# Rational Approximations to Equal Temperament

As an interesting experiment, consider starting with the equal tempered scale, then deduce the best rational numbers, that is, the best ratios of integers that will most closely match the equal tempered intervals. Now, if all integers are allowed, you will be able to get arbitrarily close to the irrational values of the equal tempered scale. To make this "study," only integers up to a specified maximum will be allowed. One question to be addressed is whether or not the just temperament is reproduced in the process.

To efficiently find rational approximations, the binary mediants search method, described here, was used. This table shows the results.

Best Rational with Maximum Integer of:
Interval Just Scale Ratio Equal Tempered Value 15 20 25 100
Unison1/11.0000 1/1 = 11/1 = 11/1 = 11/1 = 1
Minor Second25/24 = 1.04171.05946   15/14 = 1.07143    18/17 = 1.05882    18/17 = 1.05882   89/84 = 1.05952
Major Second9/8 = 1.12501.12246 9/8 = 1.125009/8 = 1.125009/8 = 1.1250055/49 = 1.12245
Minor Third 6/5 = 1.2000 1.18921 13/11 = 1.1818219/16 = 1.1875025/21 = 1.9047644/37 = 1.18919
Major Third 5/4 = 1.2500 1.25992 5/4 = 1.2500019/15 = 1.2666724/19 = 1.2631663/50 = 1.26000
Fourth 4/3 = 1.3333 1.33484 4/3 = 1.333334/3 = 1.333334/3 = 1.333334/3 = 1.33333
Diminished Fifth 45/32 = 1.4063 1.41421 10/7 = 1.4285717/12 = 1.4166724/17 = 1.4117699/70 = 1.41429
Fifth 3/2 = 1.5000 1.49831 3/2 = 1.500003/2 = 1.500003/2 = 1.500003/2 = 1.50000
Minor Sixth 8/5 = 1.6000 1.58740 8/5 = 1.6000019/12 = 1.5882419/12 = 1.58824100/63 = 1.58730
Major Sixth 5/3 = 1.6667 1.68179 5/3 = 1.666675/3 = 1.6666722/13 = 1.6923137/22 = 1.68182
Minor Seventh 9/5 = 1.8000 1.78179 9/5 = 1.8000016/9 = 1.7777825/14 = 1.7857198/55 = 1.78182
Major Seventh 15/8 = 1.8750 1.88775 15/8 = 1.8750017/9 = 1.8888917/9 = 1.88889100/53 = 1.88679
Octave 2/1 = 2 2.00000 2/1 = 22/1 = 22/1 = 22/1 = 2

This table shows that in many cases the equal tempered interval is well approximated with the just interval, but there are many cases where there is a better choice of integer ratios that can be used to approximate the equal tempered scale than those of the just scale. Such approximations might show up if you were trying to design an (approximate) equal tempered keyboard using gears, or any similar technique, that naturally produces integer ratios, and those integers must be within a practical range.