### Kepler's Music: Pitch Ranges of the Planets

Each planet has a warbling pitch, varying according to the solar angular velocity of the planet, nu-dot, as the planet moves around its elliptical orbit.

#### Pitch varies according to the 2nd law

According to the 2nd law, nu-dot varies inversely to the square of the distance from the sun, rho:
 nu-dot = K/rho2
The constant of proportionality, K, depends on the units chosen. And for Kepler, the units of choice were arc-minutes per mean-earth-day, or more simply, minutes per day,
[4 mean] K1 = 21600/P minutes per day
where P is the period of the planet in mean-earth-days.

#### Pitch ratios

The highest pitch for a planet occurs, therefore, when the motion is most rapid, and hence when rho is at its minimum, rho-min, at perihelion. Similary, the lowest pitch will occur when the motion is slowest, and hence when rho is at its maximum, rho-max, at aphelion. The ratio of the highest pitch to the lowest is thus,
high / low = {K/rho-min2} / {K/rho-max2}
= rho-max2 / rho-min2
= (rho-max / rho-min)2
But rho-max = 1+e, and rho-min = 1-e, for a planet with orbit eccentricity, e. So the expression above becomes,
high / low = = (1+e / 1-e)2
depending on eccentricity only. And so we may easily construct this table of high-to-low pitch ratios, taking modern values of e again from (Moore/Hunt, 1983; p. 440).

Planet e, modern Pitch ratio Interval
Mercury 0.2056306 2.304 9/4, Ninth
Venus 0.0087826 1.036 Unison
Earth 0.0167175 1.069 Unison
Mars 0.0933865 1.454 3/2, Fifth
Jupiter 0.0484681 1.214 4/3, Fourth
Saturn 0.0556125 1.250 5/4, Third
 We may compare these values, based on our modern data (Moore/Hunt, 1983) with Kepler's values, based on data from Copernicus and Maestlin (see Katz, 1993, p. 375). Even today, the orbital ratios are within a few percent of the musical ratios. Revised 05 jan 2002 by ralph abraham