The peak, about 1820Hz, is the note I'm whistling, and pretty much everything else is noise the recording happened to pick up.

My whistling range runs from ~659Hz (E5) to about ~2637Hz (E7). A soprano typically sing 262Hz (C4) to 1047Hz (C6), so whistle pitches are really very high!

What happens if we bring that way down, into the bass range, and add a few more harmonics to fill things out? Before we do anything else let's high-pass at 500Hz and low-pass at 3000Hz to pull out just the simple sine of the whistle (mp3):

Now we can make a copy down four octaves (mp3):

Down three octaves (mp3):

Down three and a half octaves (mp3):

Combine them together (mp3):

This sounds pretty good! What's going on?

For simplicity, imagine we start with a 1600Hz tone. The three notes we're generating from it are 100Hz, 150Hz, and 200Hz. These imply a fundamental frequency of 50Hz. This means we're effectively translating the ~659Hz (E5) to ~2637Hz (E7) sopranino whistle to a more pleasant 21Hz (E0) to 82Hz (E2) bass.

The next step is to figure out how to do this live, to back up mandolin. I quickly tried this out in Reaper, but the stock Reapitch plugin isn't willing to shift more than two octaves.