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Engineer’s Bookshelf

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Bellhop

5/11/2012 9:44 AM EDT

RE: 12 note musical scale. You can indeed play in any key and have it sound the ...

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# Book Review: In Pursuit of the Unknown by Ian Stewart

## 5/2/2012 3:06 PM EDT

It really is a funny old world. When I was slogging my way through math lessons at high school, and cursing the folks who invented things like logarithms, I would never have dreamed that I would one day read a book on equations "Just for the fun of it!"

But things change as you grow older. I just finished reading In Pursuit of the Unknown – 17 Equations That Changed the World by Ian Stewart, and I have to say that – on the whole – this was a very good read.

I would dare to say that the majority of us have at least a passing familiarity with history's great equations, such as Pythagoras's theorem, Newton's Law of Gravity, or Einstein's E = mc2. I would also dare to suggest that most of us don't know as much as we think we know with regard to the ways in which these equations were derived and the things they lead to.

The author has a very friendly and engaging style, and he provides detailed explanations of things I may once have known, but (if so) have long forgotten. We start with Pythagoras's Theorem, which describes how the three sides of a right-angled triangle are related. This isn’t a long chapter, but I came away thinking "Wow ... I didn’t know this ... or that ... or that!"

One thing I really liked about this book is that it's packed with nuggets of knowledge and tidbits of trivia. For example, clay tablets show that the Babylonians knew about the 3-4-5 triangle at least a thousand years before Pythagoras.

The interesting thing is how the fundamental equations introduced at the beginning of the book keep on reappearing in different guises as we stroll through more complex concepts later on. I cannot tell you how many times Pythagoras's theorem popped up throughout the book (a bit like the mathematical equivalent of Whac-A-Mole).

Chapter 2 is on Logarithms. Before reading this chapter I knew how to use these little rascals, but I had forgotten exactly what they were or how they were derived. By the end of this chapter I was squirming in my seat with excitement.

From here we plunge into Calculus, Newton's Law of Gravity, the Square Root of Minus One, Euler's Formula for Polyhedra, Normal Distribution, the Wave Equation, the Fourier Transform, the Navier-Stokes Equation, Maxwell's Equations, the Second Law of Thermodynamics, Relativity, Schrödinger's Equation, Information Theory, Chaos Theory, and the Black-Scholes Equation.

It has to be said that some of the equations and topics are more challenging than others, but all-in-all I really enjoyed myself. I discovered all sorts of things, like how the Wave Equation (which was originally developed in the mid-1700s to explain how strings vibrate) lead to huge advances in our understanding of water waves, sound waves, and light waves.

I was also captivated by the discussions on how harmonious ratios of the fourth (4:3) and the fifth (3:2) ended up providing a theoretical basis for a musical scale and led to the scale(s) now used in most Western music. The real interesting thing to me was the discussion on how each tone is derived as a ratio, and how each tone can be divided into two intervals, each close to a semitone, to obtain a 12-note scale (think of the white and black keys on a piano – there are 12 for each octave). The thing is that the way in which each tone is divided can be achieved in several different ways, yielding slightly different results. However it is done, there can be subtle but audible problems when changing the key of a piece of music. This completely blew me away – I used to play the piano and trumpet and trombone (not all at the same time, you understand), and I had always assumed that you could simply swap one key for another and continue on your merry way.

Later we run into the Scottish physicist and mathematician James Clerk Maxwell (1831-1879). Maxwell's Equations provided the first major unification of physical forces, showing that electricity and magnetism are intimately related. (You can only imagine my surprise when the Wave Equation from Chapter 8 jumped back into the picture.) This lead to the prediction that electromagnetic waves exist and that they travel at the speed of light (finally, after reading this book I understand how this all came about); in turn, this lead to the realization that light itself is an electromagnetic wave. All of this motivated the subsequent invention of radio, radar, television, and most modern communication systems.

As always, I could waffle on and on about this until you begged me to stop. Suffice it to say, however, that I think this book has something for everyone, from younger and/or less-informed readers to engineers and scientists. Even folks who consider themselves to be knowledgeable about mathematics will learn something here, such as the historical background with regard to how their prized mathematical tools came into being.

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Bellhop

5/11/2012 9:44 AM EDT

RE: 12 note musical scale. You can indeed play in any key and have it sound the same because we use the "Equal Tempered Scale", where the ratio between each of the 12 tones is the 12th root of 2. This tuning is a compromise. We make all of the keys sound the same by making all of them wrong! If you tune an instrument using fourth and fifth intervals, it will sound delightful in that key, but will sound increasingly out of tune as you move to more distant keys. Many musicians today object to the equal tempered scale on the grounds that it is never right. Listen to recording made using other tunings. They are distinctive! A MIDI keyboard is an ideal platform for experimenting with alternate tunings as you can quickly change from one to another.