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2009.02.12
It's easy to see that the maximum error of an equal-step tuning
for an arbitrary dyad target is 1/2 the step size. The average
error over a large enough set of dyads is 1/4 the step size.
But for chords, dyadic errors can add, up to the full step size.
For instance, if I want a 4:5:6 triad and the error of 4:5 in my
temperament is -0.49 steps and the error of 5:6 is +0.49 steps,
then the error of 4:6 will be 0.98 steps. Alternatively, I can
choose a different approximation of 4:5, the one with a +0.51
steps error. Then the error of 4:6 will only be 0.02 steps and
the total error of the chord will be lower than when I used the
best available approximations of 4:5 and 5:6.
Calling the 4:5 error x and the 5:6 error y, the 4:6 error is
then x-y, and the total error of the chord is
|x| + |y| + |x - y|
How big can any one term of this expression be when the value of
the total expression is minimized? The answer appears to be 2/3
of a step.