Irrational numbers and cabinet design

A diverting extract for non-mathematicians!

Numbers can be divided into different classes:

• Natural numbers (1,2,3,4...)
• Integers: the natural numbers along with their negatives and zero (....-2, -1, 0, 1, 2...)
• Rational numbers: these are any number which is exactly equal to the ratio of two natural numbers: for example, 1/2, 5/8, 13/29...
• Real numbers: these are all the rational numbers, and also all the numbers in between them which are not equal to the ratio of (non-infinite) integers.
• Complex numbers, and so on.

The real numbers which are not rational numbers are, funnily enough, called irrationals. These include pi, the square root of two, e (the base for natural logarithms), and so on.

The Golden Rule, or Golden Mean, is a ratio which can in a sense be described as the most irrational number of all. You can understand this by looking at how any number can be described in terms of what is called a "continued fraction". For example, you can write pi as Now you could say that 1 plus 1 divided by (292+something) is almost the same as 1, so pi is approximately equal to So, if some irrational number has a large number in its "continued fraction" this means that there are two integers whose ratio is very close to it.

The Golden Mean g is equal to , and its continued fraction is The fact that there are no numbers greater than 1 in the continued fraction for g means that it will take you longer to find a rational number as close as you like to the Golden Mean by this method than it does for any other irrational number. In a sense therefore g is the "most irrational" of all irrational numbers.

As it happens, the Golden Mean is related to the Fibonacci Series 1, 1, 2, 3, 5, 8, 13, 21... , where each number in the series is the sum of the two before it. The ratio of consecutive Fibonacci numbers becomes closer to the Golden Mean the further you go along the series, so 5/3, 8/5, 13/8, 21/13, 34/21 and so on converges towards g. The Fibonacci Series occurs with surprising regularity in the natural world - for example in pine cones, sunflowers and ammonite shells.

How does this come into loudspeaker cabinet design?

If you build a speaker in the shape of a cube, the internal resonances will be at frequencies f where f = c/2d, f = c/d, f = 3c/2d.... where c is the speed of sound, and d is the size of the box. At every one of these frequencies, there will be three coincident resonances in the box, with waves travelling along each of the three axes of the box. Now if you have the dimensions of the box in some irrational ratio to one another, the resonant frequencies corresponding to standing waves between one pair of faces will not overlap with any resonance in one of the other directions. Now, the idea is that if you can't damp out resonances in your cabinet, you might as well spread them out as evenly as possible, and this happens with the sides in the ratio 1/g:1:g. Whether this irrational number "sounds" any better than any other is a different matter.

The idea of avoiding coincident resonances also applies to diffraction on the speaker baffle. Here the part of the sound wave that travels sideways from the driver along the baffle surface interacts in a complex way with the discontinuity at the edge of the baffle. This results in suckouts and peaks in the frequency response at various frequencies related to the distance between driver and baffle edge and, possibly more harmful to sound quality, delayed reflections. If you put a driver in the centre of a circular baffle, the whole of the baffle edge is the same distance from the driver, and so there will be a strong effect at a frequency whose wavelength is related to this distance. A square baffle is a little better. You can improve matters either by putting large-radius curves at the edges of the baffle as in Lynn Olsen's Ariel to reduce the discontinuity there (you have to use at least a 5cm radius for any useful effect), by using non-parallel sides for the baffle, or by placing the drivers asymmetrically on the baffle - The Ariel design uses the Golden Mean here again.

Interestingly, some manufacturers make cabinets with a pentagonal cross-section for a related reason - the geometry of a pentagon is related to the Golden Mean, and the resonances of a cavity with this shape are particularly complex (actually, the spectrum has some of the aspects of a fractal), which is said to be subjectively preferable to those of a rectangular cabinet.

Alex Megann, May 1997

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