While cleaning up my home office, I found an old, yellowed sheet of lined paper. It held a diagram I drew decades ago, before I had access to computer graphics. I posted it in a page on sheldonbrown.com.

The diagram was novel in how it showed the ratios between adjacent sprockets of a cassette or freewheel. Plotting the sprocket pairs and ratios as a two-dimensional graph makes it possible clearly to see which pairs have nearly the same ratio -- and to identify good sequences.

Jeremy Mak took a look and wrote some Javascript which created a professional-looking version of the diagram that plots any combination of cogs you choose.

You enter tooth counts of freewheel or cassette sprockets at the top of the graph. In the area below, each pairing of sprockets is represented by a horizontal line with a sprocket at each end. The scale at the left identifies ratios established by pairs of sprockets, and the scale at the right, tooth counts.

For now, double shifts need to be calculated manually. For a half-step system, find the average step ratio of the rear cluster 'y', trace (y-1)/2 across to where it intersects your ideal small ring tooth count, and the nearest hyperbolic curve will give the tooth increment from small to big ring that will be ideal for double shifting (probably 4-6 teeth larger).. For a multi-range system, the step between chainwheels should generally allow a chainwheel shift to take the drive ratio up or down a couple steps of the cluster, to be readjusted with a rear shift. A wide-step syetem should have a chainwheel ratio equal to the step between the two or three largest sprockets of the cluster. For explanations of these terms, see our gear-theory article.

Jeremy has suggested that he might make further improvements including chainring input boxes with double-shifting graphics. We'll let you know when he does.

The original diagram is below. Jeremy's version inverts it. In the notes below the diagram, you'll see that I had suggested that inverting it would work better. The notes also describe the several sprocket sequences shown.

Now let's look at sequences that I have highlighted with colored lines in the original diagram.

The red stair steps near the upper left represent a "corncob" cluster, with one-tooth jumps up through 20 teeth, then the descending dashed line goes to where the 20-tooth sprocket is paired with the largest one, with 22 teeth

The pattern in green shows the 7-sprocket sequence on my Cannondale road bike, 13-15-17-19-22-26-32. Notice how the steps get smaller from 13 through 19, but the 19-22 tooth jump is almost the same as the 13-15. A 31 at the bottom end would make for a more even progression than the 32, but on the other hand, I have paired this cluster with a 50-34 compact double crankset. Using all 7 sprockets with the 50-tooth chainring covers almost all my needs except for steep climbs. The double shift to the 34-tooth chainring and 26-tooth sprocket gives me a step nearly the same size at the shifts above and below. The overall range is 28 to 104 gear inches, with an easily remembered sequence and no "surprise" steps.

The remaining four sequences,

• 13-16-19-23-28-34,
• 13-16-20-25-31 (a 31 was once made by Regina in Italy; you'd have trouble finding one today.);
• 14-17-21-26-32,
• 12-15-19-24-30-38,

are intended for use in half-step plus granny systems. The steps in the sprocket cluster are large, but all similar enough to be "half-stepped" with a pair of chainrings only a few teeth different. A half-step system with a third, small chainring for "granny" gears allows an extremely wide range and small, even steps with only 5 or 6 sprockets on an older frame with narrow rear dropout spacing. Half step used to be a pain to use with all the double shifts, but it works nicely with indexed bar-end shifters.