Tuesday, January 3, 2012

Subdivision of the Octave

How did we arrive at at a music scale with seven notes in an octave? The process seems to have been more Art than Science. The Pythagoreans new that the octave couldn't easily be bisected into two equal proportional steps. The reason for this is that for step, x, two steps would equal an octave and we would have x2 = 2 or x = √2 = 1.414... For the octave between C256 and C512 the midpoint would be at a frequency of 362.039... Hz. Looking at the plot below one sees that 4/3 is a reasonably good approximation for √2 but slightly smaller. Following this reasoning we arrive at the concept of the tetrachord.

For the tetrachord the step is x = 4/3 and two steps would be x2 = 16/9. The missing factor need to complete the octave which doubles the frequency of the initial note is a = 2*9/16 = 9/8. So we see that the octave can be split up into two tetrachords and a "tone" of 9/8. If we now try to subdivide the tetrachords using the tone as our "unit" we find that it does not subdivide the tetrachord evenly. One division leaves a remaining factor of 32/27 which is approximately evenly divided by 12/11.

So we can try to divide the tetrachord into two approximately equal steps b and c which will be in the neighborhood of 12/11. Examining the factors of 32/27 we see that it can be divided into b = n/9 and c = 32/3n.

Solving for n we find that it is approximately equal to 10 which gives b = 10/9 and c = 16/15. The product abc = 4/3, the step corresponding to the tetrachord. The order of the factors in a tetrachord does not affect the total step so a number of possibilities are possible.

Looking closely we see that one of Ptolemy's scales is similar but not quite the same as Helmholtz's.

Using the correct arrangement of factors we arrive at Helmholtz's sequence of ratios.

One can check Wikipedia for more information about the mathematician John Wallis.

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