Crossovers are filters made from capacitors and inductors, The drivers are connected by network of different capacitors and/or inductors. These introduce different frequency dependent phase shifts for each driver connected. Is there a way to counter that or is this not leading to an audible effect?
Speaker designers try to take those phase shifts into account – you don’t design a crossover in isolation, it’s part of the whole integrated loudspeaker system. At least, that’s how a good speaker is designed.
The crossover circuit design, choice of drivers, driver alignment, interactions with baffles and with the internal volume of the cabinet are all factors in optimising the frequency response and phase alignment according to the craftsman who designed and built the speakers I use.
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Well, even if making a phase cohereant speaker was important (it isn’t for sound quality), that phase alignment only exists for one point or a single plane in space for non-coincident sources (multi-driver speakers) and all of the early reflections and room sound is not.
Passively, you can do first order crossovers or very specific transient perfect filler driver. Speakers like Dunlavy, Thiel, Vandersteen and a few others used first order (acoustic) filters. There are big concessions in distortion, dynamic range and other factors versus using a steeper filter.
However, using DSP or a computer (and some basic measurement equipment) you can correct the phase response of an existing speaker using free tools like rePhase, or paid ones like Acourate or AudioLense. You could even use these to A/B with and without the phase correction stuff in some cases., if you want to understand the sound quality impact.
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Thanks for your reaction and clarification Chris! Step by step I’m trying to understand how the conversion of electric signals to audible sound waves works. From what you said in your reaction, I realise that the complexity of designing speakers is even a lot more complex than I imagined! Especially when, next to the position of the drivers in the speaker enclosure, you also have to take into account the position of these drivers to the floor and, if they are close to the speaker, the walls… I wonder, if it is possible to design speakers in a completely scientific / mathematical way? I guess that a lot of experience and “feel” also is necessary. (I’m not sure if I use the word “feel” correctly, but hope that you guys will understand what I mean 
)
at the current state of the available technology, there is a certain amount of art as well as science involved - which just translates to experienced designers who have tried many different ways of mounting and driving speakers knowing what is likely to work and what isn’t.
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At this point I’ll jump in to mention a product that in a single box does speaker correction digitally, namely DEQX. DEQX stands for Digital EQ and Crossover. I’m a beta tester for their so-called 4th gen product line. I’ve been using their 3rd gen for about ten years (their HDP5) to actively correct and crossover my 3-way speakers (i.e. a digital input such as 44.1/16 (or some variant of 44.1 or 96 as input converts to three coax digital outs for bass, mid and treble, hence I require three separate dacs and 6 channels of amplification). You can take a look at some technical information at Technical - DEQX. The 4th gen line will handle 4-way active systems via the one Pre-8 box. Dan.
Update: There’s a dirt cheap way of playing with active speakers using a Behringer DCX2496. This provides six dacs each of which can provide crossover and EQ (there’s no attempt to provide speaker correction, e.g. impluse response correction). That’s what I used for a few years before the DEQX unit (BTW, the 3rd gen DEQX unit had three built in BurrBrown dacs in it and the Pre8 has the equivalent of 8 ESS channels - I decided that I’d use my own dacs (all Directstreams. I’ve discovered that setting up the parametric EQ is challenging, to say the least!))
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Plus one for the DCX2496, at least for experimentation, but to add, it does have a microphone input that can be used to time align the different drivers. It works, though is worth repeating a few times and taking averages of the resultant driver delays / phase flips.
It’s a “budget” device, and tremendous fun to play with, as are Behringers’ other “2496” offerings. They’ve been available for 20 odd years now and still keep selling
I’d add that the dBx Venu360 LMS (Loudspeaker Management System) is an excellent choice as well. It provides EQ, crossover, driver time alignment and room correction. Using this to bi-amp my Sanders Sound Systems Model 10e hybrid 'stats has been revelatory. I also use it to feed my subwoofer which has its own parametric EQ.
Without these adjustments and flexibility I’d never have achieved the sound quality of my current system. The other beauty of digital devices is that they don’t drift due to time, temperature, etc. and you can save multiple configurations. The dBx device may be a pro audio product which some sneer at but this thing is essential to my listening pleasure.
I’m sure that’d be a great improvement over the Behringer kit.