* Depending on one’s definition of “planet,” Pluto and some other bodies that orbit the Sun may or may not have that status. Hence “eight or nine.”
* An example of a third on the piano is the interval from C to E (major third) or C to E-flat (minor third). An example of a sixth is the interval from C to A (major sixth) or C to A-flat (minor sixth). These are intervals that modern ears are most likely to hear as “beautiful” and easy to listen to.
* The Tychonic system had the Sun and the Moon orbiting the Earth, and all the other planets orbiting the Sun. It was the geometric equivalent of he Copernican system, but retained the unmoving Earth.
* Kepler’s first law of planetary motion: A planet moves in an elliptical orbit and the Sun is one focus of the ellipse. Kepler’s second law of planetary moton: A straight line drawn from a planet to the Sun sweeps out equal areas in equal times as the planet travels in its elliptical orbit.
* A half-step is the interval between any note on the piano and the one immediately to either side of it, regardless of whether that is a white or black key.
* Imagine you are standing across the corridor from a moving walkway in an airport. A man is walking along the walkway, from your left to your right, but he is going the wrong way and so is actually losing ground. Say he is walking at 5 miles per hour and the walkway is moving, in the opposite direction, at 10 miles per hour. From your vantage point, you see the combined movement, and the man appears to be moving 5 miles an hour toward the left. A woman is walking faster, 8 miles per hour, but also in the wrong direction. Eight miles per hour is not sufficient to avoid losing ground against the 10-mile-an-hour walkway that is moving in the opposite direction, so, again, from your vantage point, you see the combined motion, and this woman appears to be moving 2 miles per hour toward your left. You cannot be faulted for thinking that the man (who appears to be moving 5 miles per hour toward your left) is moving faster than the woman. If the walkway stopped you would find out what the true velocity of each one was, and your finding would contradict your initial impression. Likewise, Kepler concluded that if the daily rotation of the heavens had stopped, Pythagoras would have seen that Saturn is the slowest of the planets, and should be sounding the lowest tone.
* In German, dur in music still means “major”; moll is “minor.”
* You get the same result by playing scales using only the white keys on a piano but starting on different notes. The Ionian mode (start on C) is the same as the major scale, the Dorian mode (start on D), the Phrygian mode (E), the Lydian mode (F), and the Mixolydian mode (G). The Aolian mode (start on A) is the same as the minor scale.
* The three-man cell was used again in Vietnamese communism, in Algeria in the 1950s, and in the USSR in the late 1960s.
* Picture a collection of coins. Call it Set A. A collection of coins is an example of a “set” that cannot be a member of its own set. In other words, a collection of coins is not a coin. Picture, then, another set (call it Set B) that contains all things that are not coins. This Set B itself is not a coin, so it must be a member of itself. In other words, Set B is a member of Set B. Now picture a third set—Set C. This one contains all the sets that are not members of themselves. Is Set C a member of itself or not? You will find that it is if, and only if, it is not.
* However, the simplest form of a problem is not always the form in which it is first encountered. If it were, the history of mathematical and scientific discovery would have gone much more smoothly!
The Music of Pythagoras Page 40
