Adding low bass channel to a passive radiator design

Hi all!

I'm new to loudspeaker design. Here is the problem I'm trying to resolve and would love to hear some expert opinions:

I want to add a low bass (< 60Hz) channel to a speaker with a passive radiator (PR) and don't know how to do it to integrate it smoothly.

Some input data:
The PR loudspeaker is 10" Tannoy Red + 10" Scan Speak 25W/8565-00 as PR in 89 liter, damped box. Tannoy uses its native crossover, ScanSpeak completely unconnected.
This is a sample L+R response in my room, amplitude + phase:

Sample Dannoy LF response.jpg


It doesn't look very good but sounds very seducing. I listen near field in a small room, classical repertoire. The bass is really great if not outstanding but a 10" Tannoy cannot obviously produce a lot of it and I like to have more power in the lowest registers and some of that "LF breathing" especially with big orchestral music.

The guy who conceived the speaker tried to add a bass channel to it in a form of a sealed boxed with 15" driver, IIRC low passed 4th order at some 50-60Hz, simultaneously high passing the above design but the point and order is unknown to me. It didn't work, the low bass was "detached" and did not blend in. The suspicion was that the complicated phase behavior of the PR design did not blend in with the sealed box plus the high passing made it even worse.

Does anyone have any experience of designing/adding subwoofers to PR designs? I want to add/reinforce the lowest registers without affecting the great bass I have right now. One idea I have is to add a PR sub with the same ScanSpeak 25W/8565, one active, one as PR, same 89l box. And running the Tannoy PR unfiltered.

Any PR experts/practitioners can comment? Thanks in advance :)
 
I want to add/reinforce the lowest registers without affecting the great bass I have right now. One idea I have is to add a PR sub with the same ScanSpeak 25W/8565, one active, one as PR, same 89l box. And running the Tannoy PR unfiltered.

Any PR experts/practitioners can comment?
A fourth order (24dB/oct) 50-60Hz HP on the PR cabinets effectively "throws away" the PR output.
The phase behavior of the PR design would make little difference if the 15" sub had flat response and also had an acoustic crossover of the same slope.
A sealed boxed 15" driver could not put out as much output as two 12" PR below 50Hz unless it's excursion was much greater, and would probably require 5-20dB of boost to be flat to 20Hz or so.
Integration is as much (or more) room dependent than phase related.

An additional sub using the ScanSpeak 25W/8565, one active, one as PR, in the same 89l box using a LP filter around would require a LP filter, and would only have -6dB output potential as the other two PR.
Depending on room (and listening) position, that may be enough to flatten the response from ~25-40Hz.

For the most part PR phase response is very similar to bass reflex.
Looks like your PR tuning (Fb) may be ~25-28Hz, below that, it's output goes out of phase with the driver, hence the null below.

Verifying the Fb (the driver excursion and impedance will be near minimum, the PR excursion near maximum) and a near field measurement equidistant between the driver and PR would make further suggestions easier.

Room dimensions furnishings, openings and speaker and listening placement also are important considerations-all those can make +/-12dB (or more) differences in response, while ~4-5 dB difference can sound half or twice as loud at 30Hz.

Art
 
Thank you Art for your input! I'll try to clarify a couple of things, but please keep in mind you are talking to a newbie :)
[LONG REPLY ALERT]

A fourth order (24dB/oct) 50-60Hz HP on the PR cabinets effectively "throws away" the PR output.

The idea is not to high pass the existing PR cabinets and leave them as they are not to spoil the unique bass. Only adding external, stand alone, low passed subs.

The phase behavior of the PR design would make little difference if the 15" sub had flat response and also had an acoustic crossover of the same slope.

Interesting point. I was not doing the 15" sub experiment nor have that particular driver, was only told about the (negative) results. The quirky PR phase behavior was just my speculation :)

A sealed boxed 15" driver could not put out as much output as two 12" PR below 50Hz unless it's excursion was much greater, and would probably require 5-20dB of boost to be flat to 20Hz or so.

Here I'm a bit confused. There was no two 12" drivers in the picture. I think it's my mistake. To make it clear, the existing PR loudspeaker is:

10" Tannoy Red + its native crossover as the active part
10" SS 8565 unconnected as PR
both in 89L.

I'm thinking of adding another, stand alone cabinet of the similar size (89L), hosting a dedicated sub.

An additional sub using the ScanSpeak 25W/8565, one active, one as PR, in the same 89l box using a LP filter around would require a LP filter, and would only have -6dB output potential as the other two PR.

Again I think I led to a confusion. There will not be "two PR" in the existing cabinet, it stays like it is (10" ScanSpeak driven by the 10" Tannoy). The lowest bass channel would another cabinet. I did play a bit in WinISD, simulating such a potential external sub candidate: 10" 8565 active + 10" 8565 PR in 89L. Here is a sample result with a rather unconventional 1st order Bessel LP at 40Hz:

Amplitude:

Bass_LowPassed_1order_40Hz_Amplitude.jpg


Phase:

Bass_LowPassed_1order_40Hz_Phase.jpg


I have way too little experience to interpret the simulations in the context of my question.
Verifying the Fb (the driver excursion and impedance will be near minimum, the PR excursion near maximum) and a near field measurement equidistant between the driver and PR would make further suggestions easier.

Unfortunately I cannot find TS parameters of the old 10" Tannoy Red to calculate Fb. But looking at the measured phase response, I'd assume Fb at 28-29Hz, this is where one phase jump happens. The other should be at Fs of the PR which in this case is 20Hz. In reality it is a bit higher as seen from the plot.

Room dimensions furnishings, openings and speaker and listening placement also are important considerations-all those can make +/-12dB (or more) differences in response, while ~4-5 dB difference can sound half or twice as loud at 30Hz.
The room is small, awkward and close to hopeless to be honest. Most probable location of the subs would be hanged from the ceiling (standard 8.5' ceiling) due to the space constraints and to appreciate the boundary boost.
 
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Here I'm a bit confused. There was no two 12" drivers in the picture.
I made a typographical error substituting 12" for 10", and had assumed you had a stereo system, one PR in each, two PR total.
I want to add/reinforce the lowest registers without affecting the great bass I have right now.
The "lowest registers" of your speaker appear to be 25Hz and above, about as low as the low note on a grand piano.
I did play a bit in WinISD, simulating such a potential external sub candidate: 10" 8565 active + 10" 8565 PR in 89L.
That response has a hump at around 45Hz, and rolls off at around 30dB per octave (about 5th order..) below 40Hz.
It won't add low bass (< 60Hz) as you stated you wanted, but would potentially add up to 6dB of upper bass.
 
The "lowest registers" of your speaker appear to be 25Hz and above, about as low as the low note on a grand piano.
Thank you Art for your comments. Yes, the existing PR design goes low but obviously has little energy there (one 10" speaker + 10" PR cannot produce loud bass). I'm looking how to reinforce the lowest registers by adding a sub (perhaps I should have posted in the sub section, my mistake).

That response has a hump at around 45Hz, and rolls off at around 30dB per octave (about 5th order..) below 40Hz.
It won't add low bass (< 60Hz) as you stated you wanted, but would potentially add up to 6dB of upper bass.
Thank you for explaining that. I assume the steep roll off is due to the notch at Fs (20Hz), characteristic of the PR. There is another idea to use a sealed box with two 10" ScanSpeak in parallel instead of 1 active +1 PR. This is how it looks like in WinISD (the blue trace, low passed 1st order at 40Hz):

Sealed box.jpg


The question is if it will blend with the main speaker? Sealed box has a nice smooth phase response, while the main speaker has that quirky phase wrap at Fb (28Hz) and Fs (20+Hz)
 
I assume the steep roll off is due to the notch at Fs (20Hz), characteristic of the PR.
The steep rolloff is typical of PR or bass reflex below Fb (box tuning), which is quite high in your sim.
Little useful output is available below Fb, as the port or Fb phase rapidly goes to 180 out.
The notch is not due to Fs, which increases in a box.

There is another idea to use a sealed box with two 10" ScanSpeak in parallel instead of 1 active +1 PR.
The ScanSpeak 25W/8565 has only 6.5mm Xmax, so output is quite limited compared to drivers designed primarily for low frequency extension.
It's Xmech (the limit of travel due to suspension or voice coil smashing the back plate) of 12mm is less than the Xmax of relatively low grade subwoofers. Passive radiators should be at least double the displacement of the driver.
Doubling excursion (or displacement) allows 6dB more output, which sounds more than twice as loud ~30Hz.

Purpose designed passive radiators with 50mm Xmech/Xmax are available for considerably less than the cost of the ScanSpeak 25W/8565, and can have a tuning (Fb) below 20Hz in a box under half the size you are considering.
Screen Shot 2024-05-28 at 3.18.59 PM.png

The SLAPS peak to peak excursion is 4inch, incorrectly stated as Xmax, which is one way.
Screen Shot 2024-05-28 at 3.22.06 PM.png

Anyway, if you want lots of low output in a small box (or two) displacement is key.
The question is if it will blend with the main speaker? Sealed box has a nice smooth phase response, while the main speaker has that quirky phase wrap at Fb (28Hz) and Fs (20+Hz)

A sealed box equalized to the same response as a ported or PR box will have similar phase response.
That said, the output of a port or PR lags the response of the driver.
Take a look at post #160 to see what goes on:
https://www.diyaudio.com/community/threads/why-are-sealed-box-woofers-out-of-fashion.412687/page-8

and #204
https://www.diyaudio.com/community/threads/why-are-sealed-box-woofers-out-of-fashion.412687/page-11

And STVs #222
https://www.diyaudio.com/community/threads/why-are-sealed-box-woofers-out-of-fashion.412687/page-12

There are some minor differences between ports and PRs response (PRs have advantages in small cabinets) but they both work using a similar Helmholtz resonance model.
You should be able to get an idea of what actually is going on in their phase and transient response from those posts.

Be aware there is a lot of murky water in the rest of that thread ;)

Art
 
Wow, thank you Art! That's something to digest. I try to stick to ScanSpeak bass as it has some particularly nice quality.

A sealed box equalized to the same response as a ported or PR box will have similar phase response.
This is something that I don't understand. I have the main speaker with the phase response like below. It has three phase jumps at LF.

Sample Dannoy LF response.jpg


I try to supplement it with a LF sealed box that has e.g. such a phase response:

Sealed LP 60Hz phase.jpg


How can they be made similar? Would appreciate shedding some light :)
 
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This is something that I don't understand. I have the main speaker with the phase response like below. It has three phase jumps at LF.
The "jumps" are just a result of visualizing ~1080 degrees of phase on the same page.
You are measuring the phase of the speaker in the room, not the speaker.
Both frequency and phase response will be vastly different when you move the measurement mic, but the speaker response has not changed..

We don't yet have a good idea of what you want to match response to, assuming that would even help you achieve what you want.

To repeat post #2:
Verifying the Fb (the driver excursion and impedance will be near minimum, the PR excursion near maximum) and a near field measurement equidistant between the driver and PR would make further suggestions easier.
Using sine wave tones, persistence of vision makes it easy to see (or feel by hand at low volume) the changes in cone movement vs PR movement. The driver excursion will be at maximum ~1/3 octave above Fb, and rapidly go up again just below Fb.

Do three close measurements ~1 inch (25mm) from the center of the driver, center of the PR, and one midway between the two, and you will see three very different frequency and phase responses.
I try to supplement it with a LF sealed box that has e.g. such a phase response:
Note that your simulated phase responses are roughly an inverse of their frequency response.
 
Hi Art,

Here are the near field measurements. All taken at the same level (which sometimes was generating overload alert, some times level too low one, depending on the mic position).

Tannoy Red L:

Mean field L Tannoy.jpg

Middle way L:

Mean field L middle.jpg


Passive radiator L:

Mean field L PR.jpg



Right speaker similar plots:

Mean field R Tannoy.jpg

Mean field R middle.jpg

Mean field R PR.jpg
 
Looks to me that your speakers simply have too much level ~60-100Hz (+8dB) relative to 30Hz and 300Hz.

Screen Shot 2024-05-31 at 1.54.24 PM.png

+10dB in the upper bass range is over twice as loud sounding as it should be to sound like what has been recorded.
Adding subs won't correct that, but a -8dB low "Q" cut filter would, along with a more narrow width (higher Q) +2-3dB boost ~30Hz near Fb to bring low bass up to the midrange level. The filters would look like an inverse of the response that needs to be corrected.
Q.png


Below Fb the response drops due to the PR and driver going 180 degrees out of phase with each other, and can't be corrected with EQ, a HP filter ~25Hz would prevent useless excursion and wasted power.

If the PR peak to peak excursion is greater than ~20mm (3/4") at 30Hz at your desired listening level, then additional subs (as well as the corrective EQ) may be needed.

Four times the displacement (excursion) is required for each halving of frequency to maintain the same level.
If you desire the response to extend flat an octave below your speaker's Fb, using the ScanSpeak 25W/8565 as a PR, the box size would need to be (roughly) doubled, and as many as 8 of those boxes would be required.

Note that one 10" SLAPS PR would be greater displacement than four ScanSpeak 25W/8565 used as a PR, and could be tuned to 15Hz in a box less than half the size of your present cabinets.

Art
 
Hi Art, thanks!
Adding subs won't correct that, but a -8dB low "Q" cut filter would, along with a more narrow width (higher Q) +2-3dB boost ~30Hz near Fb to bring low bass up to the midrange level.
I am not sure I want to correct it Art :) However it measures, it sounds gorgeous in the upper bass. My goal is rather to augment the lowest register rather than to try to make the response ruler flat. I should have probably posted in the subs forum, sorry for that (moderator feel free to move the thread :) )

You convinced me that a sub with a single PR 25W/8665 will not add anything due to the steep roll off and of course 8 big boxes are a utopia. What about a sealed box with two 25W/8565 per channel? Placed close to the ceiling to get a boundary boost. Or it's influence on the lower bass will be also negligible? Again, ruler flat is not my goal...I think ;)
 
However it measures, it sounds gorgeous in the upper bass.What about a sealed box with two 25W/8565 per channel?
A sealed box with two 25W/8565 would have far less output potential at 30Hz than your present set up.
Run your sims at Xmax and Xmech for the PR, you will see how much less.

The response you say sounds gorgeous is similar to a "loudness contour" that should continue for another octave from 60Hz down to 30Hz or whatever you decide you would like to add low bass (< 60Hz).
To do that will require EQ, whether you do it to your existing cabinets or additional sub (s).
 
A sealed box with two 25W/8565 would have far less output potential at 30Hz than your present set up.
Run your sims at Xmax and Xmech for the PR, you will see how much less.
Indeed! I cannot simulate my main speaker lacking TS parameters for the 10" Tannoy Red. But I can simulate two hypothetical versions of the sub:
2x 25W/8565 in PR configuration (red trace) vs sealed (blue trace):

The cone excursion of the active scanspeak @10W

Cone excursion sealed vs PR.jpg


Cone excursion of the passive scanspeak in the PR version:

Cone excursion passive radiator alone.jpg


Now I'm a lost again :) The cone can travel much more at LF in the PR sub but the electromechanical filtering is cutting the output at LF much steeper than in the sealed configuration. Here is the amplitude transfer again of the two versions (sealed sub LP @60Hz, PR sub LP @40hz):

Amplitude sealed 60Hz vs PR 40Hz.jpg


So the PR sub is preferable because of the higher cone excursion despite the steeper roll-off? Or simply none of the two versions of the sub is going to augment the existing main speaker well anyway?

The response you say sounds gorgeous is similar to a "loudness contour" that should continue for another octave from 60Hz down to 30Hz or whatever you decide you would like to add low bass (< 60Hz).
To do that will require EQ, whether you do it to your existing cabinets or additional sub (s).
Very good observation, thanks! Of course EQ'ing the subs is inevitable, I just don't want to touch the existing cabinets.
 
Now I'm a lost again :) The cone can travel much more at LF in the PR sub but the electromechanical filtering is cutting the output at LF much steeper than in the sealed configuration.
What you call "electromechanical filtering cutting the output at LF" is the result of the the PR and driver going 180 degrees out of phase with each other below Fb (the Helmholtz resonance..) and can't be corrected with EQ, since the excursion of both the driver and PR much below Fb cancel each other out.
So the PR sub is preferable because of the higher cone excursion despite the steeper roll-off? Or simply none of the two versions of the sub is going to augment the existing main speaker well anyway?
Higher cone excursion without increased output is not preferable.
At relatively low SPL, either PR, ported or sealed could augment the existing main speaker well if equalized (or designed to augment the low frequency) properly.

What U want&got.png


The output of both sealed drivers are in phase with each other at all low frequencies, EQ can be applied to shape them to fit whatever response you'd prefer, but since four times the displacement (excursion) is required for each halving of frequency to maintain the same level, the sealed speaker (relatively) undistorted output is excursion limited to 6.5mm Xmax, where harmonic distortion will reach ~10%
Used as a PR, when Xmech is reached nasty noises (pup-pup, clack-clack) will occur.

Judging from your near field response result, the existing cabinets are near the same Fb as your sim.

Tannoy & Driver.png

Of course EQ'ing the subs is inevitable, I just don't want to touch the existing cabinets.
If you are already reaching excursion limits of the Tannoys and hearing low bass modulate upper frequencies (AM distortion), I understand.
That said, since you think the Tannoys "sound gorgeous in the upper bass", I'd guess they may have enough headroom left for filling in the low end without requiring additional subs, as long as you don't try to boost below Fb.
If the speakers or amplifier don't have enough headroom, at least you can estimate how much additional LF will be required.

Checking how much corner placement (or any acceptable location) changes relative response at the listening position compared to your Tannoy's position is another test you could do with an existing speaker to help estimate how much additional LF will be required.

Short out the terminals (or leave them connected to an amplifier) of any unused speakers to keep them from affecting response as "drone cones".

Art
 
Art, I owe you a more than a couple of beers :)

At relatively low SPL, either PR, ported or sealed could augment the existing main speaker well if equalized (or designed to augment the low frequency) properly.

What U want&got.png

This explains a lot, thank you! You say I need the "hump" (the SPL rise) to continue down from 60Hz if the sub is to match the gorgeous bass of the main speaker. So it's not about the strange phase response of the main speaker. If this is so looks like neither of the 25W/8565 based designs, be it sealed or PR, in a reasonable box will not do the task. Out of curiosity I simulated SPL, assuming max excursion at 20Hz. The sealed version (2x 25W8565) reached that at 200W power which is the max for two speakers in parallel. The PR variant reached that at 24W. Here are the excursions:

Max excursion sealed vs PR.jpg


and the SPL:

SPL at max excursion sealed vs PR.jpg


The sealed box generates +22dB more at 30Hz than the PR, but the peak is at some 60Hz, while I need it around 30Hz. From what I know the (accidental) creator of the main design tried joining it with a sealed 15" Vitavox AK151 but to no luck (IIRC the feeling was the Vitavox bass was "disconnected" from the main speaker but I think he was additionally high-passing the main design at some 50-60Hz and did not try playing it unfiltered). Maybe good 15" drivers, with interesting bass, soft clipping like the 25W/8565, is the way to go anyway?

If you are already reaching excursion limits of the Tannoys and hearing low bass modulate upper frequencies (AM distortion), I understand.
That said, since you think the Tannoys "sound gorgeous in the upper bass", I'd guess they may have enough headroom left for filling in the low end without requiring additional subs, as long as you don't try to boost below Fb.
If the speakers or amplifier don't have enough headroom, at least you can estimate how much additional LF will be required.
I don't think I'm reaching the limits, especially that my room is small and I listen near field, some 2m away from the Tannoys. I listen quite loud and could play louder but there is simply not enough bass energy, no wonder given a single 10" Tannoy per channel, hence my need to augment it. What I miss is some volume at the low registers, that slam of a big orchestra, some LF "breathing" pressurizing the room, added sense of space and dimensionality.
Checking how much corner placement (or any acceptable location) changes relative response at the listening position compared to your Tannoy's position is another test you could do with an existing speaker to help estimate how much additional LF will be required.
I wish I could do that but there is no other sensible location for the Tannoys than their actual one. I did rise them from the floor some 20+cm some time ago, which had a very slight effect on the bass, removing a little bit of boominess. The perspective subs, on the other hand, can be hung by the ceiling, above the Tannoys, potentially benefitting from the boundary boost.
 
You say I need the "hump" (the SPL rise) to continue down from 60Hz if the sub is to match the gorgeous bass of the main speaker.
Just going from what you say.
Personally I would not go for that much LF gain unless listening at very low SPL, ~60 dBA (80 dBC).
At 30Hz, only takes around +/- 4dB to sound twice as loud, while at 1kHz, +/-10dB is required.
At your listening position, a lift of 3-5dB ~30-50Hz may be enough.
Try it and see, nothing to loose.
So it's not about the strange phase response of the main speaker.
It's not about the room's phase response.
There is nothing "strange" about the phase response of your main Tannoy/PR speaker.
If this is so looks like neither of the 25W/8565 based designs, be it sealed or PR, in a reasonable box will not do the task.
Either could work with EQ, different EQ would be required for either.
Nothing you have provided gives us any idea of how much excursion headroom you have in your system presently, so can't say what will be required for the task.


Out of curiosity I simulated SPL, assuming max excursion at 20Hz. The sealed version (2x 25W8565) reached that at 200W power which is the max for two speakers in parallel. The PR variant reached that at 24W. Here are the excursions:
The 25W/8565 as a sealed driver should not be expected to work past Xmax, as a PR, not past Xmech.
The sealed will reach X max at ~50 watts.
As stated before, a BR (bass reflex) or Fb will provide little useful response below Fb, it should use a Hp filter to avoid excessive excursion in the range below Fb.

The 25W/8565 in an 89L ported cabinet with an Fb~25- 30Hz could put out way more "gorgeous bass" in the 30Hz range than it would sealed. Easy to tune a port, not so easy to add mass to the 25W/8565 to lower it's resonance down to ~10Hz, and it's Xmech is quite low for PR use.
A port has no Xmax/Xmech limits, though will make some noise if the it's velocity is too high.

Start comparing BR and PR response.
 
At 30Hz, only takes around +/- 4dB to sound twice as loud, while at 1kHz, +/-10dB is required.
Art, why so? Fletcher-Munson curves suggest the opposite or I'm wrong? I'm learning here :)

At your listening position, a lift of 3-5dB ~30-50Hz may be enough.
Try it and see, nothing to loose.
I understand a lift compared to what I have now at 30-50Hz. I'd need a subwoofer to do that.

Either could work with EQ, different EQ would be required for either.
Nothing you have provided gives us any idea of how much excursion headroom you have in your system presently, so can't say what will be required for the task.
Art, how do I check that headroom? There are no TS parameters for Tannoy III LZ (I've found only Fs=27Hz). Subjectively, if I play as loud as my ears can stand there is no audible distortion, but the bass is not up to the level, the music sounds loud but compressed.

You also mention:
"I'd guess they may have enough headroom left for filling in the low end without requiring additional subs, as long as you don't try to boost below Fb."
Is that by making a bigger box to lower Fb?




The 25W/8565 in an 89L ported cabinet with an Fb~25- 30Hz could put out way more "gorgeous bass" in the 30Hz range than it would sealed.
Ah, I'm not a fan of BR bass to be honest :)
 
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Art, why so? Fletcher-Munson curves suggest the opposite or I'm wrong? I'm learning here :)
Yes, you are wrong to suggest low frequencies require more level difference to sound twice or half as loud as upper frequencies.
The more current ISO 226: 2003 equal loudness curves are similar to the old Fletcher-Munson, but it's easy to see the level differences for equal loudness are much less at low frequencies.
Also easy to see that relatively less low bass is required at higher SPL:
Screen Shot 2024-06-02 at 5.53.55 PM.png

That said, those curves are for average young adults, and may not be at all similar to your hearing.
I understand a lift compared to what I have now at 30-50Hz. I'd need a subwoofer to do that.
You have presented no data that you would "need" a subwoofer to reach the low frequency level you desire.
Art, how do I check that headroom? There are no TS parameters for Tannoy III LZ (I've found only Fs=27Hz). Subjectively, if I play as loud as my ears can stand there is no audible distortion, but the bass is not up to the level, the music sounds loud but compressed.
You look at the excursion of the drivers to see what headroom you have at the level you want to hear.
If the drivers exceed their rated excursion (above Fb), you probably want additional LF reinforcement.

"Music sounds loud but compressed" could be due to your own hearing overloading (or temporary threshold shift), amp clipping, the Tannoy hitting more than Xmax, which as a guess would be under 6mm, 12mm peak to peak, or the PR suspension non-linearity before reaching it's mechanical limits of ~.
You also mention:
"I'd guess they may have enough headroom left for filling in the low end without requiring additional subs, as long as you don't try to boost below Fb."
Is that by making a bigger box to lower Fb?
With a PR, you may either add weight to the cone to lower it's Fs, or increase the box volume.
An increased box volume increases efficiency, but with a PR output is limited by excursion- drive it hard with lots of driver power, or use less in a big box.
Ah, I'm not a fan of BR bass to be honest :)
In a relatively small full range cabinet, PR can avoid the upper port resonance peaks and dips, and the interference of a long and out of phase upper reflections "leaking through" the port.

In a ported subwoofer with proper filtering neither of those problems apply.
A PR occupies far less enclosure volume than a port of equal cross sectional area tuned as low.
Within their linear range, there is very little difference between the low frequency sound of a PR compared to a port.

Art
 
Yes, you are wrong to suggest low frequencies require more level difference to sound twice or half as loud as upper frequencies.
The more current ISO 226: 2003 equal loudness curves are similar to the old Fletcher-Munson, but it's easy to see the level differences for equal loudness are much less at low frequencies.
Also easy to see that relatively less low bass is required at higher SPL:
Ah, indeed me stupid! Of course, the equal loudness spacing is much narrower at LF. Thank you so much for pointing to that.
You have presented no data that you would "need" a subwoofer to reach the low frequency level you desire.
Art, honestly I'm a bit lost. Do I understand correctly that you are pointing at two possible ways to get more bass from the existing Tannoy design:

1) Giving it more power. I tried many times rising the volume, the speakers play very loud but this does not add more bass relative to the rest of the spectrum, i.e. the tonal balance stays +/- the same. The amp clipping I have never examined. It's Yamaha B2 with 150W/channel, which I thought was enough.

2) Making a bigger box than the current 89L to move the Fb down and/or adding mass to the PR.