Choosing chainrings and cogs for a 3×8transmission

Hello,

I am in the process of choosing a new set of 3 chainrings and 8 cogs for
a 28"-wheeled bicycle with a badly worn transmission (I will replace the chain as
well, of course). It will be used in an island which does not have much
hills, so there is no real need for very high or low ratios.

My priority is therefore to have regularly spaced ratios, and smooth
shifting. And not to spend more than 50 or 60 euros in that.

I think I would for instance prefer 32 and a 44-teeth rings, which have
4 possible matching teeth where shifting can occur, over 32 and 42-teeth
rings, which only have 2 matching teeth. Does that make sense?

The chainrings I have seen a
* inner: 22 and 26 teeth;
* middle: 32 and 36 teeth;
* outer: 42, 44 and 48 teeth.

I am pretty certain I will have no need for the high ratios provided by
a 48-teeth ring, so the options a
* 22, 32, 42;
* 22, 32, 44;
* 22, 36, *: large ratio gap between the first two ring;
* 26, 32, 42;
* 26, 32, 44;
* 26, 36, 42;
* 26, 36, 44.

With the number of matching teeth where shifting can occur (in
mathematical terms, this is the GCD of adjacent teeth numbers):
* 22, 32, 42: 2, 2;
* 22, 32, 44: 2, 4;
* 26, 32, 42: 2, 2;
* 26, 32, 44: 2, 4;
* 26, 36, 42: 2, 6;
* 26, 36, 44: 2, 4.

If I eliminate the combination that only have two matching teeth between
the two largest rings, I am left with:
* 22, 32, 44;
* 26, 32, 44;
* 26, 36, 42;
* 26, 36, 44.

In term of ratio increase when upshifting, these combinations a
* 22, 32, 44: +45%, +38%;
* 26, 32, 44: +23%, +38%;
* 26, 36, 42: +38%, +17%, which is really uneven;
* 26, 36, 44: +38%, +22%.

I would tend to prefer the 26-teeth inner ring, but let us keep it in
mind nevertheless, it may become useful depending on the cogs
combination.

Now to the cassettes I have seen:
* Shimano 11-28: 11, 13, 15, 17, 19, 21, 24, 28;
* SRAM 11-28: 11, 12, 14, 16, 18, 21, 24, 28;
* Shimano and SRAM 11-30: 11, 13, 15, 17, 20, 23, 26, 30;
* Shimano 11-32: 11, 13, 15, 18, 21, 24, 28, 32;
* SRAM 11-32: 11, 12, 14, 16, 18, 21, 26, 32;
* Shimano 11-34: 11, 13, 15, 18, 21, 24, 28, 34;
* Shimano 12-32: 12, 14, 16, 18, 21, 24, 28, 32.

And a bizarre one that certainly does not match my idea of regularly
spaced ratios:
* Shimano megarange 11-34: 11, 13, 15, 17, 20, 23, 26, 34!

I think I will eliminate the following ones from my selection, because
they have many adjacent cogs with a single matching teeth:
* Shimano 11-28: 11, 13, 15, 17, 19, 21, 24, 28;
* Shimano and SRAM 11-30: 11, 13, 15, 17, 20, 23, 26, 30.

Here are the remaining ones, in term of ratio increase between adjacent
cogs:
* SRAM 11-28: +17%, +14%, +17%, +13%, +14%, +17%, +9%;
* Shimano 11-32: +14%, +17%, +14%, +17%, +20%, +15%, +18%
* SRAM 11-32: +23%, +23%, +17%, +13%, +14%, +17%, +9%;
* Shimano 11-34: +21%, +17%, +14%, +17%, +20%, +15%, +18%;
* Shimano 12-32: +14%, +17%, +14%, +17%, +13%, +14%, +17%.

FYI, I have calculated that by going from the larger to the smaller cog,
dividing the teeth number of each cog by that of the next, smaller cog.
For instance, for the Shimano 11-28 cassette, the largest cog has 28
teeth, the next one has 24 teeth. 28/24 is 1.16, which means switching
from the largest cog to the next leads to a 16% ratio increase.

With these figures, I can eliminate the following cassettes, which have
some ratio gaps over 20%:
* SRAM 11-32: +23%, +23%, +17%, +13%, +14%, +17%, +9%;
* Shimano 11-34: +21%, +17%, +14%, +17%, +20%, +15%, +18%.

Remain:
* SRAM 11-28: +17%, +14%, +17%, +13%, +14%, +17%, +9%;
* Shimano 11-32: +14%, +17%, +14%, +17%, +20%, +15%, +18%;
* Shimano 12-32: +14%, +17%, +14%, +17%, +13%, +14%, +17%.

Now is time to remember that I will have three chainrings, so this is
not how I will shift. Instead, I may use the following (ring, cog)
combinations, from lower to higher ratio:
1,1 → 1,2 → 1,3 → 1,4
↙
2,3 → 2,4 → 2,5 → 2,6
↙
3,5 → 3,6 → 3,7 → 3,8

I will therefore be interested in the ratio /decrease/ between cogs 4
and 3 and between cogs 6 and 5, which a
* SRAM 11-28: -14%, -13%;
* Shimano 11-32: -13%, -17%;
* Shimano 12-32: -13%, -11%.

This is calculated by dividing the number of teeth of cog 4 by that of
cog 3, and similarly the number of teeth of cog 6 by that of cog 5. For
instance, the SRAM 11-28 has a cog 4 with 18 teeth, and a cog 3 with 21
teeth. 18/21 is .86 which is 1 - .14, which means switching from cog 4
to cog 3 leads to a 14% ratio decrease.

All these cassette therefore have an almost similar ratio decrease
between cogs 4 and 3, which will apply in addition¹ to the ratio increase
between the inner and middle ring. If I go back to my ring combinations,
that is:
* 22, 32, *: +25 or +26%;
* 26, 32, *: +6 or +7%;
* 26, 36, *: +19 or +20%.

¹ This is not really (and really not!) an addition. A 45% increase
combined with a 14% decrease, not matter in what order, is 1.45 (1-.14)
= 1.25 which is a 25% increase.

That would eliminate the 22, 32, * ring combination that has a large ratio
gap. Remain, with their ratio increases between each ring:
* 26, 32, 44: +23%, +38%;
* 26, 36, 44: +38%, +22%.

Now I can also look at the shift between the middle and outer ring,
downshifting from cog 6 to cog 5 at the same time. Depending on the
chosen cassette, that will be:
* SRAM 11-28 and 26, 32, 44: +20%;
* SRAM 11-28 and 26, 36, 44: +6%;
* Shimano 11-32 and 26, 32, 44: +15%;
* Shimano 11-32 and 26, 36, 44: +1%!;
* Shimano 12-32 and 26, 32, 44: +23%.
* Shimano 12-32 and 26, 36, 44: +9%.

I can eliminate the combination of a Shimano 12-32 cassette and the 26,
32, 44 rings, that have a large ratio gap. Appart from that, I am left
with an almost free choice between two rings combinations and three
cassettes:
* 26, 32, 44 or 26, 36, 44;
* SRAM 11-28, Shimano 11-32 and Shimano 12-32.

If I go back to the ratio increases between cogs, I would prefer the
Shimano 12-32 (12, 14, 16, 18, 21, 24, 28, 32) that is very even, that
has multiple matching teeth for shifting even between the smallest cogs,
and which goes well with the 26, 36, 44 ring combination.

With the following shifting policy:
1,1 → 1,2 → 1,3 → 1,4
↙
2,3 → 2,4 → 2,5 → 2,6
↙
3,5 → 3,6 → 3,7 → 3,8

it would provide the following ratio increases:
+14%, +17%, +14%
+20%,
+14%, +17%, +13%,
+9%,
+13%, +14%, +17%

with a ratio ranging from .81 to 3.67.

For those that had the courage to read through all these thoughts, does
that all make sense?

--
Tanguy

On 9/2/2020 10:19 AM, Tanguy Ortolo wrote:
Hello,

I am in the process of choosing a new set of 3 chainrings and 8 cogs for
a 28"-wheeled bicycle with a badly worn transmission (I will replace the chain as
well, of course). It will be used in an island which does not have much
hills, so there is no real need for very high or low ratios.

My priority is therefore to have regularly spaced ratios, and smooth
shifting. And not to spend more than 50 or 60 euros in that.

I think I would for instance prefer 32 and a 44-teeth rings, which have
4 possible matching teeth where shifting can occur, over 32 and 42-teeth
rings, which only have 2 matching teeth. Does that make sense?

The chainrings I have seen a
* inner: 22 and 26 teeth;
* middle: 32 and 36 teeth;
* outer: 42, 44 and 48 teeth.

I am pretty certain I will have no need for the high ratios provided by
a 48-teeth ring, so the options a
* 22, 32, 42;
* 22, 32, 44;
* 22, 36, *: large ratio gap between the first two ring;
* 26, 32, 42;
* 26, 32, 44;
* 26, 36, 42;
* 26, 36, 44.

With the number of matching teeth where shifting can occur (in
mathematical terms, this is the GCD of adjacent teeth numbers):
* 22, 32, 42: 2, 2;
* 22, 32, 44: 2, 4;
* 26, 32, 42: 2, 2;
* 26, 32, 44: 2, 4;
* 26, 36, 42: 2, 6;
* 26, 36, 44: 2, 4.

If I eliminate the combination that only have two matching teeth between
the two largest rings, I am left with:
* 22, 32, 44;
* 26, 32, 44;
* 26, 36, 42;
* 26, 36, 44.

In term of ratio increase when upshifting, these combinations a
* 22, 32, 44: +45%, +38%;
* 26, 32, 44: +23%, +38%;
* 26, 36, 42: +38%, +17%, which is really uneven;
* 26, 36, 44: +38%, +22%.

I would tend to prefer the 26-teeth inner ring, but let us keep it in
mind nevertheless, it may become useful depending on the cogs
combination.

Now to the cassettes I have seen:
* Shimano 11-28: 11, 13, 15, 17, 19, 21, 24, 28;
* SRAM 11-28: 11, 12, 14, 16, 18, 21, 24, 28;
* Shimano and SRAM 11-30: 11, 13, 15, 17, 20, 23, 26, 30;
* Shimano 11-32: 11, 13, 15, 18, 21, 24, 28, 32;
* SRAM 11-32: 11, 12, 14, 16, 18, 21, 26, 32;
* Shimano 11-34: 11, 13, 15, 18, 21, 24, 28, 34;
* Shimano 12-32: 12, 14, 16, 18, 21, 24, 28, 32.

And a bizarre one that certainly does not match my idea of regularly
spaced ratios:
* Shimano megarange 11-34: 11, 13, 15, 17, 20, 23, 26, 34!

I think I will eliminate the following ones from my selection, because
they have many adjacent cogs with a single matching teeth:
* Shimano 11-28: 11, 13, 15, 17, 19, 21, 24, 28;
* Shimano and SRAM 11-30: 11, 13, 15, 17, 20, 23, 26, 30.

Here are the remaining ones, in term of ratio increase between adjacent
cogs:
* SRAM 11-28: +17%, +14%, +17%, +13%, +14%, +17%, +9%;
* Shimano 11-32: +14%, +17%, +14%, +17%, +20%, +15%, +18%
* SRAM 11-32: +23%, +23%, +17%, +13%, +14%, +17%, +9%;
* Shimano 11-34: +21%, +17%, +14%, +17%, +20%, +15%, +18%;
* Shimano 12-32: +14%, +17%, +14%, +17%, +13%, +14%, +17%.

FYI, I have calculated that by going from the larger to the smaller cog,
dividing the teeth number of each cog by that of the next, smaller cog.
For instance, for the Shimano 11-28 cassette, the largest cog has 28
teeth, the next one has 24 teeth. 28/24 is 1.16, which means switching
from the largest cog to the next leads to a 16% ratio increase.

With these figures, I can eliminate the following cassettes, which have
some ratio gaps over 20%:
* SRAM 11-32: +23%, +23%, +17%, +13%, +14%, +17%, +9%;
* Shimano 11-34: +21%, +17%, +14%, +17%, +20%, +15%, +18%.

Remain:
* SRAM 11-28: +17%, +14%, +17%, +13%, +14%, +17%, +9%;
* Shimano 11-32: +14%, +17%, +14%, +17%, +20%, +15%, +18%;
* Shimano 12-32: +14%, +17%, +14%, +17%, +13%, +14%, +17%.

Now is time to remember that I will have three chainrings, so this is
not how I will shift. Instead, I may use the following (ring, cog)
combinations, from lower to higher ratio:
1,1 → 1,2 → 1,3 → 1,4
↙
2,3 → 2,4 → 2,5 → 2,6
↙
3,5 → 3,6 → 3,7 → 3,8

I will therefore be interested in the ratio /decrease/ between cogs 4
and 3 and between cogs 6 and 5, which a
* SRAM 11-28: -14%, -13%;
* Shimano 11-32: -13%, -17%;
* Shimano 12-32: -13%, -11%.

This is calculated by dividing the number of teeth of cog 4 by that of
cog 3, and similarly the number of teeth of cog 6 by that of cog 5. For
instance, the SRAM 11-28 has a cog 4 with 18 teeth, and a cog 3 with 21
teeth. 18/21 is .86 which is 1 - .14, which means switching from cog 4
to cog 3 leads to a 14% ratio decrease.

All these cassette therefore have an almost similar ratio decrease
between cogs 4 and 3, which will apply in addition¹ to the ratio increase
between the inner and middle ring. If I go back to my ring combinations,
that is:
* 22, 32, *: +25 or +26%;
* 26, 32, *: +6 or +7%;
* 26, 36, *: +19 or +20%.

¹ This is not really (and really not!) an addition. A 45% increase
combined with a 14% decrease, not matter in what order, is 1.45 (1-.14)
= 1.25 which is a 25% increase.

That would eliminate the 22, 32, * ring combination that has a large ratio
gap. Remain, with their ratio increases between each ring:
* 26, 32, 44: +23%, +38%;
* 26, 36, 44: +38%, +22%.

Now I can also look at the shift between the middle and outer ring,
downshifting from cog 6 to cog 5 at the same time. Depending on the
chosen cassette, that will be:
* SRAM 11-28 and 26, 32, 44: +20%;
* SRAM 11-28 and 26, 36, 44: +6%;
* Shimano 11-32 and 26, 32, 44: +15%;
* Shimano 11-32 and 26, 36, 44: +1%!;
* Shimano 12-32 and 26, 32, 44: +23%.
* Shimano 12-32 and 26, 36, 44: +9%.

I can eliminate the combination of a Shimano 12-32 cassette and the 26,
32, 44 rings, that have a large ratio gap. Appart from that, I am left
with an almost free choice between two rings combinations and three
cassettes:
* 26, 32, 44 or 26, 36, 44;
* SRAM 11-28, Shimano 11-32 and Shimano 12-32.

If I go back to the ratio increases between cogs, I would prefer the
Shimano 12-32 (12, 14, 16, 18, 21, 24, 28, 32) that is very even, that
has multiple matching teeth for shifting even between the smallest cogs,
and which goes well with the 26, 36, 44 ring combination.

With the following shifting policy:
1,1 → 1,2 → 1,3 → 1,4
↙
2,3 → 2,4 → 2,5 → 2,6
↙
3,5 → 3,6 → 3,7 → 3,8

it would provide the following ratio increases:
+14%, +17%, +14%
+20%,
+14%, +17%, +13%,
+9%,
+13%, +14%, +17%

with a ratio ranging from .81 to 3.67.

For those that had the courage to read through all these thoughts, does
that all make sense?

Wow. You may have set a record for the longest discussion of gearing in
a single post!

Back in the days of five rear cogs, some people (like me) put lots of
thought into getting tooth counts perfectly correct. I found it easiest
to do a logarithmic plot of gear inches. Eventually I wrote a Fortran
(anybody still know Fortran?) program to do that. These days I do it in
Excel. It's set up so I can just enter my tooth counts and see a graph
of the result.

In those days, typical objectives were evenly spaced gears over a wide
enough range, and minimizing duplicate gears. With only ten (or maybe
15) possibilities, it didn't make much sense to waste any by duplicating.

These days I don't think those issues are nearly so critical. Certainly
you want a low enough range for climbing whatever hills you may have to
deal with. But it's quite common for bikes to have uselessly high gears.
And with so many rear cogs, a person would have to be much more picky
than in days of yore to complain about having a 22 tooth cog when they
wanted a 23 tooth.

To me, that makes discussions of shifting order academic. I don't shift
the way a long haul trucker does when starting on a hill, going through
a sequence of gears. I choose a front chainring based on overall terrain
("This is mostly downhill, I'll get on my big ring") and fine tune by
picking one of the many rear cogs. Close enough is perfect.

And I don't worry about how easily my front shifts happen. They don't
happen all that often - unless it's a bike with ancient "half-step"
gearing, and in that case they are always easy.

Again, I think "Good enough is perfect."

--
- Frank Krygowski