"Here will we sit and let the sounds of musicI was thinking a bit more concretely about this, and I looked again at the numerical ratios that define notes and intervals. Even in my muddy vesture of decay, I thought it would be fun to extend that particular idea to the planets.
Creep in our ears. Soft stillness and the night
Become the touches of sweet harmony.
Sit, Jessica. Look how the floor of heaven
Is thick inlaid with patens of bright gold.
There's not the smallest orb which thou behold'st
But in his motion like an angel sings,
Still choiring to the young-eyed cherubins.
Such harmony is in immortal souls,
But whilst this muddy vesture of decay
Doth grossly close it in, we cannot hear it."
In the last post I mentioned the difference between equal-tempered notes and those that you get from just intonation. Equal temperament is just taking the octave (the distance between any given note and a note with twice as high a frequency) and splitting it up equally into 12 semitones. Music nerds break up the octave into 1200 "cents," where each 100 cents is one equal-tempered semitone. These are shown in the image below by the equal-sized blocks of color:
However, the notes that really sound the most harmonious don't fit exactly onto those 100-cent intervals. Instead, they're defined by simple ratios of frequency. The perfect fifth, for example, is an exact multiple of 3/2 times the frequency of the base note, or tonic. If you compute the cents of that interval, you get 701.9549560547 or so. That's pretty close to 700. Many other harmonious ratios end up being close to even hundreds in cents, and that's what made equal temperament so popular and useful. The little white lines in the image above show where many of them fall on this scale. I gave only one example -- see 590 and 610 -- where music theory folks prefer to distinguish between two notes, but an equal-tempered instrument like the piano can only play one "average" note that has to serve for both (600). You know the old joke... you can tuna piano, but you can't tune a diminished fifth!
Music blogger Gary Garrett has talked about some other bluesy notes that fall even further away from the equal-tempered boundaries...
It was an eye-opener for me to see all of these -- and to learn where in popular music they're used, too. There are so many possibilities!
But back to the music of the spheres. I realized that each planet has its own "frequency" (i.e., the rate at which it orbits the Sun), and one could make similar ratios. I gave it a try, using the Earth's frequency as the "tonic" with which all of the others are compared. Earth is the natural "home key," of course. (I guess I could go all Shakespeare again with "Man is the measure of all things.") :-) I chose to ignore octaves, since in music it's seen as insignificant, harmonically, to go up or down an octave. I also picked some interesting asteroids and one famous comet, since they're just as much fellow travelers around the Sun as the big planets. Here's where their frequency ratios fall, musically...
Notice the big clump of planets right near the Earth at the top. Far-away Pluto orbits the Sun once every 248 years, but that's close to 256, which would be exactly 8 octaves up from the Earth's orbit once every 1 year. Mercury orbits once every 0.24 of a year, which is close to a quarter (2 octaves down). All those near halves, doubles, halves of halves, doubles of doubles, and so on, fall very close to either the unison or the octave. These were also noticed hundreds of years ago and codified into something called the Titius-Bode Law. Modern astronomers don't quite know what to do with that "law," since it's not exact, and there seems to be no physical reason for the planets to be sitting at near factors-of-two separation like that. "Coincidence" may be the best explanation, since the thousands of other extrasolar planets don't seem to follow such a law.
Notice also that Jupiter falls close to a perfect fourth, which is the inverse of a perfect fifth. When I first computed these things, I used periods instead of frequencies, and Jupiter fell just about smack dab onto the perfect fifth. (There's a reason for that, since periods are "inverses" to frequencies in just the same way as the musical inverses that I mentioned above.) I almost posted the period-version of the chart, but the pedant in me knew that it's more honest an analogy to use frequencies like in music. Anyway, in my last post, I quoted Gary Garrett comparing Jupiter's weak gravitational pull to a chord with a root on the perfect fifth! More coincidence?
Who knows. But I do know that these kinds of harmonious comparisons are just the ticket for thinking about the Glass Bead Game. Hesse knew music was the key, as did Shakespeare...
"The man that hath no music in himself,
Nor is not moved with concord of sweet sounds,
Is fit for treasons, stratagems, and spoils;
The motions of his spirit are dull as night
And his affections dark as Erebus.
Let no such man be trusted. Mark the music."