What were the transfer curves of major recording technology from Wax cylinders to HiFi electronics?
This should be a comment, but I can't put pictures in a comment.

You made me curious with your question, so I downloaded a few copies of random wax cylinder recordings and had a look at them with the spectrum analyzer in Audacity.

I think you'll find that there is no "one true transfer function" for that kind of equipment.

This recording of "A Hot Time in the Old Town" by Francis McCarty has a really ratty frequency range:



It has a lot of rumble below 200 Hz, and usable audio really only from above 200Hz to close to 4000Hz.  The usable range is anything but flat.  There's like about a 3dB per octave highpass filter at 4000Hz.  Sounds like the guy is singing in a tin can.

This recording of "Butcher's Boy" by Frank Luther is much better:



It also has a lot of rumble below 200Hz, but above that the response is fairly flat.  The voice sounds almost normal if you can ignore the noise and the singing style.

The frequency analysis are made of the noise before each song - that's what you actually hear from the playback.  Given that these are recordings of the recordings, the transfer function is the total result of the following:


Original recording transfer function.
Original playback transfer function.
Modern recording transfer function.
Any post processing that may have been applied in an attempt to improve the recording.
MP3 compression artifacts.


I would make a guess that such recordings could be considered "flat" from about 200Hz to maybe 4000Hz.

I expect there was a lot of variation in the results, though.  The acoustics of the room the recordings were made in would have an influence, and I would almost bet that there were mechanical resonance and damping effects in the recording and playback machines intended to "equalize" each other.  Sort of like a mechanical "Dolby noise reduction" where you use a resonance in the recording side to make some frequencies louder to make up for a loss in the playback - or the other way around where you would use a damping effect to reduce some overly loud frequency range caused by a resonance in the recorder. That's speculation, though.