What can I do about ignorant neighbors?

I don’t think that ignorance is your problem. Their problem is autocentricity on their part. You might try talking to them, coming to an agreement about times, volumes, etc.
You might take it up with the landlord.
You might want to conduct your negotiations with the landlord present, too.

Ask the landlord to tell them when they have a problem, to talk to you about it rather than call the police. File a harassment complaint against them in the police department.

Keep a careful record of the times they have called the police on you, with the reasons and such.

How about some sort of elaborate scheme involving huge amounts of gasoline, explosives, & holding a city hostage?

There may be several things going on.
1. Are you playing your music after 10 PM and do you have your bass set on high. Sometimes if the bass is too high the vibration will give a thumping sound to your neighbors which can be quite annoying. Even if the police come over they can’t say anything if they find your music is within reasonable noise limits as set by the apartments or neighborhood. Well at least in my state.
I use to rent apartments and the cops would tell us of a noise complaint but many times by the time they got to the door the music would be off and they couldn’t just take one neighbors word against another. They actually had to be caught in the act for fine to be issued.

2. It also could be you have the wrong neighbor pegged for the complaint. And maybe they think it is you when it is your neighbor. And now your neighbor thinks it is you. I use to get people in apartments who use to think the noise was coming from one neighbor and it was actually another.

3. It also can be racially or personally motivated in which case talking to them won’t make them see past their predjudice.

In case it may be an error. One day play your music at the volume you normally do and go see your neighbor. Ask them if they would please come over to see if the music is too loud. It may be they think it is you when it is not. If it is you than lower the bass and ask them to go home and please let you know if that is more acceptable. Whatever you do, most states have a 10 Pm noise ordinance. So you may want to buy a wireless head set for times you want to play after 10.
Hopefully they will see that you are trying to resolve this issue with them and everyone can go home happy.
One more thing. If they have been as noisy as you’ve said it is possible the cops have been called on them by other neighbors and they think it is you. So now they are trying to retaliate. I use to have a neighbor who’s kid always had the cops called on him for speeding. I would get nasty glares from them but it wasn’t me. It was the neighbor next to me. I think one day they did find out it wasn’t us somehow, because they suddenly changed their attitude towards us.

You might also consider using headphones when listening to music at night.

You need visible and audio proof of their violations. best bet is to buy or rent or borrow a video camera and tape them in their moments of glory. next time there is a complaint against you, show the video to the officer and your neighbors. the video tells it like it is.

The next time they have a party, make sure you have filmed all the perpetrators have done. Then, call the neighbors over the next time you have the cops knocking on your door and show the cops your neighbors having a pissing contest on your garage door.

You said: “avoid getting another fine”. You got a fine from the officer who responded to the noise complaint? If so then you probably were being too loud. I have had police respond to neighbor’s noise complaints before and told us that we didn’t seem too loud to him, but could we try to keep it down, and then left. I can’t see a cop issuing some kind of fine for excessive noise based only on your neighbor’s statement that you were too loud.

Maybe you could try being a little more courteous, I know I have suffered from lack of sleep due to noisy neighbors in apartments many times. I am not one to call the police for such a thing, but it certainly can be very upsetting. Maybe you are entirely in the right, but try to see it from their perspective. In this case I would even recommend talking to them, seeing if you can iron out your differences without involving the police.

If they are having louder, rowdier parties than you are though, then you may as well call the police on them since they’ve started that approach. Turn about is fair play.

If you think education is expensive…. try ignorance.