That is definitely a good question. I don't know the answer, but I will say that I've seen a few speakers with a nominal impedance of 6 ohms (like mine).

Great question, especially since the "binary" choices were created way before digital audio.
A speakers impeadence varies greatly over its frequency range. A statistical nightmare. Huge maximum/minimum deviations. Very narrow max peaks. How do you chose its impeadence rating? Its minimum? Its average? Its median?
And we haven't even mentioned its phase vs impeadence characteristics. Also can be very important. Or not.

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I'm really just trying to understand why the resistance of speakers matters ...

The R of a speaker doesn't matter other than as a small component of the speaker's overall impedance. Why impedance matters is because an amplifier cannot deliver its wattage into nothing. The amplifier requires a load in order to do its work. Since voltage in an amplifier is constant the amount of current required to maintain that voltage is dependent on the impedance load the amplifier sees (works into). Lower impedance will require higher current to maintain constant voltage. Higher impedance will require less current. High current requires large power supplies. Large power supplies are expensive to build and expensive to ship.

"Impedance, for the purposes of this discussion, may be thought of as the AC equivalent of DC resistance. As a result, Ohm's law is applicable: in a circuit the product of impedance and current is voltage (conventionally represented as V = I x R) which in the case of a loudspeaker-amplifier circuit is constant. In other words, as impedance rises, the current required from the amplifier to maintain the circuit's voltage decreases. More importantly, as impedance lowers, the current required increases. This concept is important for both tube and solid state amplifiers, as will be explained in greater depth later."

How the amplifier is coupled to the load makes a large difference in how it does its work and how impedance matters to the amplifier. In a transformer coupled amplifier such as the McIntosh tube unit, the tubes are doing the work but, due to their inherently high output impedance, transformers are required to step the impedance down to a workable match (approximately 1 Ohm nominal) with a loudspeaker. If the load is approximately matched to the output tap, unlike the typical direct coupled solid state amplifier, a tube amp will produce the same approximate wattage from each tap. When the load is nominally four Ohms connecting to the four Ohm tap will produce "X" watts. When the load is nomally eight Ohms connecting to the eight Ohm tap will produce approximately the same "X" watts. Another alternative would be to parallel multiple output tubes to lower the impedance but that makes for a fairly complicated amplifier and a far more expensive design. (In a direct coupled amplifier as you find in most solid state designs, the output devices have a fairly low output impedance on the order of 0.01-0.1 Ohm average.)

In a way, a high speaker load provides the impedance to keep the amp from falling flat on it face when it tries to do its work. The impedance load maintains a stable condition (voltage) as if two people were pushing against one another. The amplifier is pushing (supplying current) to get work done while the speaker is providing the impedance to resist the work and the result is the amplifier easily maintains a stable condition (a constant voltage). As long as the impedance is high enough both parties will do fine with minimal effort. If the impedance load is suddenly lowered, as if one person had relaxed their effort, the person (amplifier) pushing the hardest (suddenly supplying large amounts of current) will fall flat.

That's a very simple idea that doesn't involve anything more than the concept of a load on the amplifier. But there is obviously more to the load the amplifier works into than just impedance to work. Most real world loudspeakers are a reactive load combining dynamic impedance, capcitance and inductance characteristics which make more complex demands on an amplifier. (Include a negative feedback circuit and the whole affair gets incredibly complicated.) If I haven't already directed you to the Symphony Sound article on tube friendly speakers, here it is; http://www.symphonysound.com/articles/tubefriendly.html

It should answer a reasonable amount of your questions about load impedance.

As to the higher impedance taps on some tube amplifiers, you'll find this arrangement mostly on older designs. It really wasn't that uncommon to find a 16 Ohm driver up until the 1960-70's, or when speakers were typically driven by high voltage/low current tube amplifiers. If you were looking to drive multiple speakers from one amplifier without using a 25 or 70 Volt distribution transformer, a series connection between speaker systems could give you the proper configuration with almost any combination of drivers. In recent years many tube amplifiers have gone to dual or even single output taps which are meant for a now more common four or eight Ohm nominal impedance load.

Making the speaker connection at the various taps on the Mac amplifier will change the output impedance of the amplifier somewhat and result in slightly lower power output with increasing distortion product as the connection moves further away from where the NFB circuit is located. Since most speakers loads are fairly dynamic and can fluctuate over a reasonably to substantially wide range trying the various taps on an amplifier can result in a slight alteration in sound quality. Running an eight ohm nominal speaker load off the four Ohm tap might sound a bit tighter to you depending on the amplifier and speaker while the sixteen Ohm tap might sound a bit sweeter.

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"Impedance, for the purposes of this discussion, may be thought of as the AC equivalent of DC resistance. As a result, Ohm's law is applicable: in a circuit the product of impedance and current is voltage (conventionally represented as V = I x R) which in the case of a loudspeaker-amplifier circuit is constant. In other words, as impedance rises, the current required from the amplifier to maintain the circuit's voltage decreases. More importantly, as impedance lowers, the current required increases. This concept is important for both tube and solid state amplifiers, as will be explained in greater depth later."

If I may

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The home for this is called the

Found it!

Crap! It needs new batteries.

First off, there is no rational reason for this "binary progression" as you call it. It is purely a matter of convention and tradition. Just like we count in multiples of ten only because we happen to have ten fingers. Or there are seven days in a week because this corresponds to the four phases of the moon, something totally insignificant in our day and age.

The taps in tube amps exist because:

1. Amplifier output impedance is not standardized
2. Speakers are designed on the assumption that amp output impedance is close to zero.

Tube amps cannot have such a low output impedance, therefore compromises have to be made. A lower-rated tap has a lower output impedance, therefore it approaches the "zero impedance" assumption better. Except that this way the amp's power output falls and the speaker cannot play as loudly.

A higher-rated tap does output more power, but does so more in frequencies where the speaker's impedance is higher and less so where it's lower. This changes the loudspeaker's frequency response, sometimes for the better, often for the worse.

So, you ask, if we choose a speaker whose response improves - or stays the same - with a higher amp output impedance, are we home free? Well, not quite! See point 2 above. Regardless of what happens to the overall frequency response, the bass alignment of the speaker will always suffer because all speakers are designed on the assumption that they will see a zero amp output impedance. If they don't, the bottom two or three octaves they reproduce will become exaggerated, lumpy and time-smeared, and there is just no way around this.

One could design a speaker in such a way that its bass works well with a non-zero amp output impedance, but this would work only for one such value and not the others. And since the vast majority of (transistor) power amps today have near-zero output impedances, it makes no sense for speaker manufacturers to cater for anything else. And even if they did, it would only work for a fraction of those few outliers, not for all of them. Hence it's not done.

Therefore, if one chooses to have a tube amp, one has to seek the tap that offers the best compromise between output capability, response uniformity and bass quality. Generally speaking, the higher the minimum speaker impedance and its uniformity across the frequency spectrum, the easier this task will be. Several decades ago, when tube amps were the norm, speaker impedances tended to be higher for the reasons outlined above, so everything was easier. Nowadays this is not so.

Nice job. I think that's what the original poster was wanting to read.