CP Method
The CP (Circular Pitch) method is probably the oldest standard and was likely derived empirically. This method isn't as precise as the ANSI method because it basically rounds the material diameter to some approximate integer value of the tool's CP.
Once again it really depends on Knowing what knurl tools you actually have and what their CP is. Most tool manufacturers have a variety of CP wheels available in various sizes and pitches. There are several out there like; Form Roll Die Corp., or Dorian Tool. Both websites helped me and the Dorian Knurling Catalog has good detail about the process.
CP Tools
My research found the three most common CP pitches talked about were 33, 21, & 14, for fine, medium and course tools. However, a huge amount of other pitches are available. For instance, Form Roll Die has a Grid of 19x5≈95 CP tools in their KP series alone, as well as another ~20 in DP tools. The KP series is the size I would use for my LMS scissor tool (3/4 x 3/8 x 1/4).
CP is specified in TPI or LPI (teeth or lines/in. around the circumference of the tool). I will use TPI. For straight knurl tools it is measured Parallel to the teeth (perpendicular to the Axis of the wheel) and called Normal TPI. The three common pitches would then be designated 33TPI, 21TPI, & 14TPI. For straight knurl tools these values can be translated to a distance value by dividing the TPI into one inch (1/33= .o303). This means if you measure the distance between the teeth perpendicular to the tool you will get .0303, .0476 & .0714 respectively. To find the total number of teeth on the tool you can take the circumference of the tool and multiply by TPI (π x D x TPI), then theoretically round to the nearest integer (whole number). More on this later.
Always keep in mind that Tools must have an Integer (whole number) number of teeth on the circumference. That is the key relationship between Tool and Material Diameter. It's real hard to make a partial tooth...not desirable...and where would we be then, trying to make "Good Knurls". @¿@
Diagonal and Diamond tools have a slightly different process that uses Transverse CP because of the angles. Transverse CP is measured Perpendicular to the teeth on the axis. It is calculated by the formula; Transverse TPI = Normal TPI x Cos (helix angle)...typically 30°=.866, so the Transverse values for the common diagonal CP tools are 28.57, 18.186, & 12.124, respectively. Note: these are not Integers.
The Conundrum
Let's take a look at this Transverse value in some detail. On the KP chart, notice that it includes the "Number of teeth". A KPR-214 is specified as 14TPI and has 34 teeth around the circumference. So if we calculate the circumference of a 3/4" tool (Cir.= π x D), we get 2.3562". Now we take that circumference and divide by the TPI (14) which equals 32.9867. Basically it should have 33 teeth, not 34??? Also note the specs for KPR-221 & 233 have 50T & 77T respectively, and do Not round to the calculated number of teeth on the circumference. The numbers for 221 & 233 are (2.3562 x 21=49.4801 & 2.3562 x 33=77.75). Technically, rounding these should have resulted 49 & 78 teeth respectively, Not 50 & 77. The final tell tale (or Tall Tale) is on Roll Forms definition page where they state that Diagonal and Diamond knurls are Specified in Normal TPI "Except" the KPR & KPL 214, 221, 233 & 235. Turns out that Form Roll makes the KP Series to "Pratt & Whitney and Armstrong" specs...which I have not been able to find other than a reference to Fed'l Spec.00-H-581a...whatever that is.
If we rearrange our formula for Transverse TPI to: Normal TPI = Transverse TPI/Cos 30 (.866) and plug in the numbers for their specified P&W/Armstrong Transverse Value, we get 14/.866=16 Normal TPI. Now take this and multiply by the Circumference (2.3562 x 16=38.0899 teeth on the circumference. Still not 34T! Without knowing the ACTUAL Transverse TPI of the "P&W/Armstrong" specs we have NO way to calculate the actual number of teeth on the tool.
Ok Then! Lets take the Circumference divided by the actual 34T (2.3562/34= .0693) to get our distance value between teeth. Then divide that into 1" to give us the CP/TPI of this tool (1/.0693=14.4300). Well that works if you get a-Round-To-It. It really means that TPI is an Approximate Value stated as an Integer, even in Transverse.
Intermission
Why is this thus? What is the meaning of this thusness? I suspect it has something to do with tooth height for proper forming, which I have not been able to find for CP Knurls. Although Dorian defines CP tooth depths (Pg. 10) as percentage of the illusive CP. Thusly I am unable to provide the meaning behind the anonymousness, nor anomaly. Over All I am more inclined with Artemus Ward and must decline on account of the Muchness of it all. @¿@
Oh! Lions, Tigers and Bears. Oh My! I really am boggled and bothered, Knowing there are Way Tooooo many standards out there, and me thinkest this could be a Doctoral Thesis. So all I can do is, reiterate my point that you must Know the specs of your Tools in order to have any luck at all attempting to create a "Good Knurl". Soooo, lets move on shall we, to how to use the CP Method to approximately determine our Material work diameter.
CP Rounding Method
Little Machine Shop technical page gives us a clue to how this works in their calculation table, however it needs some explanation as to how the values for "Pitch Factor" are derived and what it is exactly.
Calculating "pitch factor" (PF) for this operation is relatively simple. In these examples we will use a 3/4" tool and the Common CP/TPI for Course, Medium and Fine tools (14, 21, 33). Basically you multiply π x TPI, so for these tools PF will be: π x 14= 43.9823, π x 21= 65.9734, & π x 33= 103.6726.
Table 1 shows various pertinent values for these tools.
Pitch Factor - Table 1
So, What is PF?
What have we actually done here? With our formula PF=π x TPI you will notice that Pi comes in to play, meaning it is related to the Diameter or Circumference of the tool. The relation of Teeth on the circumference gave us a distance factor between teeth, but in the chart above notice that total teeth values are not integers like the TPI values, because of Pi. This is because TPI is actually an Approximate Integer value to the Actual Number of Teeth on the tool...like we saw above with the diagonal knurls.
Hopefully you remember the Dissertation on the ANSI Method and that for DP tools, Pnt = Circular Pitch on nominal Diameter which is derived by π/P. The Same formula works here if we apply the TPI Distance Value (1/TPI) for P and we get the formula PF=π /(1/TPI).
Here are the numbers: PF=π/(1/33, 21, 14TPI)=103.6726, 65.9734, & 43.9823
So PF in basic terms is the Number of times the distance between teeth goes into Pi and will almost relate to an integer ratio to the Material diameter. The rest of the procedure will tell how to apply the PF to get our Material Starting Size.
Applying PF to Determine Work Size
Once you have the PF you can now begin to calculate your approximate material diameter to start your knurling. Begin by multiplying your Pitch Factor by your material diameter, in this case we will use 1/2" stock. We get; .5 x 43.9823=21.99, .5 x 65.9734=32.9867, & .5 x 103.6726=51.8363. We now drop the decimal on these values, leaving us 21, 32, & 51, respectively.
The last step is to divide the integer values by the Pitch factor to arrive at; .4774, .4850, & .4919 for the sizes you need to turn your 1/2" stock to begin your knurling process. Here is a table showing the values calculated.
Work Size Calculator - Table 2
Epilogue
As you can see this method does give us some starting sizes for stock, but over all it is an Approximate Starting Place because of the conundrum of actual tool dimensions and the fact that it is a Rounded Value. In Practice it does pretty well in larger diameter (>1/2"), but the smaller stock diameters will be out more and more significantly because of the rounding, particularly on course or medium tools.
When using this method, if you find your knurled parts are not Good, take off .005 or .010 and try again. I will have some tips for this on the Techniques Page...if I ever get there. It's about trying to get that illusive Integer Ratio of diameters for material and tool.
Once again I reiterate that "Good Knurls" can only come from Knowing what tool you actually have and creating the proper material size to start making chips.