Diagram courtesy of Wikipedia |

## Calculating Gear Ratios

The factors that determine how fast a bicycle moves at a given pedaling speed are the size of the front sprocket, the size of the back sprocket, and the size of the wheel. In the United States, the most common way of expressing this is the size of the wheel (in inches) needed to reach this effective gear ratio assuming there were no other gears (e.g. an hypothetical penny-farthing.) This value is traditionally rounded to the nearest whole number. Consider the following example:

- A one speed bicycle with a 46 tooth sprocket in front, an 18 tooth sprocket in back, and a wheel 27 inches in diameter: The size of that gear would be 47 ÷ 18 x 27 = 71 inches.

Determining the number of teeth on front and rear sprockets is simple and unambiguous. Determining the wheel size is less so. The good news is that in most cases, road bikes (with the notable exception of folding bikes) have fairly similar wheel sizes so getting this not quite right has modest consequences.

The best way to determine the size of your wheel, if you can do it carefully enough, is to have someone hold your bicycle as you sit on it with the tires inflated to the pressure you normally use, precisely mark the position of the bicycle, move the bicycle forward one complete turn of the wheels, precisely mark the position of the bicycle, and measure how far the bicycle moved. This gives you the actual circumference of the wheel under your precise conditions. To calculate the gear inches as described above, use the formula diameter = circumference ÷ π.

I have never measured my wheel size as is described above. It requires a partner, is more fuss than I am up for, and I am not confident of my ability to make measurements precise enough to be useful. The alternative is to look up the wheel size. To do that, you need to know the tire size and the manufacturer of the tire, and you have to do some research and some calculations. Here are the considerations:

The best way to determine the size of your wheel, if you can do it carefully enough, is to have someone hold your bicycle as you sit on it with the tires inflated to the pressure you normally use, precisely mark the position of the bicycle, move the bicycle forward one complete turn of the wheels, precisely mark the position of the bicycle, and measure how far the bicycle moved. This gives you the actual circumference of the wheel under your precise conditions. To calculate the gear inches as described above, use the formula diameter = circumference ÷ π.

I have never measured my wheel size as is described above. It requires a partner, is more fuss than I am up for, and I am not confident of my ability to make measurements precise enough to be useful. The alternative is to look up the wheel size. To do that, you need to know the tire size and the manufacturer of the tire, and you have to do some research and some calculations. Here are the considerations:

- You might think that you could determine the wheel diameter from the marked tire size. For example, you might think that a 27 x 1¼ inch tire is 27 inches in diameter. It is not, it is closer to 28 inches in diameter (see below). A 26 x 1.3 inch mountain bike tire is less than 25 inches in diameter. These are not huge differences, of course, so if you do use the marked tire size, you will only be off by about one gear shift (e.g. from your lowest gear to your next lowest gear.)
- A given wheel can usually fit multiple sized tires (fatter or skinnier.) Putting fatter tires on your bike makes the wheels larger and thus all your gears higher, and putting skinnier tires on does the reverse. Again, this is a relatively small effect in most cases.
- There are two reasons you may need to know the manufacturer of the tire. Firstly, some manufacturers have made a proprietary sized tire with a standard designation. For example, a 26x1⅜ tire from a Schwinn three speed results in a wheel diameter of 26.6 inches whereas a 26x1⅜ tire from an English three speed results in a wheel diameter of 26.3 inches (and neither tire will fit on the other bike - yikes!)
- Tires are not always the size they are marked. I am considering getting Grand Bois Cypres Extra Leger tires for my May 18th brevet. These tires are marked as 30 mm wide but are in fact 32 mm wide.
- Historically, tire sizing has been completely irrational and chaotic, but things are getting better with the introduction of the ETRTO standard for tire sizing. ETRTO sizes look like 28-622; the first number is the width of the tire in millimeters, and the second the diameter of the wheel before you put the tire on (more or less.) I have noticed that more and more new tires are sold under their old size name, but have the ETRTO standard size written on the tire. If it is not, you can look it up on Sheldon Brown's website. With that in hand, you use one of the many available lookup tables; this one for example, to get a wheel size. Finally, you convert circumference to diameter and millimeters to inches as required.
- Example, the above mentioned 27 x 1¼ inch tire has an ETRTO size of 32-630. Looking that up, the final circumference of the wheel is 2220 millimeters, 2220 ÷ π gives a diameter of 706.6 millimeters which equals 27.82 inches.

My 2010 Surly Crosscheck came with 700c, 32 mm wide tires. The ETRTO standard for this size is 32-622. Looking that up and doing the appropriate conversions gives a wheel size of 27.19 inches. It also came with front sprockets of 48 and 36 teeth and rear sprockets ranging from 12 to 25 teeth. The result:

Highest Gear = 48 ÷ 12 x 27.19 = 106 inches.

Lowest Gear = 36 ÷ 25 x 27.19 = 38 inches.

This high gear was way higher than I could ever imagine using, but I would have liked a lower low gear. For comparison, the bicycle I rode through Europe (including across the Alps) when I was 17 years old had a low gear of 27 inches. An old man like me certainly needs a gear at least that low. That 27 inch low gear was calculated as follows:

The bike had sew-up tires which are ETRTO 23-622, and front gears of 49, 44, and 28 teeth paired with back gears of 13, 16, 19, 23, and 28 teeth, so the low gear was:

28 ÷ 28 x 26.63 = 27 inches.

The first change I made to my Surly was to replace the tires. I did not do this to lower the gears, but because I felt like the stock tires were too "knobby" and fat, so I replaced them with slick, 28 mm wide tires. Although this had the effect of lowering my low gear, the effect was so small as to be lost when I rounded the gear to the nearest whole number:

36 ÷ 25 x 26.94 = 38 inches.

As I was preparing for my brevet in May of 2012, I took my Surly into the bike shop to have it tuned up and checked out. Foolishly, I had gone too long with the same chain (who knew chains needed periodic changing?) and had worn out the rear gears as a result. Thus, I took the opportunity to replace the rear gears with gears that ranged from 11 to 32 teeth, which required replacing my Shimano Tiagra rear derailleur with a Shimano Deore. The result was a low gear that felt dramatically lower; it was plenty low enough for the rolling hills of the brevet:

36 ÷ 32 x 26.94 = 30 inches.

However, now I knew better. Next time I would replace the chain before the rear gears were damaged. Not. Between one thing and another, I found it difficult to be without my Surly long enough to let the bike shop at it. Finally, I gave up, bought a new chain, and installed it myself. It was remarkably easy! However, I was too late. Once again I had worn out the rear gears. (I could tell because the new chain slipped on the most worn gear.) Although the Deore derailleur was listed as having a maximum rear sprocket size of 32 teeth, Sheldon Brown notes that Shimano is very conservative with their specifications, so when replacing the rear gears, I went with gears ranging from 11 to 34 teeth. Replacing the gears was remarkably easy, and the slightly out of specification gears work great. My new low gear is:

36 ÷ 34 x 26.94 = 29 inches.

Let's now look at not just the highest and lowest gears, but all the different gears on my Surly:

Notice that the 36x15 and the 48x20 sprocket combinations result in almost exactly the same gear ratio. The same is true for 36x13 vs 48x17 and 36x17 vs 48x23. Other combinations are also close, but these are the closest. - they are really the same thing. So really, I only have 13 truly different speeds. Back in the mid-1960's, we would spend hours with pencil and paper (computers and spreadsheets had yet to be invented) designing different combinations of sprocket sizes to get as many different distinct gear ratios as possible by avoiding such duplications, known as gear overlaps.

Conventional wisdom argues that trying to adjust sprocket size to minimize gear ratio overlap is a fool's errand, especially today with 9 back sprockets; who needs more than 13 different gear ratios? The basis for this argument is that the convenience of getting from one gear ratio to another is often more important than getting to a very specific ratio. Imagine that, in order to get rid of the gear overlap, I spent $50 to replace the 36 tooth front sprocket on my Surly with a 34 tooth front sprocket. The gears I would get would be as follows:

Whereas the 36x15 gear used to overlap with the 48x20, the 34x15 is now half way in between the 48x20 and 48x23 - I got my gears back! However, suppose I am riding along in my 48x23 (56 inches) and want to shift to my next highest gear, the 34x15 (61 inches). To do that I would require two shifts, one in front and one in back. I could either shift the back, putting me into the uncomfortably high 86 inch gear, and then shift the front, just to get from 56 inches to 61 inches. Alternatively, I can shift the front first, putting me into the uncomfortably low 40 inch gear before shifting the back to get to where I wanted to be. Conventional wisdom argues that under any given condition, either the 56 inch gear or the 65 inch gear will be close enough such that it is not worth the double shift. The right way to think about the gears, say the experts, is not as 18 speeds or 16 speeds or even 13 speeds, but as two sets of 8 speeds, a high set and a low set. Mostly, one stays in one or the other set, switching between the gears in that set, but when conditions change (e.g. one heads into the wind or into the mountains) one switches to the other set.

Nonetheless, I am seriously considering purchasing the 34 tooth sprocket. The main reason I am considering this is to get an even lower low gear. If I make this change, my low gear becomes:

34 ÷ 34 x 26.94 = 27 inches

I would finally have matched the alp-riding bike of my youth! However, although getting a lower low is my main motivation, getting rid of gear overlap is a motivation as well. Partly it is an esthetic consideration, knowing I am getting all my gears is somehow satisfying. Also, although I agree that when I am shifting between gears to accomodate rolling hills or shifting winds I am unlikely to go to the trouble of getting to the intermediate gears, I can imagine being in the last 24 miles of a 200K brevet, riding into a steady headwind for mile after mile, being discouraged, exhausted, and in pain, and feeling that 56 inches makes me pedal just a little too fast, and that at 65 inches, it is just a little too hard to pedal. Being able to go into that 61 inch gear might be just the psychological edge needed to carry me to the finish. And besides, I just like to fidget with my bicycle.

Highest Gear = 48 ÷ 12 x 27.19 = 106 inches.

Lowest Gear = 36 ÷ 25 x 27.19 = 38 inches.

This high gear was way higher than I could ever imagine using, but I would have liked a lower low gear. For comparison, the bicycle I rode through Europe (including across the Alps) when I was 17 years old had a low gear of 27 inches. An old man like me certainly needs a gear at least that low. That 27 inch low gear was calculated as follows:

The bike had sew-up tires which are ETRTO 23-622, and front gears of 49, 44, and 28 teeth paired with back gears of 13, 16, 19, 23, and 28 teeth, so the low gear was:

28 ÷ 28 x 26.63 = 27 inches.

The first change I made to my Surly was to replace the tires. I did not do this to lower the gears, but because I felt like the stock tires were too "knobby" and fat, so I replaced them with slick, 28 mm wide tires. Although this had the effect of lowering my low gear, the effect was so small as to be lost when I rounded the gear to the nearest whole number:

36 ÷ 25 x 26.94 = 38 inches.

As I was preparing for my brevet in May of 2012, I took my Surly into the bike shop to have it tuned up and checked out. Foolishly, I had gone too long with the same chain (who knew chains needed periodic changing?) and had worn out the rear gears as a result. Thus, I took the opportunity to replace the rear gears with gears that ranged from 11 to 32 teeth, which required replacing my Shimano Tiagra rear derailleur with a Shimano Deore. The result was a low gear that felt dramatically lower; it was plenty low enough for the rolling hills of the brevet:

36 ÷ 32 x 26.94 = 30 inches.

However, now I knew better. Next time I would replace the chain before the rear gears were damaged. Not. Between one thing and another, I found it difficult to be without my Surly long enough to let the bike shop at it. Finally, I gave up, bought a new chain, and installed it myself. It was remarkably easy! However, I was too late. Once again I had worn out the rear gears. (I could tell because the new chain slipped on the most worn gear.) Although the Deore derailleur was listed as having a maximum rear sprocket size of 32 teeth, Sheldon Brown notes that Shimano is very conservative with their specifications, so when replacing the rear gears, I went with gears ranging from 11 to 34 teeth. Replacing the gears was remarkably easy, and the slightly out of specification gears work great. My new low gear is:

36 ÷ 34 x 26.94 = 29 inches.

## Unusable and Overlapping Gears

With two gears in the front and 9 gears in the back, there are 2 x 9 or 18 possible combinations of front gears and back gears, so my Surly was sold as an 18 speed bicycle. However, best practice says that I should not be using the two "extreme" gears; the 48 tooth sprocket in front with the 34 tooth sprocket in back nor the the 36 tooth sprocket in front with the 11 tooth sprocket in back. The reason is that in these two cases the chain is going from the far right on the front to the far left on the back or vice versa, putting it at a sharp angle relative to the front and rear sprockets. According to conventional wisdom, this generates a lot of friction which both wastes energy and is bad for the chain and the sprockets, so one is advised not to use these combinations. In my experience, these extreme gears work just fine, but they are not all that useful, so if I fogo them, my 18 speed bicycle becomes a 16 speed.Let's now look at not just the highest and lowest gears, but all the different gears on my Surly:

The gear ratios were calculated as above, based on a 28-622 size tire which has a diameter of 26.94 inches. |

Conventional wisdom argues that trying to adjust sprocket size to minimize gear ratio overlap is a fool's errand, especially today with 9 back sprockets; who needs more than 13 different gear ratios? The basis for this argument is that the convenience of getting from one gear ratio to another is often more important than getting to a very specific ratio. Imagine that, in order to get rid of the gear overlap, I spent $50 to replace the 36 tooth front sprocket on my Surly with a 34 tooth front sprocket. The gears I would get would be as follows:

Whereas the 36x15 gear used to overlap with the 48x20, the 34x15 is now half way in between the 48x20 and 48x23 - I got my gears back! However, suppose I am riding along in my 48x23 (56 inches) and want to shift to my next highest gear, the 34x15 (61 inches). To do that I would require two shifts, one in front and one in back. I could either shift the back, putting me into the uncomfortably high 86 inch gear, and then shift the front, just to get from 56 inches to 61 inches. Alternatively, I can shift the front first, putting me into the uncomfortably low 40 inch gear before shifting the back to get to where I wanted to be. Conventional wisdom argues that under any given condition, either the 56 inch gear or the 65 inch gear will be close enough such that it is not worth the double shift. The right way to think about the gears, say the experts, is not as 18 speeds or 16 speeds or even 13 speeds, but as two sets of 8 speeds, a high set and a low set. Mostly, one stays in one or the other set, switching between the gears in that set, but when conditions change (e.g. one heads into the wind or into the mountains) one switches to the other set.

Nonetheless, I am seriously considering purchasing the 34 tooth sprocket. The main reason I am considering this is to get an even lower low gear. If I make this change, my low gear becomes:

34 ÷ 34 x 26.94 = 27 inches

I would finally have matched the alp-riding bike of my youth! However, although getting a lower low is my main motivation, getting rid of gear overlap is a motivation as well. Partly it is an esthetic consideration, knowing I am getting all my gears is somehow satisfying. Also, although I agree that when I am shifting between gears to accomodate rolling hills or shifting winds I am unlikely to go to the trouble of getting to the intermediate gears, I can imagine being in the last 24 miles of a 200K brevet, riding into a steady headwind for mile after mile, being discouraged, exhausted, and in pain, and feeling that 56 inches makes me pedal just a little too fast, and that at 65 inches, it is just a little too hard to pedal. Being able to go into that 61 inch gear might be just the psychological edge needed to carry me to the finish. And besides, I just like to fidget with my bicycle.

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