Bi-amping Rules

But gain aside, I still don't quite understand the need for an outboard crossover to control the frequency between the HF and LF sections since both amps are feeding full range and each section of the speaker still operates within its own design frequency range.

kalali,
the advantage of an external x-over lies in the fact that you can tweak the x-over slopes to suit your listening needs &/or your room.
In an in-built (in the speaker cabinet) x-over the designer does this job for you using his design skill & keeping in mind 99% of the population so that his speakers work in a wide range of room sizes. So, it's often the best compromise.
With an external x-over you can change the slope (-6dB or -12dB or -24dB or -48dB & so on) make the speaker tuned to your room.
Plus, some of the factory x-overs are very power hungry. Using an ext x-over you can overcome this shortfall & make it easier for your amp to drive your speakers.
You can also upgrade all the components in your ext x-over & make it as transparent sounding as your wallet will allow.
Plus another major thing these days - some digital x-overs will also allow you to do room correction & bring your speaker closer to being a time-coherent speaker (which is a big deal, i.e. being time coherent). The merits of room & speaker correction towards making the speaker time-coherent are huge. One listen & you will know.
hope this clarifies.....

A couple of comments and clarifications regarding today’s posts in this thread:

In the case of passive biamping, i.e., biamping without an electronic crossover "ahead" of the power amps, it is of course correct that what has to be matched in some manner is the gain of the two amps (gain being the ratio of an amplifier’s output voltage to its input voltage, for a given load impedance), not their maximum power ratings. While amplifier gain is often unspecified, if Stereophile has reviewed the product the measurements section of the review will usually indicate the gain, or it may be determined by contacting the manufacturer, or it can be calculated to a reasonably good approximation from the max power rating and sensitivity specs that are usually provided, per the methodology indicated in the second of my posts dated 3-10-2012 in this thread.

Passive biamping will greatly minimize the amount of low frequency CURRENT and POWER that is supplied by the high frequency amp, and will greatly minimize the amount of high frequency CURRENT and POWER that is supplied by the low frequency amp. However the VOLTAGE that will be supplied by each of the amps will correspond to the full frequency range of the signal. As I indicated in the last paragraph of my post earlier in this thread, a consequence of that (which would be avoided if an electronic crossover is used) is that if there is a **large** disparity in the power capabilities of the two amps, much of the power capability of the higher powered amp will not be able to be utilized without driving the lower powered amp into clipping.

Regards,
-- Al

In the case of passive biamping, i.e., biamping without an electronic crossover "ahead" of the power amps, it is of course correct that what has to be matched in some manner is the gain of the two amps (gain being the ratio of an amplifier’s output voltage to its input voltage, for a given load impedance), not their maximum power ratings.

In the context of a passive crosover, while it seems as a simple thing to just match the gain of the two "different" amplifiers used in biamping, it is much more than that. Two different amplifiers will never have the same timing and phase response in the music reproduction. So in effect you are feeding two differently timed signals to the two drivers (LF and HF) and also both signals vary in their respective phasi-ness. The final output will be a sound where the drivers do not integrate and sing as one. The HF and LF will be all there but they will sound like individuals. All the work done at the crossover to marry the drivers are nearly lost. The same happens when biwiring with different cables for LF and HF.

If you bring in the tonality aspect it gets even more complex. Imagine the lower notes of the piano coming from one amplifier and the higher notes from another amplifier of a different make (which means different voice)! Ultimately it will also mean the fundamental notes are generated by one amplifier and the higher order harmonics by a totally different amplifier. Will the fundamental and its harmonics now sound like it is coming from the same note ? It is ultimately a "cooked" sound to say the least.

Pani, to be sure it's clear, my comment that you quoted was not intended to imply anything inconsistent with your comments.  And personally I am in essential agreement with everything you have said above.  From my post in this thread dated 7-25-2016:

While anecdotal indications are that there are certainly **some** cases where audiophiles have achieved fine results by bi-amping, in both passive and active configurations and by using different as well as identical amplifiers, I agree with the foregoing comments and the article that in the great majority of circumstances the corresponding funds are likely to be applied more constructively in other ways. And presumably and hopefully the designers of most high quality speakers have gone to great pains to try to make them sound as coherent as possible throughout the frequency range. Why risk undoing that by using different sounding amps?

Regards,
-- Al