Usually only drivers which have high efficiency, low Qts (0.25 - 0.4), Qes (0.3 - 0.4) and medium to low Fs values are suitable for tapped horn systems.  Tapped horn alignments also tend to be more suited to pro audio duty rather than home subwoofer duty, due to the size requirements needed to execute a "proper" tapped horn, where the gain at the higher end of the passband is enough to ensure that any out of band response peaks are not significantly higher than the passband. Nevertheless, there are several DIY examples of tapped horns with significant out of band response peaks available on the Internet. I assume the designers planned to deal with the high out of band response peaks by using steep crossovers and/or DSP.  One of the problems with taking that approach however is that those high out of band response peaks also amplify any harmonic distortion components that happen to occur at the same frequencies, and those cannot be eliminated by those methods. 

My recommendation therefore is, if you're aiming for a subwoofer design that is not at least 98dB/2.83v/1M in its passband, and preferably higher than that (to ensure that those out of band peaks are not too high above the passband), then consider using an offset transmission line design instead.  The net required volume will be smaller and you won't have to deal with the out of band response peaks. You will be giving up a bit of efficiency in the process however.

Tapped horns are a bit more complex than transmission lines to design and therefore you should spend a lot of time making sure that your simulation is correct before committing it to wood to make sure that you're not disappointed with the results.

Design Notes:
For these design notes, I will step you through the design process for a simple single-expansion tapped horn. My tool of choice for designing tapped horns is Hornresp.  There may be better tools out there, but Hornresp is simple to use (once you learn how to use it!) and the results are close enough to be useful for most purposes.

Choosing the driver:
For this example, I'm going to use the Dayton Audio PA310.  This is a cheap but decent quality 12" pro audio driver with a nice low Fs and Qts, making it suitable for a number of different designs. 

The specifications of the PA310 are as follows:
Sd = 530.9 cm^2
Re = 5.2ohms
Fs = 39 Hz
Vas = 99.27 l
Qes = 0.31
Qms = 7.51
Qts = 0.30
Le = 1.4 mH
Xmax = 5 mm

First, we're going to start with a simple transmission line alignment for this driver. "Classical" transmission line theory suggests that an appropriate transmission line configuration for this driver would be one that is long enough to resonate at 39 Hz, and has a constant cross-sectional area of 530.9 cm^2.  The following image shows the parameters for a Hornresp sim of this "classical" type of transmission line.  Note: to enter Cms, Mmd, BL and Rms for the driver, simply double-click on the "Sd" text box - a dialog box will pop up asking you for the t/s parameters for the driver, and Hornresp will use these to calculate the other parameters it uses. 

A brief outline of what's shown in the sim above:
Ang - for comparison purposes, this shouild always be set at "2.0 x Pi", which basically represents a speaker sitting on the floor
Eg - the voltage that's being applied to the sim, 2.83V
Rg - the resistance of the cable (which I've set to zero for the purpose of this sim)
S1 - the cross-sectional area at the start of the transmission line (in cm^2)
S2 - the cross-sectional area where the driver is located (in cm^2)
Par (1) - the distance between S1 and S2 (in cm)
S3 - the cross-sectional area at the end of the transmission line (in cm^2)
Par (2) - the distance between S2 and S3 (in cm)

So far, so good.  At this point, we can select "Tools...Loudspeaker Wizard", and check what this sim is going to look like

Ok, that looks like what a end-loaded 1/4 wave resonator would look like, with the driver (the red circle) at one end and the vent at the other. Now, let's see what the predicted response of this sim looks like...

As is expected with the use of an end-load design like this one, there's this massive notch in the response between 100 Hz and 200 Hz. 

Now, let's see if we can deal with that notch by using a tapped horn alignment instead. 

First, let's change the Hornresp sim to that of a tapped horn. To do this, we're going to add values for S4 and L34, change the L34 expansion to "Par" and change the type of simulation from OD (offset driver) to TH (tapped horn).

Let's have a look at what the response looks like, by using Hornresp's Loudspeaker wizard.  It should not look much different from the end-loaded transmission line's response, because we haven't really moved the driver to the best location yet to extend the passband.

From this point, we're going to "move" the driver's location into the line, by adjusting the values for L12 and L34, using the simulated schematic as a guide. The result will look something like the diagram below.

Now, we need to adjust the length and taper of the line to achieve the flattest frequency response. To do this, we're going to set the calculation for S2 and S3 to "Auto" (by double-clicking where it says "Manual"), and basically adjust S1, S4 and L23 (and L12 and L34 if necessary) until we get something useful, e.g. a reasonably flat response, decent power-handling and out of band response peaks that are not significantly higher than the usable passband.

Illustrated below is just one possible tapped horn alignment for this particular driver that meets most of these requirements.

The tapped horn alignment illustrated above is large compared to an offset transmission line alignment with the same low frequency extension using the same driver, but it also is quite a bit more efficient. The out of band response peaks are around 6dB above the passband (I would aim for them to be no higher than that), and the passband itself is not quite flat, so it's not perfect, but it's a good compromise. A tapped horn alignment with smoother response can be achieved using this driver, but this will require further tweaking of the Hornresp simulation.  For example, converting the simulation into a dual-expansion tapped horn (by defining values for S5 and L45 in the "Input Parameters" window) and adding a bit of damping to the first half of the line, the response illustrated below can be achieved.

Generating a good-looking simulation of a tapped horn in Hornresp is not that difficult to do.  Converting the simulation into an actual build is where the real challenge lies.  Usually it's a lot easier to just start with a known tapped horn "fold" (basically a tapped horn folded into the form of a box) and modifying it to work with the driver that you want to use.  Under the "Horn Folding" section of this website, I've provided several workbooks that describe some of the more common tapped horn folds - these workbooks can be used to modify the dimensions of the box and the location of the internal panels to produce a tapped horn that will get the best results from the driver that want to use. The workbooks will automatically generated the required parameters for a Hornresp simulation of the tapped horn that is illustrated in the workbook.

Tapped horn resources and projects on the internet:

• POC #3 - another tapped horn
• POC #3.1 - a minor upgrade to POC #3
• POC #4 - the "B1" tapped horn
• Tapped Horn Experiments (Cowan Audio)