Subs & Group Delay

I'd go group delay first as it is no use having low distortion if there is a whole load of resonance or ringing from your sub - remember your room is going to resonate too! So less resonance or the sooner the sub stops ringing (overdamped) the better IMHO. Ideally you want both low group delay AND low distortion AND high SPL output but it is expensive to get all three...

a good rule of thumb is less than 25 ms group delay at 20 Hz This corresponds to 20 feet added distance between something you hear at 20 Hz and its upper harmonics at 80 Hz (which may only have a 2 msec delay). It is probably not audible at this level so bass will be nice and tight and sound integrated with the rest of the music (kick drum is real tangible and there). Delays of 50 msec are probably becoming audible (nearly a full cycle) but this is all hotly debated. The problem is that the tweeter sound of the kick drum beater hitting the skin may last a mere 2 msecs - so if the deepest sounds arrive delayed by 50 msec then it is going to sound like delayed room or concert hall reverberation rather than the instrument itself - so while this may be an added and pleasing effect on dinosaur footsteps it may cause the kick drum to sound distant and less "integrated"or lacking "punch". (You can look up Hass Effect to understand how our brain separate sounds that arrive much later - we simply interpret these late arrivals as being reverberant reflected sounds and NOT the primary sound.)

As a rule of thumb, ported subwoofers generally have twice the phase shift and twice the group delay of sealed subwoofers and oscillate twice as long after the signal stops. That is why I claim that "as a rule of thumb" sealed subs will be tighter and better sounding even if they have poor SPL output. (Of course a ported sub that is tuned way way low at 10 or even 5 HZ is likely to sound pretty damn good and close to sealed sub - so it is only a "rule of thumb")

I'm aware of a controlled listening test in which 30% second harmonic distortion was statistically undetectable

Second harmonic is fairly benign but 3rd harmonic is bad - so it depends on the type of distortion. The other issue is that 1% distortion at the 3rd harmonic of a 20 Hz fundamental will sound nearly equally as loud as the fundamental note itself due to your increased hearing sensitivity at 60 Hz compared to 20Hz (see Equal loudness contours). So in the end, a distorting subwoofer simply populates the bandwidth where your hearing is much more highly sensitive from 60 to 180 Hz with audible response. This is may be quite the opposite of your original intention when buying the sub - which was to augment the bottom octave rather than overwhelm the 60 to 180 Hz range.

1)An SVS 13" Ultra with its ports sealed shows app 5% THD at 20hz at 100db anechoic. In a real world listening room, that's presumably a crushing spl, so could one safely assume distortion is no issue?

It is pretty impressive yes but a far cry from a crushing SPL due to the fact we can barely hear 20 Hz. For comparison it would sound as loud as a 40 db SPL note at 1 Khz...a little above the rustling of leaves. The good news is that if your sub can achieve 115 db SPL then it suddenly becomes much more audible - as 120 db SPL will sound like 80 db SPL at 1 Khz - so clearly audible even above a conversation. Also there is not much at 20 Hz anyway so the biggest benefit is the lack of distortion at higher harmonics rather than any bone crushing sound.

The reality is that a great sub will NOT sound devastating on a 20 Hz pure tone at 100db SPL - is will be mostly inaudible if it is a good high quality sub!!!

Is there any benefit to looking for a lower # here?

probably not but lower is always better - it just may not be audibly perceptible.

I assume that the sub generates this distorted output as a by-product of its response to a 20hz input signal.

Exactly - in most subs playing a 20 Hz tone most of the sound you are actually hearing is distortion.

I'd also note that A0 on a piano (typically 27.5hz)
registers as LOUD!

You will hear the 2nd and higher harmonics of the piano loudly.

Now for the weird part.... you can completely remove the fundamental and play
just the harmonics and your ears will hear the correct note (but timbre will be
off)....hearing is a weird science!!!!

If you notice above I mentioned a pure 20 Hz tone or test tone (not a note from
an instrument as any instrument will have higher harmonics too)

but my assumption is that the same signal in a 15' x
20' room should be perceived as much louder than in the (quasi) anechoic
measurement environment used by HTshack. Do you believe that this is a
good assumption (at least below the primary cancellation frequency)?

Absolutely - you will get at least a 3 to 6 db boost and more or less at room
mode peaks/troughs.

Finally, do you have a sense as to which test sweep
(they go in 5db increments from 90db to 115db) in those subwoofer tests
might be the best proxy for LOUD listening - (music) in a real room.

I can't say but I think you need roughly 110 db SPL in the parking lot at 30 Hz
to convincingly represent peaks from real live instruments (like a drum set) in
a large sized room. This will have you covered for rock and dance club tracks
but classical can be handled with 10 to 15 db less. You can probably get away
with 100 db SPL or less at 20 Hz - as I think I mentioned I'd rather not have
20Hz at all if it was at all distorted.

One or two JL F113's would have it covered for most people in a large room
(if you ask around I think you will find several people own two of
these monsters and are very happy)

I am actually having a hard time digesting this. If you
would like to try re-explaining this to me I would be obliged. I am not sure
how I want to counter to those statements.

My comments are entirely based on equal loudness contours. A deep bass
pure tone at 20 Hz (with no harmonics) is quite soft sounding at 100 db SPL
(in fact you may hear rattling of windows etc. louder than the pure tone - i.e.
you are more sensitive to harmonics or vibration induced by the 20 Hz tone).
Basically we hear 1 KHz tones with some 60 db better hearing sensitivity. As
you get louder and close to 130 db SPL at 20 Hz then the equal loudness
curves flatten significantly and that pure tone would sound as loud as 100 db
spl at 1Khz ( a mere 30 db drop in sensitivity as our hearing response
"flattens" as you go louder) As you can see from those curves - even a
tiny bit of harmonic distortion (say 5%) from your sub on a 20 Hz pure tone is likely to be more audible than the test tone itself - just because of our poor hearing sensitivity at 20Hz.

Bob,

The general idea is that low frequencies are much less audible compared to midrange frequencies...with the biggest loss as you go down from 60 to 20 Hz. I may have exaggerated it some but it certainly "significant" whichever way you look at it.

The implication is that harmonic distortion in the 3rd and higher harmonic of a 20 Hz note may be detrimental - even in tiny amounts - making the sound louder than intended on the recording. See what Seigfried Linkwitz has to say - scroll to the bottom of the page.

This is a review of an important AES paper on subwoofers that appeared in Stereophile - The Science of Subwoofing.

Subwoofer

distortion guidelines are offered: Harmonic distortion will not be audible if the second harmonic is below 3%, the third around 1%, and higher harmonics no greater than 0.1–0.3%.

and

The authors sadly report that it is evidently believed in the loudspeaker industry that woofer distortion is not particularly important; as a result, there are very few drivers built that possess adequate linearity. All of this bodes ill for our chances of finding the ideal subwoofer from a commercial vendor.

For details see: Louis D. Fielder & Eric M. Benjamin, "Subwoofer performance for accurate reproduction of music", JAES, Vol. 36, Number 6, pp. 443 (1988).

Most of us are listening to distortion on our subwoofers...