I know a little bit, but not as much as I'd like. I have to pass on this one.
Speaker cone and compression driver diaphragms and tweeter dome materials all have the same goal:
-be lightweight
-be rigid
-be damped
-be durable
-be sexy
-be affordable
Paper is an amazingly good balance of these things with a little weakness on durability, and less sexy factor. Thankfully paper cones can be coated with water-resistant treatments that make them very durable. I am still absolutely amazed that one of the least expensive cone materials is also one of most well-rounded performers too (paper).
All the other materials are basically a mixture of those five qualities above. There are too many materials to analyze here but I think some of the the more exotic materials like beryllium and graphene have higher performance because they have the trifecta: lightweight, rigid, damped. The high price and exotic nature makes them sexy too.
Synthetic materials are great for mobile audio and marine and motorcycles where the environments are harsh. All kinds of plastics and polymers and composites are used because of their chemical and ultraviolet resistance. They don't always have the lightest weight, but they are generally affordable and very durable.
I'm not sure what you mean about odd order harmonic distortion though. When a cone is playing a high enough frequency it starts to distort in shape and has standing waves and bending modes but I don't think they add harmonic distortion to the mix. All the cone breakup and bending modes are the same frequency so you will get some off-axis response changes and some higher frequency roll-off, but I'm not sure there will be extra energy at other frequencies. See my post above with Dr. Russell's website and his animated gifs. They are sensational!
I think the thing you are hearing that is undesirable is the different damping coefficients. Better damping means the breakup modes are more controlled. Also certain material and geometry combinations can lead to dramatic or less dramatic breakup mode shapes. Scanspeak for example does a lot with geometry to control the way the breakup modes form and which standing waves are supressed. I'm going to borrow a picture from Erin's site below.
Attachment 9807
https://www.erinsaudiocorner.com/dri..._4.5_midrange/
Something else to consider: every cone or diaphragm will have a breakup mode at a high enough frequency. The clever speaker designers will either make those breakup behaviors gentle and polite such as the Scanspeak speaker above, or the designers will try to push those breakup behaviors well above the intended frequency range of the speaker.
The Audiofrog GB60 is a great example of this because it is intended to be crossed over at 2khz but the spike in frequency response from bending modes occurs at about twice that frequency.
The Peerless SLS 6" is one of my favorite midbass speakers. The frequency response falls apart above 500hz'ish due to bending modes. I'm guessing the manufacturer didn't worry about that because the specs on that driver scream "strong midbass for a 3-way system". It's all about the intended use in the end!
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edit: this is fun! thank you for the questions