JohnR covered most everything.
I would add, from my experience, the advantages of a first-order crossover are in clarity, dynamics, timbre/texture, and imaging- the whole gamut. Along with less amplifier interaction.
There are 1st-order filters out there which are very complicated, from the designer's choice of drivers and cabinet design. Anytime one adds more parts, low-level signals are lost, nuances of pitch and timbre are obscured, and dynamic contrasts are reduced. It sounds laid back and hazy.
A simple 1st-order circuit is very difficult to design, because the drivers cannot require any "correction", nor can the cabinet be a factor in the resulting tone balance.
If you are ever going to judge a speaker quickly, put on an artist whose notes span a given crossover point. You'll hear everything change/collapse/pinch/bite/become grainy/sharp in that tone range. Don't let anyone tell you "it's the recording". To know that for yourself, listen to it on a decent pair of headphones- which have very little phase shift as a rule. You'll hear nothing wrong with "that" recording.
I know of no crossover circuit, except the first-order ones, that is time-coherent. Meaning that the woofer and tweeter are not only "in-phase" on every tone shared, but also that those two drivers start and stop together.
Higher-order filters (2nd and up) inject a time delay between the drivers, a delay that is different at every frequency. It is for this reason alone that people's opinions of those speakers change depending on the music and who is doing the listening. The time delay means that depth of image is foreshortened in that tone range, that a tweeter's image is "separate" from the woofer, that some notes are piercing, others are smeared, and that the higher-range instruments do not occupy the same acoustic space as what is heard from the woofer/mid.
Because the different orders have different time delays, the audible effects are different. If the crossover point to the tweeter is 2-4kHz, you hear a 2nd-order adds a graininess, like the older AR speakers from the late 70's/early 80's. A third-order circuit is very soft in its imaging in that tone range, and dynamically not quite right. A 4th-order crossover, implemented "by the book", will sound pinched in that tone range, and quite forward. To reduce those side-effects, many 4th-order speakers have their crossover points between woofer and tweeter separated- the tweeter crossing over a little higher than "textbook" and the mid a little lower. A dip in the frequency response is thus induced at that frequency, when measured with test tones or pink noise. Most times, this technique also "tones down" the tweeter's LF-reflection off the surrounding cabinet. That is, if you think that a sonic splash off the front is ok for the music.
To do its timing thing correctly, a first-order crossover circuit requires that the woofer, mid, tweeter lie at the same distance (acoustically) from your ear. If they do not, then that crossover, with those drivers, in that cabinet, is still not a minimum phase shift design- sort of close, but no cigar.
I know of no co-axial speaker that is time coherent, nor of any single-driver cone speaker.
Full-range single panel speakers are/can be minimum phase- and IMO why they are enjoyable, even at the expense of sitting dead on center because of that large panel vs. the short treble wavelengths.
Hope this helps!
Best,
Roy Johnson
Green Mountain Audio
