Given two frequencies, how do I determine the middle point between
them?

Example: Lo freq: 160; Hi Freq: 1250; Width: ~3 octaves.  Middle freq:
???.

Example: Lo freq: 1600; Hi Freq: 5000; Width: ~1.5 octaves.  Middle
freq: ???.

Context: I'm creating a few ultra simple excel spreadsheets to help me
calculate settings for Sound Forge's multi-band audio signal
compression. (not data compression)

Hi,  donphiltrodt-ga 

Thanks for the interesting question.

In terms of musical theory the "average" of two tones is
mathematically the geometric mean of their frequencies.  The simplest
case would be a range of two octaves (which is a ratio of 4 between
the frequency extremes), and the midpoint would be one octave from
either endpoint (a ratio of 2 = sqrt(4) from each).

To answer your two examples:

Between 160 Hertz and 1250 Hertz (cycles per second) the "midpoint"
is:

sqrt(160 * 1250) = 447.21 Hertz

Between 1600 Hertz and 5000 Hertz the midpoint is:

sqrt(1600 * 5000) = 2828.4 Hertz

Note that the "width" quoted for your first range (3 octaves) slightly
overstates the actual width (a ratio of 7.8125 rather than 8), while
the stated "width" of the second range slightly understates its value
(a ratio of 3.125 rather than 2.828).

I don't think for your stated purpose (sound compression) that a very
precise value for the midpoint will be significantly better than a
rough one.  If you'd like to use a "standard" tone for each of
midpoints, I'd pick:

440 Hertz =  "A4," the fourth A from the bottom of the piano keyboard

2793.8 Hertz = "F7", the seventh F from the botton of the piano
keyboard

The standard piano tuning is based on the "equal temperment" scale and
anchored by the tone 440 Hertz at A4.

For a short introduction to the theory and history of the musical
scales, see:

http://www.silcom.com/~aludwig/The_musical_scale.htm

This is a vast subject and an interesting one, but not essential to
your immediate question.  If I can clarify the answer further, please
ask!

regards, mathtalk-ga