I feel a long post coming on, so I apologize in advance to all the subscribers with dial-up connections. I also apologize if it comes across as didactic, but I feel like spending some text on describing the terms "polarity" and "phase" before I even start commenting on the initial question, or the other points that have come up as the thread developed.
Polarity and phase aren't the same thing, and I prefer to be careful how I use the terms. It helps me keep clarity in my thinking, which is a challenge at the best of times. Many people in our field use the terms rather interchangeably. I guess it isn't technically wrong to say that when something's 180 degrees out of phase when it's reversed polarity, but I don't think it helps keep the issues straight if we toss the terms around loosely.
Imagine if you will, a loudspeaker some distance from a listener. (The listener could be a microphone.) The loudspeaker is fed a pulse such that the initial movement of the cone is outward, toward the listener. The first part of the waveform radiated toward the listener will thus be a compression wave, that is an increase of pressure. Of course that's followed by a rarefaction, or decrease of pressure, and the positives and negatives keep alternating until the sound dies away. Now imagine that we turn the loudspeaker around 180 degrees so it faces away from the listener. When we feed it the same pulse, the cone moves away from the listener first, and the first part of the waveform approaching the listener will be a rarefaction, or decrease of pressure. We could accomplish the same thing by reversing the connections to the loudspeaker. That's polarity. We haven't changed anything having to do with time, only direction.
Now imagine two loudspeakers side by side. We feed a low frequency tone to both loudspeakers. They're wired the same, so they are producing positive pressure changes simultaneously, and negatives simultaneously, and the output of the two speakers adds nicely. It's easy to imagine what happens when we reverse polarity of one speaker, whether we do it acoustically (by turning the speaker around) or electrically (by swapping the wires.) But what happens when we move one speaker ten feet farther away? We haven't changed direction; they are still producing positives simultaneously, but the positives aren't reaching our listener simultaneously. The positive from the farther speaker is reaching our listener about 8.8 ms later than the same positive from the near speaker. We have introduced a shift in time, and now we are in the realm of phase.
There's a lot to talk about when we start considering phase relationships (and phase is always a relationship.) But the thread started with polarity, so let's stick with that for a while.
Starting with Steve's question about how to handle a mic that's wired pin 3 hot, when the standard for mics is pin 2 hot: you said, "the sax sounded great, no matter the polarity." In this case, that's your answer. However, I think the owner of the mic would be well advised to rewire it. There will be plenty of times it gets used in conjunction with other mics in the same room at the same time, and it's good engineering practice to start with everything in a standard condition. Klaus has (in other threads) suggested that every studio should have a polarity popper, and use it to get the whole system consistent. I wholeheartedly concur. There are enough things to do on a session without worrying about unintended polarity reversals.
There can be good reasons to reverse the polarity of a mic when you're using it in conjunction with other mics. Just to give one example: I was fortunate enough to learn from Fred Catero, one of the all time greats. He insisted on checking the polarity of each mic on a drum kit by ear, once he had them all reasonably well placed. It's easy to do...just flip the polarity switch on the console, one channel at a time, while the drummer is playing. You'll immediately hear that one way's fuller, the other way's thinner. Generally the fuller way is preferred. As you'd expect, when you've got one mic on top of a snare pointing down and one underneath pointing up, you'll almost always wind up flipping one. But you might be surprised to find out how often the kick mic is out from the rest of the kit.
Now on to the other topic: is correct polarity audible, or what? As David Satz pointed out, the actual waveforms of some musical instruments are asymmetrical, meaning that the positive-going part of the wave is different from the negative-going part. He mentioned the muted trumpet, and indeed that's one of the most exaggerated cases. In the mastering lab I always have the scope on, so I'm spending hours each day looking at waveform displays of music. I have come to the realization that asymmetry is very common. I almost always see it in vocals. Generally it's noticeable in all wind instruments. Occasionally I see something that's an electronic artifact from defective equipment, but mainly I'm talking about properly recorded stuff.
So music can be asymmetrical...can we hear that? Before getting into that, let me say here that I don't understand Bill Yacey's comments: "Phase changes numerous times with distance to the source being varied, so I don't understand what the absolute reference is with only one mic being used...with higher frequencies the phase changes many times over a very small distance between the mic and sound source." My understanding is: if a sound source puts out a pulse such that the first part of the waveform is positive-going, the initial arrival will be positive-going at one inch, at one foot, at ten feet. If I'm missing something, please explain.
Anyway, my experience with listening for polarity is: I hear a difference sometimes, under some circumstances. The more coherent the speakers, the more likely I am to hear it. When I tried on mid-80's JBL four-ways, where the crossovers flipped polarity at every crossover point, well...no. When I listened on Auratones...probably the first phase-coherent speaker that was widely used in studios...yes, I could hear it fairly often.
When I have heard a difference, and when I've had a preference, and when I've been able to determine which way was presenting it in the same polarity as the natural sound, I found that my preference had usually been for the polarity that matched that of the natural sound.
By the way, if you say you don't hear it, I'll be the last guy on earth to argue with you. But if you say I don't hear it, I have to ask, "How do you know?"
I try to keep polarity consistent in my recordings, but it can get complicated. The first time I got really interested in this was when a colleague, Elliot Mazer, decided to check the whole chain in a studio where I was working at the time. This was a big multi-room facility, so it was a big job, but the results were interesting. Generally the polarity remained consistent throughout the system in each room, and from room to room...it was a very professionally built studio, and in those days just about everything was balanced, so we didn't run into problems from a lot of balanced-to-unbalanced connections. The surprising thing happened when we checked the tape machines. They all gave back the same polarity we fed them...either in input or off tape. But when we moved the tape from one machine to another and played it back, it was anybody's guess how it would come back. All 24 tracks would be the same polarity on any one machine, but it was a crapshoot whether it was the same polarity as the machine that had recorded the tape. And these were machines from one manufacturer. They thought it was important to make it consistent on throughput, but they didn't have a standard for what polarity they were actually printing on tape. I realized then that as a project moved from room to room (or if the tape machines got swapped around for any reason) that we'd be putting down overdubs that may or may not have been in the same polarity as the basic tracks. Of course it was already common at that time for a record to be done in a bunch of studios in different cities, so the polarity of all those tracks would be pretty well randomized by the time tracking was done. Play one of those recordings and flip the polarity (of both channels simultaneously of course...we're not talking about left vs. right here) and either way should be about half right.
Well, I've rambled enough for one post. Better save the rest for the box set.