods in the following simplified way. If we suppose the measuring of the two planetary periods to be carried out not by the same time-scale, but each by the time-scale of the other, the formula becomes:
r_J³ / r_E³ = (t_J / t_E) / (t_E / t_J) = period of Jupiter measured in Earth-years / period of Earth measured in Jupiter-years.
Interpreted in this manner, Kepler's third law discloses an intimate interrelatedness of each planet to all the others as co-members of the same cosmic whole. For the equation now tells us that the solar times of the various planets are regulated in such a way that for any two of them the ratio of these times, measured in their mutual time-units, is the same as the ratio of the spaces swept out by their (solar) orbits.

Further, by having the various times of its members thus tuned to one another, our cosmic system shows itself to be ordered on a principle which is essentially musical. To see this, we need only recall that the musical value of a given tone is determined by its relation to other tones, whether they sound together in a chord, or in succession as melody. A 'C' alone is musically undefined. It receives its character from its interval-relation to some other tone, say, 'G', together with which it forms a Fifth. As the lower tone of this interval, 'C' bears a definite character; and so does 'G' as the upper tone.