I've been aware for a while about the the controversy surrounding the standard A note and whether it should be set to \(440 \text{Hz} \) (as it now is) or changed to \( 432 \text{Hz}. \) I'm trying in this post to look at the mathematical properties of \( 432 \).
- \( 432^2 = 186624 \) is close to the speed of light as measured in miles per second. Wolfram Alpha gives a figure of \( 186282 \) miles per second for the speed of light in a vacuum which is \( 99.82 \text{%} \) of \( 432^2 \).
- It also turns out that the area of an equilateral triangle whose numerical area is equal to its perimeter is given by \(12 \sqrt{3} = \sqrt{432} \).
- \( 432 \) sits between the twin primes \( 431 \) and \( 433 \)
- The factors of \( 432 \) are \( 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, 216 \text{ and } 432 \). The sum of these divisors is \(1240 \).
- \(432 \) is a 3-smooth number, one that is of the form \( 2^i*3^j \text{ where }i,j>=0 \) or to put it less mathematically it is a number that can be written as a power of two times a power of three, specifically \( 2^4×3^3 \). Such numbers have been called harmonic numbers. Here are the harmonic numbers up to \( 1000 \):
- \( 432 \) is the sum of four consecutive primes: \(103+107+109+113 = 432\)
- \( 432 \) is the sum of two positive cubes: \( 6^3+6^3=432 \)
- OEIS lists \( 2944 \) entries for the number \( 432 \)
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