Pictured is a freewheel with 15, 16, 17, 18, 19, and 20 tooth sprockets. What I don't understand is why such small increments are worthwhile. Surely the point of having gears is the variance that they provide? Please enlighten me.
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The point is efficiency.
Cyclists actually have quite a narrow optimal power band. Most of us can bang away at a cadence of 50 rpm, up to about 90 rpm. Some of us pedal faster - 90 to 120 rpm. At those lower cadences (50) it feels like we're producing lots of power and at the high cadences it feels like we're just breezing along. But that is confusing torque with power. The optimal power band is where we produce most power most efficiently.
How narrow is that power band? Take me as an example: 50+ years of road cycling, leg muscles well trained to buzz along at 100 to 120 rpm. My optimal power band is 105 to 115 rpm - on a flat ride I can do that for hours. If my cadence is dropping on a rise or into a headwind and I'm starting to grind, I know I'm down around 100. On a hill, after I run out of gears, if I can maintain a cadence of 90 or above I'm happy. But it's not as efficient.
So there are three factors to consider: the rider, the terrain, and the wind.
Terrain. On a flat road, close spacing gives us scope to choose the optimum gear. On a hilly road, we might want wider spacing at the bottom of the range to help with climbing. For MTB, AFAIK such a cluster would be unsuitable, but I'd welcome input from an MTB rider.
Wind. On a circuit, wind is like hills. Except that we always seem to have a headwind. Whatever. In wind a cluster with wider spacing helps.
The Rider. You. Where is your power band, and how wide is it? With training it will change. You'll hear many riders say that they "top out", meaning they couldn't pedal any faster (look at the tables below and see what that means). More factors come into it now: training and pedals. Pedaling fast takes practice, not just to pedal, but to do it smoothly so you don't bounce uncomfortably. And you have to be "clipped in" - no flat pedals.
But I digress. A general purpose 10-speed road cluster for a strong rider might be
11, 12, 13, 14, 15, 17, 19, 21, 23, 27
The difference between each gear at the top end (11 to 15 teeth) is between 6% and 9%. At the bottom end, the difference is about 10%, except for the "granny gear". Taking our narrow power band into consideration, and assuming we have 39x53 front chain rings, that means we'll be able to find a comfortable gear at a wide range of speeds.
On the 39T chain ring, at 90 rpm (as an example; it's the ratios that matter, not the exact cadence):
19 kph on the 23T sprocket
21 kph on the 21T
23 kph on the 19T
25.8 kph on the 17T
27.5 kph on the 16T
29.3 kph on the 15T
31.4 kph on the 14T
34 kph on the 13T
37 kph on the 12T
40 kph on the 11T
And still on the 39T chain ring, at 110 rpm:
23 kph on the 23T sprocket
25.5 kph on the 21T
28 kph on the 19T
31.6 kph on the 17T
33.5 kph on the 16T
35.8 kph on the 15T
38.3 kph on the 14T
41.3 kph on the 13T
45 kph on the 12T
49 kph on the 11T
So, if you were riding at 30 kph, you could choose from the 15T cog, the 16T or the 17T. One of them will (in my case the 16T) be closest to your optimum cadence.
Switching up to the 53T chain ring, at 90 rpm:
26 kph on the 23T sprocket
28 kph on the 21T
31 kph on the 19T
35 kph on the 17T
37 kph on the 16T
40 kph on the 15T
43 kph on the 14T
46 kph on the 13T
50 kph on the 12T
54 kph on the 11T
And still on the 53T chain ring, at 110 rpm:
32 kph on the 23T sprocket
35 kph on the 21T
38 kph on the 19T
43 kph on the 17T
46 kph on the 16T
49 kph on the 15T
52 kph on the 14T
56 kph on the 13T
61 kph on the 12T
66 kph on the 11T
So now, considering that we have both chain rings available, you can see that at 30 kph you have 2 more possible gears. So depending on the terrain ahead, you would choose the big or small ring and the appropriate back cog.
For even stronger riders, the bottom gears are only 21 and 23, or maybe as low as 25.
Hope that helps!
From personal experience, having just changed to a cassette with less range, I have to say that smaller steps definitely offers some advantage. I went from an 8-speed 11-32 to a 12-23. Having a single tooth of difference between adjacent gears means that it's more likely that you will be in the "right gear". The "right gear" is the gear in which you aren't spinning too fast, and aren't pushing too hard for the given conditions (slope, wind, etc.). Sure a large gear range is desirable, but there is such a thing as too much range, where some of the gears (usually the easiest gears) are never used. Personally, I found that I never needed the extra range in my ride, especially considering I have triple chainrings, so I went with smaller increments which made the range smaller, while allowing me to be in the "right gear" more often.
FWIW, that is a junior racing freewheel from back in the day. Juniors have a limited high gear ratio they can use.
Now as to whether a corn cob ( that was the nickname for it in the USA) really helps or not, every racer certainly believes that they do. And with power meters you can prove it. All I know is that when you're suffering at the back, having just a slightly lower gear to keep the legs spinning helps a lot mentally.
There is also the intimidation factor of lining up with special gears that prove you're really a racer. Back when there were only 5 gears on the back and the smallest available chainring was a 42t, that was much more of a real choice than with 10/11 speed setups common now. In the 70's showing up at a race with a big cog bigger than 21t meant you weren't a "serious" racer.
It is about what range do you need? If you are riding flats and only need a 20 as the lowest and 15 as the lowest then might as well pack in a tight spacing. One is a variance.
By that argument what is the benefit of a 7 speed 12x32 versus a 10 speed 12x32? It is just a closer variance. You have never been riding a stretch where one gear is too high but the next is too low?