I have heard about games and seen some discussions on using Logarithmic Scaling for ground scales to compress the ranges so that long range weapon platforms (notably artillery or tank guns) can be used on the same gaming board with short range weapons. It is a straight forward mathematical exercise to do this, but the compression factors will create some weird visual anomalies and necessitate using sophisticated Die Modifiers (DM) adjustments at those long ranges that seem a bit out of kilter with the actual measured tabletop range.
What follows is my analysis of the M16A1 rifle used in Vietnam as an example (all mathematical errors are clearly mine and reflect the 35 years since I did real math) and trying to develop an effective logarithmic tabletop scale for 28mm games.
Wikipedia has a nice article on M-16s referencing army weapons data when fired from rest cited in this report (http://www.dtic.mil/dtic/tr/fulltext/u2/a228398.pdf) and duplicated in the M-16 Wikipedia entry (https://en.wikipedia.org/wiki/M16_rifle#Range_and_accuracy).
Let’s apply the probabilities to a D20 roll, where you need to roll 20 or less and we will apply a positive die modifier based on the range. So for the case of 300 Meters the M16A1 DM is Zero, for 600 Meters the DM is +5, and for 800 Meters is +12. Per the report, we will ignore ranges beyond the maximum lethal range of 900 Meters for the M-16A1.
Now translating those ranges into log values where the 10 times the log of the range in meters will be represented as inches on the table, we get:
Now to plot them and use a polynomial trend-line to estimate the DM at 900 Meters (max lethal range), the estimated DM for 900 Meters (lethal range) is +17 as shown below by the orange data and trend-line:
So for the M-16A1 at Tabletop Ranges 0 to 25 inches DM = 0, 26 DM +1, 27 inches DM +3, 28 inches DM +5, 28.5 inches DM +9, 29 inches DM +12, 30 inches DM +17 and cannot shoot past 30 inches on the table.
The net is that there is nothing wrong with using Log ranges, but the game system needs to use appropriate DMs. This leads to some very wild DM for a small squad of 28mm minis on 1 inch (25mm) bases. Out to 26 inches, ignore the DM for range and then suddenly a mini at 30” has a DM = +17 to miss while his mate 3 figures over is still at DM = 0. This would make playability very difficult.
Alternatively, you could just use this data to estimate the Range DM =0 to 26 inches and make that the maximum range. So what is interesting about that is that the US army treats max range as 300M (which is 26 inches in our 28mm log scale) and does not train to shoot the weapon further than that. To qualify you shoot a series of pop-up targets at 50, 100, 200, and 300 meters with most at the shorter ranges AND need to hit 50% of them.
So taking this a bit further, let’s include estimates for shooter accuracy. What the data above does not account for is the ability of a soldier to accurately aim the weapon at range nor the ability of the shooter to bring the weapon to bear. As range extends, two things happen – the shooter needs to compensate for bullet drop and the flight path of the bullet is an expanding cone due to barrel movement. (The barrel movement reflects things like how well the weapon fits the shooter, the stance, how rigid the gun is held, nerves, etc.) Added to the effects of visual acuity (mirage effects) and the effectiveness of obscuring cover, camo, and target darkening (target darkening – have you ever noticed that the bright red car 20 feet in front of you turns very dark when it is several hundred meters down the road). These are in essence accuracy effects that degrade linearly. Out to 50 meters, the shooter should have no issues seeing the target, holding his aim so that the flight path cone intersects a man-size target, and the target will still be fully visible. So for an average shooter using regular vision and iron sights shooting off-hand, let’s assign a hit probability of 80% due to nervousness in combat at 100M and by the time we get to 300M we will assume that the hit probability has decreased to 50%. Since this is a linear degradation, the DM becomes 20 at ~640M. (You can adjust this to reflect how accurate you think your Green, Regular, Veteran, and Elite troops will shoot in YOUR game. I took a swag at it based on my observations of Regulars shooting in preparation for war and then how nervousness effected them.)
So with this combined data, on our 28mm log range tabletop we get these DMs:
The net is that a normal shooter cannot hit past 27 Inches and that you get dramatically changing DMs for all ranges past 20 inches.
Personally, I prefer to use realistic ground scales that match the minis. That means a 4 foot board is really only about 100 meters and that anything other than short-barreled pistols are always in range. As long as plenty of cover and terrain exist this is not a factor. However, this does mean that artillery ends up on the table as an objective instead of a realistic threat to infantry (unless it is shooting over open sites).
This also means that any DMs for range should be linear. Of course, we usually use range bands, but since the DMs are linear, if you use the average DM for the range band it is still a close estimate.
So if you are trying to figure out if the rules you use reflect some real ballistics and shooter ability, you can perform a similar analysis to see if the DM make since. If not, then it is a game not a simulation. Both can be fun and for the case of Toy Soldiers a game seems adequate to the purpose.