House Rules - Logarihmic/Geometric Scales

[Hopkins Vioreo]

This is my first time posting on this forum. I'm actually quite new to GURPS, so part of the reason of writing this post is to see how these House Rules attempt fits within GURPS (it could be the case that they just don't work well within the structure of the game).

MOTIVATION
GURPS is intended to be a generic role-playing game, able to grasp a huge variety of scenarios to work with. Sometimes, some of these scenarios just demand really off-the-chart attributes. One example is when attemp to run an anime-based scenario, such as Naruto. Other examples involve truly ludicrous Marvel characters, such as The Hulk, or Kaijus such as Godzilla. This page, for example, had to give Godzilla a ST of 7500; ridiculous for a human, but then again, it IS Godzilla.
This is the gist of the problem I'm attempting to tackle here: how to make GURPS better work with such huge demands, if needed. The basic idea? Logarithmic/Geometric scaling.

THE IDEA
By "logarithmic/geometric scaling", I mean a scale that progress more in terms of geometric proportion (think the Richter Scale). The idea is making each 1 additional point in an attribute (or skill, or task difficulty) represent an improvement of X% (I like to use 25%) from your current levels. I will use the Strenght as an example to better explain.

The idea is that each 1 point of Strenght is a 25% increase from the level before. So ST=11 is 25% stronger than ST=10, which is itself 25% stronger than ST=9. But notice the geometric progression between ST=11 and ST=9: 11 here is 25%*25% stronger than 9, which is around 56% stronger than 9. To graps the difference between 2 levels, you just need to do 1.25^(LevelA-LevelB).

The power of logarithmic scales is that they can get very big very quickly. We can see this by looking at Basic Lift levels when applying this scale. According to 4e rules, with a ST=10, your character can lift 10kg above its head, so let's use this as the origin point.
ST=11 is 25% stronger, so your character would lift 12.5kg
ST=18, considered the pinacle of human achievment, would mean a lift of ≈ 60kg.
ST=21 lifts 116kg, 10x stronger than average human
ST=28, such as C31R07, is able to lift ≈555kg
Now, let's crank up these numbers to see what we get
ST=31 is ≈1084kg, so this is already 100x stronger than average human
ST=100 and you already can lift 5 billion kg, the weight of Mt Everest according to some sources.
ST=248 is on the order of 10^24kg, the same scale as the weight of planet Earth
ST=310 lifts on the order of 10^30kg, the weight of the Sun.
ST=547, and you can lift 10^53kg; this is the weight of the entire visible universe.
So, thanks to the power of logarithmic scales, in order to have a character able to lift the entire universe, you just needed to give him an ST=547. It's high, but consider the Godzilla with ST=7500 from earlier and we'll see we can work with more tameable values; things only really explode in scale with these values in the range of the hundreds, within the 10-18 range of "normal human characters" it ranges from 10kg to 60kg, which I think it is reasonable.

DISCUSSION
So, above I showed the gist of my idea: to interpret the Attributes progression in a logarithmic fashion. But as I said on the beggining, I'm quite new to GURPS, and don't have a full grasp of the entire system yet, so it is possible this approach can end up bringing more problems than solutions to the table.
For example, I used ST as an example, but if I used IQ, someone with an IQ=18 would be ≈6 times more intelligent than the avg human... does that make sense? Also, being guided by a formula like 1.25^(Att-10), if you had ST=0, you would still be able to lift 1.25^(0-10)≈1kg above your head; you can get lower values, but you would have to deal with negative Attribute values, I'm not sure if they make sense with GURPS.
As I said, this is just an attempt to solve the scalability problem that some players might find when attempting more ludicrous campaigns. Maybe the best solution for such cases are the ones suggested on that Naruto forum: Rescale and Refactor.
But in any case I'm posting this here, just to see how good could it work.

[Anthony]

You might want to look up Knowing your Own Strength, in Pyramid 3/83.

[whswhs]

In standard GURPS, ST 10 gives Basic Lift of 20 pounds, and ST 20 gives Basic Lift of 80 pounds. If you take that as a logarithmic scale, then each +5 ST gives x2 Basic Lift. So then +50 ST would give x1000 Basic Lift: ST 60 is BL 20,000 lbs., ST 110 is 20,000,000 lbs., and so on.

The fifth root of 2 is about 1.15, so a 15% increase for +1. I would prefer that to your 25% increase, I think. The series is +15%, +30%, +50%, +75%, and +100% . . .

[Anthony]

Originally Posted by whswhs

The fifth root of 2 is about 1.15, so a 15% increase for +1. I would prefer that to your 25% increase, I think. The series is +15%, +30%, +50%, +75%, and +100% . . .

25% is roughly one decibel (actually 25.89...%) giving x10 per +10.

[whswhs]

Originally Posted by Anthony

25% is roughly one decibel (actually 25.89...%) giving x10 per +10.

Yes, I know that. I prefer x4 per +10.

[Anthony]

Originally Posted by whswhs

Yes, I know that. I prefer x4 per +10.

The advantage of a decibel scale is that x2 (3.01), x4 (6.02), x5 (6.99), x8 (9.03), and x10 (10) are all extremely close to integer values. On a x4 per 10, x2 (5), x4 (10), and x8 (15) remain near integer values, but x5 (11.6) and x10 (16.6) are not (x4 per +10 handles x3 slightly better -- it's 7.92, whereas it's 4.77 on a decibel scale -- but multiples of 3 are used a lot less than multiples of 5 and 10).

[Pursuivant]

Reworking GURPS ST in various ways is a popular house rule. A web search for topics like "variant GURPS ST" or "alternate GURPS ST" will bring up some interesting ideas for you to consider when designing your own variant.

It's not Log/Geometric Scales, but one way to handle massive ST scores is to use "decade" (D-scale) or "century" (C-scale) ST, HP and DR, where you divide those traits by 10x or 100x, respectively when creatures of similar scale are fighting each other.

Divide other applicable values, like weight, as well when they apply to the character.

E.g., A Gojira vs. Rodan slugfest would use C-Scale ST and damage, giving Gojira just ST 75, HP 75 and DR (massive number)/100. Building weight, HP and DR also gets divided by 100x, allowing the dueling kaiju to smash those puny human obstacles.

[dataweaver]

Originally Posted by Anthony

25% is roughly one decibel (actually 25.89...%) giving x10 per +10.

Originally Posted by whswhs

Yes, I know that. I prefer x4 per +10.

(Brief note: this second one is identical to ×2 per +5.)

I think the scale to choose ought to aim to resemble existing progressions in the vicinity of ST 10: say, from somewhere around ST 9 to ST 16, more or less, the LogST should have roughly the same lifting capacity as the RAW ST. The former grows far too quickly for that.

That said, I also prefer to use scales that are defined as ×10 per +n, where n is at least rational and preferably an integer. That's for the rather pragmatic reason that if you don't, your scale will never fall into a repeating pattern.

Mathematically, I also like n to be a multiple of 3, because these progressions often deal with cube roots; and when switching to a log scale, taking the cube root of the multiplier is the same as dividing the scale factor by 3. It's one reason why I like the Speed/Range progression, which is ×10 per +6. Another is that square roots amount to halving the scale factor; so n being even is nice, too.

However, ×10 per +6 is far too rapid a growth rate for ST: in GURPS RAW, ST 16 can lift roughly 2.5× as much as ST 10; the Speed/Range scale replaces that with 10×. Even decibel is too fast, with ST 16 lifting 4× as much as ST 10. A twelve-step progression does a bit better with ST 16 lifting a little over 3× as much as ST 10.

I'd say that a best-fit progression would be 15 steps. Here's ST 10–25 according to this progression, along with the RAW ST that would have the same lifting capacity:

ST 10=×1.00 (RAW ST 10.0)
ST 11=×1.17 (RAW ST 10.8)
ST 12=×1.36 (RAW ST 11.7)
ST 13=×1.58 (RAW ST 12.6)
ST 14=×1.85 (RAW ST 13.6)
ST 15=×2.15 (RAW ST 14.7)
ST 16=×2.51 (RAW ST 15.8)
ST 17=×2.93 (RAW ST 17.1)
ST 18=×3.41 (RAW ST 18.5)
ST 19=×3.98 (RAW ST 20.0)
ST 20=×4.64 (RAW ST 21.5)
ST 21=×5.41 (RAW ST 23.3)
ST 22=×6.31 (RAW ST 25.1)
ST 23=×7.36 (RAW ST 27.1)
ST 24=×8.58 (RAW ST 29.2)
ST 25=×10 (exact; RAW ST 31.6)

…and after that, it repeats an order of magnitude higher:

ST 25=×10.0 (RAW ST 31.6)
ST 26=×11.7 (RAW ST 34.1)
ST 27=×13.6 (RAW ST 36.9)
ST 28=×15.8 (RAW ST 39.8)
ST 29=×18.5 (RAW ST 43.0)
ST 30=×21.5 (RAW ST 46.4)
ST 31=×25.1 (RAW ST 50.1)
ST 32=×29.3 (RAW ST 54.1)
ST 33=×34.1 (RAW ST 58.4)
ST 34=×39.8 (RAW ST 63.1)
ST 35=×46.4 (RAW ST 68.1)
ST 36=×54.1 (RAW ST 73.6)
ST 37=×63.1 (RAW ST 79.4)
ST 38=×73.6 (RAW ST 85.8)
ST 39=×85.8 (RAW ST 92.6)
ST 40=×10 (exact; RAW ST 100)

…and scaling down:

ST –5=×0.10 (RAW ST 3.16)
ST –4=×0.12 (RAW ST 3.41)
ST –3=×0.14 (RAW ST 3.69)
ST –2=×0.16 (RAW ST 3.98)
ST –1=×0.19 (RAW ST 4.30)
ST 0=×0.22 (RAW ST 4.64)
ST 1=×0.25 (RAW ST 5.01)
ST 2=×0.29 (RAW ST 5.41)
ST 3=×0.34 (RAW ST 5.84)
ST 4=×0.40 (RAW ST 6.31)
ST 5=×0.46 (RAW ST 6.81)
ST 6=×0.54 (RAW ST 7.36)
ST 7=×0.63 (RAW ST 7.94)
ST 8=×0.74 (RAW ST 8.58)
ST 9=×0.86 (RAW ST 9.26)
ST 10=×1 (exact; RAW ST 10)

For contrast, here's what a decibel scale would look like, for ST 10–20:

ST 10: ×1.0 (RAW ST 10)
ST 11: ×1.26 (RAW ST 11.2)
ST 12: ×1.58 (RAW ST 12.6)
ST 13: ×2.00 (RAW ST 14.1)
ST 14: ×2.51 (RAW ST 15.8)
ST 15: ×3.16 (RAW ST 17.8)
ST 16: ×3.98 (RAW ST 20.0)
ST 17: ×5.01 (RAW ST 22.4)
ST 18: ×6.31 (RAW ST 25.1)
ST 19: ×7.94 (RAW ST 29.1)
ST 20: ×10 (exact; RAW ST 31.6)

And ×4=+10 does this:

ST 10: ×1.00 (RAW ST 10.0)
ST 11: ×1.15 (RAW ST 10.7)
ST 12: ×1.32 (RAW ST 11.5)
ST 13: ×1.52 (RAW ST 12.3)
ST 14: ×1.74 (RAW ST 13.2)
ST 15: ×2 (exact; RAW ST 14.1)
ST 16: ×2.30 (RAW ST 15.2)
ST 17: ×2.64 (RAW ST 16.2)
ST 18: ×3.03 (RAW ST 17.4)
ST 19: ×3.48 (RAW ST 18.7)
ST 20: ×4 (exact; RAW ST 20)

My issue, to the extent that it matters, is that lifting capacity is nearly a full point behind RAW ST throughout most of this range. If you find that acceptable, and you're willing to either accept that you'll never hit exactly a power of 10 or, with a slight adjustment, are actually using a ×1000=+50 sequence, this isn't half bad.

[dataweaver]

Originally Posted by Anthony

You might want to look up Knowing your Own Strength, in Pyramid 3/83.

The real fun is if you can also get this to work with Conditional Injury, which puts damage on a log scale.

[Anthony]

Originally Posted by dataweaver

The real fun is if you can also get this to work with Conditional Injury, which puts damage on a log scale.

Well, the problem is that CI doesn't actually do that. It puts wounding on a log scale but still relies on a linear scale for DR.