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What Effect Does Impedance Have On Tone?

There may be a difference in tone because no two speaker designs will ever be identical. Many parameters will be different based on how the variables interact.

Does it matter? Maybe.
Maybe? Not so sure about that. Arriving at one’s preferred tone is a major consideration for most, but probably the most overlooked consideration is how a cab is going to function in the environment it’s being used.
 
So, does a tube power amp stage (with separate 4/8/16Ohm outputs) provide roughly half the dampening capability when correctly connected to an 8Ohm cab compared to being correctly connected to a 16Ohm cab? (i.e. would it be in that regard similar to linear/everyday SS power amp?)
With tube amps, the damping factor stays relatively constant across the different loads due to the output transformer.

I once was the proud owner of a Glockenklang Heart-Core mk2 and the power amp section of these beasts is 2.7-Ohm-cabable. So when recording we hooked up three differen cabs just to find out which one we like best. And then we hooked up all three of them at once and miked them all (each mic'ed up alone in the isolation booth). After a lot of reamping and repositioning etc., everybody could hear, that any bass cab (single 12s and single 15s) run at by the power amp stage toiling at a 2.7Ohm load would sound different from the identical cabs when connected alone thus having the amp only coping with one load and at 8Ohm.
Maybe so.
 
Some solid state amps do have an impedance switch, so we have to be careful about generalizations.
I couple of years ago I worked on an old Peavey solid state amp that had a transformer-coupled (well, autotransformer) output, with separate outputs for various loads. (The output transistors were toast, which leads me to suspect that someone might have ignored the speaker cab impedance labels on the rear panel.)
 
I couple of years ago I worked on an old Peavey solid state amp that had a transformer-coupled (well, autotransformer) output, with separate outputs for various loads. (The output transistors were toast, which leads me to suspect that someone might have ignored the speaker cab impedance labels on the rear panel.)
There were some oddball designs of course, this wasn’t all that uncommon during the early days of solid state amps (especially in the germanium era, where voltages were very limited)
 
There were some oddball designs of course, this wasn’t all that uncommon during the early days of solid state amps (especially in the germanium era, where voltages were very limited)
The Peavey I worked on was of 1980's vintage, approximately. All silicon transistor. I have seen some older odd designs, like the late 60's Quad 303 power amplifier (45 watts-per-channel stereo) with a single-ended power supply and capacitor-coupled output. It was one of the first all-silicon amps. (The name "Quad", by the way, doesn't come from "quadrophonic" but rather "Quality Unit Amplifier, Domestic". The company is best known for their electrostatic loudspeakers.)
I recently worked on a guitar amplifier (I think) that had a single-ended power supply and capacitor-coupled output, but I forget what it was.
 
Then there's the actual impedance curve of a speaker. We call a cab with a single driver in it "8ohm", buts it's hardly that. That's determined over an average of it's total response.
IMG_2167.jpeg
That rising impedance as frequency goes up details why drivers lose top end response. The higher the impedance, the less power provided by the amplifier.

Then you add a port and crossover and things get even more complicated. As others have mentioned, this is why a reactive load is totally different than a resistive load.
IMG_2168.jpeg
Pretty neat stuff.