Can a speaker use a first-order passive crossover if high power is not used? Are higher order XOs used for something else besides keeping undesired high powered signals out of the tweeters?
High order is needed when drivers are used to the extremes of their operating range. Often high order notch filters are used with modern light weight metal or ceramic drivers - this is an attempt to reduce the out of band "ringing" - it often works although at higher output levels ringing can still be audible issue.

High order filters and notch filters may introduce phase shifts that can be detrimental, however, they seem to be common these days. In the past designers would stick to first order filters.
Most Coincident speakers use 1st order and are happy with 3-10 watts due to 94-102db efficiency.
The answer is most likely yes. A first order XO on a tweeter limits the low frequency signal sent to the tweeter, below the crossover point, at the rate of 6db per octave. Second order is 12 db rate, third is 18db rate, and forth is 24db rate of cutoff. If your crossover point is well within the operating range of your tweeter power is not a problem. So if you need to use the full operating range of your tweeter, as in a two-way speaker, a higher order might be better. From all I've read there are pros and cons to every type and order of crossover. I'm partial to third order, but don't ask me to explain why.
Think of crossover parts like speed bumps, the less you have in the speaker's crossover, the easier it is for the partnering amplifier to drive the loudspeaker as there is less for the amp to push through.

Therefore, next to speakers which use no crossover (Fostex, Lowthers, AER, etc.) a first order crossover is the most low power amplifier friendly.
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Bob, based upon my own work in designing crossovers, I strongly disagree with your assertion.

In fact, your own links do not well support your position. By that, I mean no offense, so let's talk things through...

First, for the obvious reasons, throw out wikipedia altogether.

Onto a more cogent example, in the first link, the Zobel network is discussed. I realize my opinion is the minority one, but I am definitely not a fan of the Zobel whatsoever. Again, I return to my statement that crossover parts are like speed bumps. Zobels clearly rob the music of its life (though some systems actually need that, and is why the Merlins use them and the Walker HDLs are as popular as they are - otherwise things would come across as too lively); this is the truest measure of the validity (or, lack thereof) of my statement about crossover parts soaking up the power an amplifier is delivering to a loudspeaker.

If the argument is that the Zobel is less damaging to "the numbers" than the other avenues of dealing with the impedance rise of the woofer/midrange's voice coil, then I guess I agree. Though, that is no more an accomplishment than being the tallest midget in the circus. The supposition that this impedance rise needs to be "corrected" is a fatally flawed idea in the first place. Despite the conventional wisdom (funny how that was all developed with "perfect" solid state amplification), impedance peaks on a loudspeaker often should be celebrated, as that is exactly what a tube amplifier wants to see. That is where tube amplifiers work best.

In fact, this is the very reason that you see me here on Audiogon presenting the contrarian idea that tube amplifiers are normally more bright than solid state amplifiers - they're so happy with the higher impedance presented to them by the upper regions of a woofer/midrange driver that they're putting TOO much power out in that area, making the overall sonic presentation come across as overly forward. No amplifier is more prone to this than the OTL tube amplifier, which put out twice the power when impedance doubles, and why many feel they are overly forward or bright - that is, when they are used with the wrong (most) loudspeakers.

I don't want to bore everyone with a similar analogy of the notch filter beyond saying the correct way to design a crossover is one that is far enough above the tweeter's resonant frequency as to render even that impedance spike irrelevant. I say this from the standpoint of wanting to protect the tweeter, but even the objectivists will applaud as we will have conquered the numbers as well in this case. But, again, the main point I want to stress is that more parts in the crossover equates to more required power. A speaker with a notch filter will not be as efficient as one without.

And, now, I will backtrack, and throw a caveat into what I have just been espousing. I haven't spent enough time on the matter to really have a total handle on the "why", but I definitely find not including an inductor, which has become fashionable on various loudspeakers of late, on the woofer/midrange definitely zaps a lot of the life out of the music as well. It may be that in driving the woofer/midrange flat out, one is forcing it to operation outside of its optimal range, and the negative impact is far more than some have realized. But, perhaps most surprising drawback is the weak bass output of this configuation, which, as I have said, I have not yet really put in the requisite time to figure out.

I also feel the need to go back and clarify my position on first order crossovers in general, as it is sweeping to the point of being incorrect. SIMPLE first order crossovers (apart from no crossover at all) are the easiest to drive. Some designers, such as Dunlavy and Thiel, believe in first order crossovers with the components you speak of, and wind up with some of the more difficult to drive loudspeakers out there.

I will say that an active crossover can be far more complex, and have no deleterious effects in terms of power loss.

Finally, I want to praise the wisdom of the folks who have come before us and laid the foundation for what exists today. The knowledge of such people is far more than we give them credit for, and impressive viewed in any light. They had all of this crossover topology in front of them, and fact is that they overwhemingly employed simple first order crossovers in the loudspeakers of their day. Perhaps, we need to come to grips with the fact that it was not for the reason that they didn't know any better, but because they did...
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