Dunno and I ain't gonna argue with you but I don't think music wire is just wire.
just

Mmmhmmmm this is interesting, definitely taking notes

contact a spring company, and buy a few pounds of music wire, of the correct diameter. while on the phone, ask them about the time/temp required for heat treating in your kitchen oven. youll need a winding tool of some kind to form the music wire as its very stiff. remove from oven and allow to air cool. no quenching involved. the spring company will have someone there thatll be able to answer your q's, and get you started. dont over complicate this thing.

have you looked at the thread in wilderness trapping about building a wolf trap?

I've seen some good spring making stuff in old gunsmithing books from my Library, so you could check that out. Heating springs takes the springiness out. Trick is to harden them again by heating them up red hot until they lose their magnetism and then quenching them quickly. Some say quench in oil, some say water is fine, I don't know what's best.
Once the spring is hardened it'll be super brittle, so you have to temper it around 600° (again, according to a book I read once, not experience). Lead and salt both melt at that temperature. One method I've seen is putting it in a crucible full of salt in a furnace and heating it until the salt melts, at which point you'll know it's reached the right temperature. Leave it there for a bit and then take it out and quench it. My dad worked in a ball bearing factory as a teen and they tempered some parts with that method.
I'm not an expert, but if you try any of this I hope it works for you!

I said above" I don't know"
but here is a thought.
Spring metal has a memory. It wants to go back to its "original" form. When you coil a newly wrapped spring and then put pressure on it ,s in a trap as it tries to return to straight it cannot but its trying so it becomes stronger.
When you heat it you rearrange the moiecules and retrain it to another "original " position.
Make sense?
just

Making coil springs is pretty simple , no heat involved , Check out Mr Pete 222 on youtube, he has a good video on it. Wolfdog ,if you PM me your address i will send you some 1/8 and 1/16 music wire to experiment with its pretty easy with a home made mandrel and a vise. it might be a few days before i see a pm but let me know. Free of charge.

Music wire is just one alloy of spring steel.

Ok so cold forming is good gotcha