First an electronic pitch pipe, then a traditional reed pitch pipe. Finally I understand why the electronic pitch pipe sounds so hollow, and why it is so much harder to get my pitch from it: It's missing half of the harmonics!
The Bb at about 470 Hz is the lowest distinct line (around the 0.5k mark). So the electronic pitch pipe is missing every other overtone of that fundamental. That means you're hearing the 5th in the next octave up and the 3rd in the next octave after that, but not hearing the actual octaves around 1000 Hz and 2000 Hz. Clearly those are present in the reed overtones.