Monday, December 26, 2005

Repeating Scale

This article proposes a modified Strength Scale which follows a repeating pattern of values, and offers a general formula for deriving your own Scale. Some people are happy with Scale as it is, and this article is not intended for them. Also, some math is involved in deriving the formulas, but you can safely skip it.

Abstract

This article proposes a modified Strength Scale which follows a repeating pattern of values, and offers a general formula for deriving your own Scale. Some people are happy with Scale as it is, and this article is not intended for them. Also, some math is involved in deriving the formulas, but you can safely skip it.

If It Ain't Broke...

Fudge Mass Scale is a very elegant mechanic, and it already works just fine. There are some rounding errors in the numbers, and the question of whether it describes Strength or Mass is both vague and complicated... but Fudge is not a precise game, and it gets the job done with little fuss. So if it works, why mess with it?

Because it is table-based.

Fudge is a very intuitive game. The rules are quickly memorized and easily applied with simple rules-of-thumb. You can run everything out of your head... everything except Scale. As nice as the Scale mechanic is, it requires a table lookup. The table is not easily memorized, and the rounding errors are large enough that a pocket calculator cannot be used as a reliable substitute.

Looking at the Fudge Scale table in the rules, you'll notice sequences like 10-15-25-40 that seem to be repeated in various places. But the locations seem to be irregular, and the numbers don't always match.

I have been unable to find a clear pattern in the standard table, but the illusion of a pattern inspired me to try to create a Scale progression that does follow a repeating pattern. This article is the result.

The Fundamentals

Mass Scale in Fudge is described in terms of Mass, but defined in terms of Strength. For purposes of this article we do not care about the Strength versus Mass question, but we do need a basic definition to build upon. This can be found in Steffan's design notes:

"I'm sometimes asked why I chose such an awkward set of numbers for the Strength and Speed scales. Why not some more common logarithm? The answer is: reality test. I did a lot of research into human strength and speed, and found that the strongest human is about five times as strong as the average human..."

"So if the strongest and fastest are Legendary, and the average is Fair, that means Legendary Strength should be five times stronger than Fair Strength"

(As an aside, note that this assumes that Legendary is the top of the scale for a realistic game, not Superb. If you consider Superb to be the maximum human limit, and Legendary to be superhuman, then you should adjust the scale so that scale +3 has a 5x value multiplier. More on that later.)

Ideally, then, we'd like the Fudge Scale to provide x5 Strength at Scale +4. This is our foundation.

Standard Scale

Mass Scale is exponential, based on the general formula that each +1 Scale indicates a Strength 1.5 times greater than the previous value. Reduced to a formula,

  • strength = 1.5 ^ scale

If you remember your logarithms from high school, you can also perform the inverse calculation,

  • scale = log(1.5) strength

(Note: Excel spreadsheet formulas are provided at the end of the article)

Inspiration from Hero

The Hero System has a similar scale mechanic for strength: it is exponential and allows creatures of greatly differing sizes to interact meaningfully. But the Hero strength scale has one feature Fudge does not: it is regular, doubling its value over every interval of 5. That means that +5 strength means "twice as strong", +10 means 4x as strong, etc. This is a very handy feature, especially since multiples of 5 are by far the most common strength values in the game.

The Hero strength scale can be reduced to a formula just as the Fudge Scale can:

  • lifting capacity = 2 ^ ((STR - 10) / 5)
  • STR = 10 + 5 x log(2) lifting capacity

Designing a New Scale

Now we need to come up with a new formula. Following the Hero System example, and using our base assumption of x5 Strength at +4 Scale, an obvious and simple formula is:

  • strength = 5 ^ (scale / 4)

This provides all Scale values, and is nearly identical to the original, except that every +4 Scale is x5 Strength. But it does not repeat in an obvious pattern; its predictability is really limited to multiples of +4.

If you remember your laws of exponents from high school, the formula can be rewritten like this:

  • strength = (5 ^ (1/4)) ^ scale

That looks complicated, but what we've done is isolate the scaling constant to a single term that can be pre-computed:

  • strength = constant ^ scale
  • constant = 5 ^ (1/4) = 1.495349

so,

  • strength = 1.495349 ^ scale

Each level is 1.495 stronger than the previous level. We can take this a step further and create a generalized formula. If we want Strength increase by (multiple) over a given (range) of Scale values,

  • constant = mult ^ (1 / range)

For example, if Strength increases by x5 for every +4 Scale, mult = 5 and range = 4. I'll use the notation mult/+range (5/+4) to represent this.

Now that we have a generalized formula, let's think about what sort of progression is the most useful.

The 5/+4 scale is better than the standard 1.5/+1 scale, because it is easy to remember that every +4 Scale is five times stronger, and the 1.5/+1 is still close enough to be a useful approximation. This is not only true for Scale values of 0/5/10/etc, but any value: Scale +7 is five times stronger than Scale +3, and Scale +1 is five times stronger than Scale -3. Similarly, every +8 means a multiple of x25, and +12 means a multiple of x125.

So it's now a lot easier to "ballpark" a Scale, but still pretty rough. We could improve the ball-parking if we are willing to fudge the x5 aspect. A very nice scale that would work well for Fudge is 2/+2 (every +2 Scale means twice as strong). This puts Legendary at only x4 Strength, but it is VERY easy to estimate nearly any Scale value you need, especially if you work with computers and are familiar with the powers-of-2 progression. A dragon 100 times stronger than a man? Well, 128 is 2^7 which equates to +14... so 13 or 14. If you need an in-between value, the multiple is roughly 1.4. The downside is that this Scale diverges quite a bit from the standard Fudge Scale, especially at large values: Scale +20 is x1024, instead of x4000 using the standard Fudge Scale. This is not a problem if you write your own material or estimate everything, but not so good if you use published or shared material.

Better than either 5/+4 or 2/+2 would be a Scale based on powers of 10, because increasing the Scale one interval merely adjusts a decimal place. Looking at the standard Fudge Scale, we see powers of 10 at +6 and +17. A smaller interval is more useful and easier to work with, so let's try a Scale based on 10/+6.

Using the formula, our constant is 1.467799, which is close enough to 1.5 to fudge the difference. The Scale diverges a bit from the standard Scale, but within at least +1 or -1 out to Scale 20 or -20. Few campaigns will ever exceed that range. Best of all, this is a repeating Scale: the values shift one decimal place every six intervals. Here is the basic Scale:

Scale Value
+0 1.0
+1 1.5
+2 2.0
+3 3.0
+4 4.5
+5 7.0
   
+6 10
+7 15
+8 20
+9 30
+10 45
+11 70
   
+12 100
+13 150
+14 200
+15 300
+16 450
+17 700

Each group of six Scale levels is a power of 10:

Scale Value
-18 0.001
-12 0.01
-6 0.1
+0 1
+6 10
+12 100
+18 1,000
+24 10,000
+30 100,000
+36 1,000,000
+42 10,000,000
+48 100,000,000

(The table above is rounded for easy memorization, but you can compute and use the precise values without losing the repeating aspect.)

Estimation is easy: adjust by a multiple of six, read the value, and multiply back out.

  • Scale +17: subtract 12 (x100), Scale 5 is x7, result is 700
  • Scale +45: subtract 42 (ten million), Scale 3 is x3, result is 30 million
  • Scale -22: subtract 18 (x1000), Scale 4 is 4.5, result is 0.0045

This is also a nice estimator, since we often think in terms of twice (+1), three times (+2), five times (+4), or ten times (+6).

For Those Who Do Not Use Legendary

If you don't use Legendary levels and consider Superb to be the the maximum human ability, then Strength should be x5 at Scale +3. The most obvious adjusted Scale in this case is 5/+3, or a Scale constant of 1.71. This one is not so easy to repeat (if you like that feature), although 10/+4 is fairly close. Anyway, this article should provide you with some tools to explore this further if you are interested.

Excel Spreadsheet Formulas

  • Standard Mass Scale:
    • value = POWER(1.5, scale)
    • scale = LOG(value, 1.5)
  • Generalized Scale
    • mult = Y
    • range = X
    • constant = POWER(mult, 1/range)
    • value = POWER(constant, scale)
    • scale = LOG(value, constant)
  • Standard Fudge Scale: mult = 1.5, range = +1
  • Superb Scale: mult = 5, range = +3
  • Precise Scale: mult = 5, range = +4
  • Repeating Scale: mult = 10, range = +6