### Kepler's Music: Mean Pitches of the Planets

Mean: As a planet moves around an elliptical orbit with the sun at the lower focus and semi-major axis scaled to one, the solar distance, rho, varies from rho-max = 1+e at aphelion (top) to rho-min = 1-e at perihelion (bottom). The arithmetic mean or median is obviously rho-mean = 1.

Average: The time-average distance, rho-avg, is generally larger than 1, as the planet spends more time near aphelion where its speed is slowest.

Mean distances: In any case, we may regard rho-mean=1 as determining the basic pitch of the planet, which then increases and decreases around this mean or median value. Further, this is the distance which is given in tables of planetary data as the mean distance of the planet from the sun.

Mean pitches: Soner or later we must choose a mean pitch for one planet, then all the others will be determined by ratios, relative to the chosen one. Let us choose, for example, the mean pitch of Saturn, the slowest planet (of the six known to Kepler) as 20 Hz, near the bottom of the human range of hearing. Then we may compute mean pitch values, based on values for R, the mean distance (semi-major axis) for the planets in millions of kilometers, Mkm, from the formula,

mean pitch = factor * 20 Hz, where
factor = (R-Saturn/R-planet)2
The results, using R values taken from (Moore/Hunt, 1983, p. 440), are shown in this table.

Planet R, Mkm Factor Pitch, Hz
Mercury 57.91 607.221 12144
Venus 108.21 173.908 3478
Earth 149.60 99.990 2000
Mars 227.94 39.193 784
Jupiter 778.34 3.361 67
Saturn 1427.01 1 20
Note that more than 9 octaves are required, as 29 = 512.
Revised 05 jan 2002 by ralph abraham