--- In tuning-math@y..., jpehrson@r... wrote:
> Well, my big "mean" math question has to do with the idea of
> the "root mean square" (RMS) method of finding averages that Graham
> Breed was talking about on the "fat" list...
>
> I'm actually intrigued by this, since I'm not understanding why
> squaring everything, adding it all together and then taking the
> SQUARE ROOT of the sum is going to lead to an accurate average...
>
> Why is this done this way again?? This is pretty interesting,
> actually...

You don't want to take the _straight_ average because it might be
zero just from positives and negative signs canceling out.

The two simplest alternatives are to take the _maximum_ error, or to
take the average of the absolute values of the errors (called MAD,
for Mean Absolute Deviation).

The RMS is known as the Standard Deviation in statistics. It's the
standard measure of error in science and engineering. There are
several reasons for this. Let me give you a rough idea of why it
makes some sense in this context.

Look at the dips in the harmonic entropy curve. Notice how they
are "rounded" at the bottom. Any curve with a round minimum like this
(not getting too technical) approximates a parabola more and more
closely the more you zoom in on the minimum. A parabola is just the
curve representing squared error. So if you sum the squared errors,
you're summing the dissonances, in a sense. And then you have to take
the square root at the end so that the result is comparable with the
units for a _single_ error. For example, in the 3-limit there's only
one interval to evaluate. Let's say it has a 2-cent error. So any
sort of _average_ over this one interval would have to be 2 cents. It
wouldn't make much sense to say the average was 4 cents when there's
only a single 2 cent error, would it? So that's why you have to take
the square root after summing. If you want to get more technical,
check out a statistics book.