[crazygringo]

I've always wondered... is there some way to mathematically "solve" for this with multiple microphones and multiple speakers?

Like with 2 of each, or 3 of each, where you play the same waveform through every possible pair of speaker and microphone, you can solve some kind of system of matrix equations to determine the only possible combination of responsiveness at each device at each frequency?

Or do you just need a reference microphone with known characteristics, period, end of story, because math can't do it?

(Obviously from a practical perspective you want the reference microphone... I'm just curious about in theory.)

[analog31]

Yes, multiple microphones is how microphones can be calibrated in the first place. You make a particular kind of microphone that also functions as a speaker, with certain testable assumptions about how they work. Then you point one at the other and vice versa. The result is a reference microphone that can then be used to calibrate other microphones.

You don't do it every day, which is why an outfit like Bruel & Kjaer can charge a lot for their gear. ;-)

[hunter2_]

I believe it's hardly different from trying to deduce perfect distances from multiple rulers of dubious precision: you need to compare them to one of extreme precision. Arranging 4 rulers into a perfect square proves that they have equal lengths, but you still don't know their offset from standard length.

However, if you ignore tolerances and assume that every microphone of a given model number has equal response, then it's simply a matter of having that known response information available, similar to a hypothetical brand of ruler being known for coming up short.

[crazygringo]

> but you still don't know their offset from standard length.

But that's fine for microphones -- the question here isn't to determine their absolute volume, which is of course unsolvable. It's to determine the relative "volume" (response) at each frequency. It's the shape of the curve that matters, not its offset.

And again, I'm not looking for a practical solution (like getting the info from a manufacturer) -- I'm just curious about it in theory. If it's inherently solvable or not.

[gmueckl]

This isn't solvable. Each loudspeaker and microphone has its own frequency response and will always measure the product of any two of them. This does not result in a unique solution for any single frequency response, even when you know the clean source signal. There is always a degree of freedom of how much each device in a pairing contributes to the final response.

[dwiel]

what if you move the microphones and speakers at varying but precise speeds so that doppler shift can be used to shift frequencies? you could play a tone on the speaker, shift the relative velocity (spin the microphone really fast?) and calibrate a frequency range of the microphone. with a calibrated mic frequency range, you can now calibrate that range of the speaker. repeat. each calibration step is going to accumulate error. to be clear, not a practical solution, but fun to theorize.