1st,2nd,3rd order crossovers,etc

Lots of info already here on that. SEARCH, my friend.
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1st order = 6 db per octive electronic slope
2nd order = 12 db per octive electronic slope
3rd oder = 18 db per octive electonic slope
and so on. The steeper the slope the less bleed through you get from the frequencies you want to filter out. As for pros and cons....I will leave that to someone more knowledgable.

Cheers,

Rich

I'll bet many here are in the dark about such matters. I ain't got the brain to understand; so in my case the prof is in the pudding--listening.---YET, not knowing ahead of time how they work;what to look for and such.My only criteria is the pudding.

If you're asking out of a need for education, then this is not the forum - your question is way too encompassing and requires a book for an answer as Sean states above.

However, if you're asking with an eye towards purchasing speakers and you're worried their crossover design might not be "the best", then consider that there is no crossover design that can be called across-the-board superior. The selection of crossover by a speaker designer depends on a lot of factors. Some designers, such as Thiel, feel that a first order is superior for the sound they're looking towards and will build around that. Others will find a 4th order Linkwitz is the way to go for a particular driver/enclosure combo. Whatever they decide will be based on what they are looking to accomplish withing a certain price point. Suffice to say that whatever speaker you are looking at has been designed to the best of the mfr's ability to meet the goal of the speaker. Whether they've succeded is a different story. Listen critically, and don't worry about what makes it spin.

Please do a search on phase and time coherence in loudspeakers on this forum. Many threads on it which address just such questions.