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It appears that the world is very lucky with the 12 tones.

Why do we not use a ten-tone or twenty-tone equal-tempered scale? Is there something special about twelve?

The answer is: Yes, the twelve-tone equal-tempered scale is remarkable. The nearly perfect intervals seen in the table above are not typical of other equal-tempered scales. Consider the six basic consonant intervals less than an octave (described above): 3/2, 4/3, 5/4, 6/5, 5/3, 8/5. The twelve-tone equal-tempered scale is the smallest equal-tempered scale that contains all six of these pure intervals to a good approximation - within one percent.

Let's compare the twelve-tone equal-tempered scale to some other scales.

    * All equal-tempered scales with 14 notes or fewer (except the twelve-tone equal-tempered scale) contain at most only two of the six basic intervals within one percent.
    * Several equal-tempered scales with between 15 and 30 notes (notably the 19-tone and 24-tone scales) contain all six basic intervals, but in none of these scales are the intervals more nearly pure than in the twelve-tone equal-tempered scale.
    * The 31-tone equal-tempered scale has all six basic intervals to a good approximation, some with better accuracy than the twelve-tone scale, but the most important fifth (3/2) interval is less accurate than in the twelve-tone scale (218/31=1.495). Some Indonesian music actually uses a 31-tone equal-tempered scale.
    * The 41-tone equal-tempered scale is the first with a better fifth (3/2) interval than the twelve-tone scale (224/41=1.5004).
    * The 53-tone equal-tempered scale has all six basic intervals with a better accuracy than the twelve-tone scale (231/53=1.49994).

by das monde on Wed May 14th, 2008 at 05:38:42 AM EST
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