Correction. He HAD a tire machine. He informed me, to my horror, that he had sold it the month before. What to do now? I had read about a new mounting procedure that some people had tried, I figured now was a good as a time as any to give it a try, so I broke out my tire irons and got to work.
First, obviously, you will be removing the wheels from your bike. Not having to take my wheels to the shop to have them mounted means I can do one at a time, which is a lot easier than pulling both wheels off the bike at the same time.

With the wheel removed, I set it on a small piece of carpet to prevent scratching or damage to the wheel.

I use a valve core removal tool to unscrew the valve core from the valve stem. Careful - there is a lot of air pressure in there that will try to fire the valve core out of the valve stem like a bullet! This will rapidly deflate the tire.

Next, I use a bead breaker to break the bead on the tire. This basically means you are peeling the tire's bead away from the tire rim. It has been held there with a fair amount of pressure for some time, so it takes a bit of force to break it free. This cheap and simple bead breaker from Harbor Freight makes the job simple. Alternatively, Motion Pro has a set of tire levers that can be hooked together to also function as a bead breaker. You need to break the bead all the way around the tire, on both sides.

We will need some tire lubricant. Cars and shops use a gooey paste for this. This goopy lubricant is incompatible with the balancing beads that I use in my tires, so I need something that does not leave behind residue, and this Purple Power Lubricant does the trick. Another option is Ken Tool Bead-Eze. Don't use dish soap or detergent - anything alkaline or acidic can etc and corrode your aluminum rims, causing air leaks.

I put this lubricant in a spray bottle - it's fairly thick, so you need a decent sprayer that can handle it. It is water-based, so it will evaporate - you may need to reapply it if you take some time to do this job.

With the bead of the tire separated from the rim, I spray bead lubricant all around the tire. I then push the side of the tire down, away from the rim edge.

In the middle of the rim is a deeper channel. The tire must be pushed down so that it can rest in this channel. This allows the tire to shift when we start to demount it on the opposite end.

Next, 180 degrees away from where the tire has been pushed down into the channel, I push these rim protectors, also from Motion Pro onto the rim. These are important! Without them, you will gouge and damage the rim, and the tire bead will no longer seal against it.

While pushing the opposite end of the tire down into the channel, I use the spoon end of a tire iron to push up under the rim and hook the edge of the tire from the inside.

Once you have hooked the tire with the tire iron, pivot it back over the rim to pull the tire up and over like the edge. Notice the tire is actually pulled away from the rim - this is why the opposite end of the tire must be pushed down into the channel. If it wasn't, the tire would not slip enough to give you the free play in order for it to do this. It's also why you want to use lots of lubricant, to allow the tire to slide this way.

While holding one of the tire irons in place, repeat the process with a second one, again pulling the edge of the tire up and over.

Slide one tire iron over and work the tire up and over the rim there as well.

Slide one of the rim protectors over, and repeat the process. When you get about half of the tire off the rim, you can pull the rest off by hand.

Now one side of the tire has been pulled completely up and over the rim.

Looking at the bottom, you can see how the tire can be slid up along the rim.

After applying more lubricant, reach in with the tire irons to hook the far side of the tire the same way.

Looking from underneath, you can see how the far side of the tire has been hooked by the tire irons and is being pulled up and over the rim. There is a fair amount of pressure holding the tire against the rim (it's a tight fit!) so the lubricant is necessary to get it to slide up and over the rim as it is removed.

Now that the tire is removed from the rim, it's a good time to clean the rim.

It's also a good idea to remove the valve stem and replace it - valve stems should be replaced regularly. The tire must be removed to replace the valve stem, so now is the perfect time to do it. For rubber stems, I use a valve stem tool. For metal stems, they normally have a retaining nut that must be removed.

Screw the puller onto the old rubber valve stem, then use a block of wood for leverage, pull the stem up and out of the tire:

Lubricate the new valve stem, and push it up through the hole from the inside:

Screw the puller tool onto the new stem, and pull it the rest of the way into place:

Mounting the New Tire
This is the new technique that I read about and thought I would give a try. To start, I bought four of these, each packages contains two lashing straps. These were from Harbor Freight, but I now realize that I could have actually gotten them cheaper on Amazon - you can buy a package of ten for $20 there.

I cut the ends off of the 12 foot straps to make them about 2 feet long, and strapped them around the new tire evenly.

Next, going around in a circle, I squeezed the tire together so the beads touched one another, then tightened the straps to hold them this way.

With all the straps tightened and the beads touching each other all the way around, we have two benefits for mounting this stiff new tire: first, the inner diameter of the tire opens up slightly, making it easier to mount. Second, we only have to mount the tire to the rim ONCE, instead of doing it twice (once for each bead). Before starting, I now put the tire out in the sun for ten minutes. This warms it up and softens the rubber, making it again a bit easier to mount.

Putting a bit of lubricant on the tire ONLY (and not on the rim) I was able to easily lever the new tire onto the rim the same way I took the old one off. Start with one end of the tire shoved down into the rim's center channel, and work the tire beads over the rim gradually, the exact reverse of how the old tire was taken off.

IMPORTANT NOTE!! Motorcycle tires are usually directional!! Make sure you check the rotation of the tire (it will have an arrow on it showing the direction of rotation) before mounting it!

You don't want to realize you did it backwards and have to take it off again! It might be a good idea to put a temporary mark on the rim before taking the old tire off, just so you don't mix it up:

Aluminum rims pit over time, and as they do, they begin to leak. It's very annoying to have to refill your tires with air every time you want to go riding! I use this bead sealer compound to stop this from happening. It will seal the worst, pitted rim.

It comes out of the can as a thick goo, but dries quickly (within a minute) after applied. I wear rubber gloves, dip my pinky finger in the goo, and spread it over the INSIDE of the rim, where the tire bead will contact it. I then wipe out any spilled or excess from the outside of the rim. Because the straps on the tire are holding the bead away from the rim, it dries easily.
After 15 minutes the sealant is well cured, so I loosen the straps one by one and pull them off the tire.

Now the moment of truth: Sealing the bead. To start with, I use my finger to spread a little bit of tire lubricant along the exposed area of the rim between the tire bead and the rim. This helps the tire slide along the rim until its bead meets up with the rim. You want the tire to do this evenly so that the bead is fully seated all the away around on both sides.
I use these clip-on valve chucks - however I use a pair of needle nose pliers to rip out the valve actuating piece in the middle of them. Basically you want to get as much air into the tire as fast as you can. There is no valve core in the valve stem to actuate, so the actuator doesn't need to be there obstructing airflow. I clip it to the valve stem, then hook up the air compressor to blast air into the tire. As the tire inflates, it pushes outwards, sliding the tire bead out toward the rim (which is why we need the lubricant). As the bead contacts the rim, you will hear a loud bang - one for each side of the tire.

If the bead doesn't seat right away, I will unclip the hose and let the tire deflate, check where it has not seated, apply a bit more lubricant, and try again.
Once the bead has seated, I screw the valve core back into the valve stem, and fill the tire to the prescribed pressure. I check both beads, making sure the tire is perfectly even all the way around. Manufacturers usually mold a line around the circumference of the tire near the bead. As the tire spins, make sure the distance from this line to the wheel rim does not change. This indicates that the tire has seated evenly all the way around. I'll also spray some soapy water around the diameter of the tire where it meets the rim, as well as on the valve stem, checking for bubbles that might indicate a leak.
The tire is now completely mounted and ready for use once the wheel is reinstalled. For our pre-GL1800 Goldwings, when replacing the rear tire, don't forget to re-lubricate the rear splines with proper Moly paste (not grease!) before reinstalling the rear wheel.
So how did this new mounting method work? Amazingly well! I was astounded at how easily the tire popped onto the rim. The added benefit of only having to work one bead over the rim instead of two makes the process even quicker.