Golden Rules 1.
2.41421 . . . = 1 + √23.
1.41421 . . . = √2 4.
1.32471 . . . , which is the real-number solution to the equation x3
- 1 = 0 .5.
1.27202 . . . = √1/2 + √5/2, which is the real-number solution to the equation x4
- 1 = 0. This rectangle was first found by Robert Ammann.Tuning Up1.
This six-note scale, which the Greeks called a hexachord, has only five distinct intervals: 9/8, 3/2, 27/16, 81/64, and 243/128. The scale was constructed by repeatedly multiplying 128 by 3/2 and dividing by 2 when necessary to keep all pitches within one octave. 2.
Four distinct half steps. From smallest to largest: 25/24, 135/128, 16/15, and 27/25.3a.
The 16-note equal-tempered scale includes a note that best approximates the major third in a 12-tone scale in just intonation: 1.24186 . . . 3b.
The 29-note equal-tempered scale has a note that best approximates the perfect fifth in a 12-tone scale in just intonation: 1.50129 . . . 3c.
The 41-note equal-tempered scale has a note that best approximates a major third (1.24580 . . . ) and a perfect fifth (1.50042 . . . ). Working the Angles
Shown in the figure at right are all the ways to bounce the ball off all four sides of the table and reach each of the three upper pockets. There are two solutions for the middle top pocket.
To figure out which of the paths to pocket B is shorter, imagine that the sides of the table are mirrored walls. If you were to look into the mirrors, you would see an infinite array of reflected tables. Draw mirrored copies of the pool table on all sides extending out two tables in all directions. A bouncing path on the table is equivalent to a straight path through the mirrored tables: Each path continues past the table's edge at the points where the ball has bounced off one of the sides. For the path to bounce off all four sides, it must end in one of the four yellow tables, yielding the four solutions to the original problem. Of the two paths to the red pocket, the dotted path on the left is slightly shorter, since the ball starts toward the left side of the table.
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