Previously I've talked about the logarithmic scales that Explore uses for measurements. An example of a logarithmic scale is 1, 2, 4, 8, 16. This is a very useful system for scaling things up/down, in particular by choosing the scales carefully you get +1 on one scale gives you +1 on another scale. For example, +1 weight corresponds to +1 height and also +1 lift.
I've been working on tweaking the scales I use, which has required coming up with some new progressions. Anyone playing the game doesn't have to care how I came up with the numbers, but over at Gaming Ballistic, Douglas Cole has been talking about why he loves the GURPS logarithmic scale so I thought I should briefly discuss how I came up with my scale rather than any other version.
In fact one observation I have from playing Explore is that the players don't need to even think about the scale to play the game, it's just the underlying reason why the game fits together without any sharp corners to get snagged on.
In fact one observation I have from playing Explore is that the players don't need to even think about the scale to play the game, it's just the underlying reason why the game fits together without any sharp corners to get snagged on.
Doubling Every N
The first approach to creating a logarithmic scale is one where it doubles every +N (e.g. 2, 3, 4, 6, 8, 12… doubles every +2).
The scale I choose then is:
N, N+1, N+2, … 2N-1
2N, 2(N+1), 2(N+2),… 2(2N-1),
4N, 4(N+1), 4(N+2),… 4(2N-1),
For example, for doubling every +4 the scale is:
4, 5, 6, 7,
8, 10, 12, 14,
16, 20, 24, 28,
You then extend this in the reverse direction to low values:
1, 1 1/4, 1 1/2, 1 3/4,
2, 2 1/2, 3, 3 1/2,
This has several good points:
- It has integers for high values, nice fractions for low values.
- Each scale contains all the earlier scales, so all include 1,2,4... and all scales apart from the first two include 10, 20, 40 etc.
The downside is that at high values the scale has values such as 1024, instead of easy to work with numbers such as 1000.
Tenfold Every N
The alternative approach is to a tenfold increase every +N (e.g. in GURPS you have 1, 1.5, 2, 3, 5, 7, 10, 15, 20 etc.).
With this approach, however, if N is large (i.e. there is a slow increase) you get a lot of decimals instead of nice fractions. In our earlier doubling every 4 scale, we would be x10 every +13. This gives:
10, 12, 14, 17, 20, 24, 29, 35, 41, 49, 59, 70, 84, 100
You can tidy this up a bit, e.g. round to nearest 5 or 10, but below 10 the values become nasty decimals.
1, 1.2, 1.4, 1.7, 2, 2.5, 3, 3.5, 4, 5, 6, 7, 8.5, 10,
(10), 12, 14, 17, 20, 25, 30, 35, 40, 50, 60, 70, 85, 100
This isn't an issue in some games, but a slow increase like this is exactly what has proved to be necessary in Explore.
Combining The Approaches
Combining The Approaches
The solution I'm favouring is to use the doubling scale until you get to a value which is 10x a previous value and then switch to the 10x scale. For example you can switch at 10, 20, 40, or 80. (Note you cannot switch until after the scale has only got 1/2 fractions, i.e. above 2=>20 in the example.)
1, 1 1/4, 1 1/2, 1 3/4,
2, 2 1/2, 3, 3 1/2,
4, 5, 6, 7,
8, 10, 12, 14,
16, 20,
and then switch to 10x...(20), 25, 30, 35,
40, 50, 60, 70,
80, 100, 120, 140,...
Swapping from feet to inches, pounds to ounces
When you get to small sizes it can be useful to switch to a smaller unit, such as from feet to inches, or pounds to ounces. All scales contain 16, and almost all contain 12. Hence you can start the scale using the small unit (e.g. inches), and when you get to one large unit (e.g. foot) you start again at 1.
For example, there are 16oz to 1lb, so we can have
1oz, 1 1/4 oz, 1 1/2 oz, 1 3/4 oz,
2 oz, 2 1/2 oz, 3 oz, 3 1/2 oz,
4 oz, 5 oz, 6 oz, 7 oz,
8 oz, 10 oz, 12 oz, 14 oz,
16oz = 1lb
… and then restart at 1 on the lb scale.
(1lb), 1 1/4lb, 1 1/2lb, 1 3/4lb,
2lb, 2 1/2lb, 3lb, 3 1/2lb,...
Sometimes you might have to move the join point to be higher up so that the fractions are units on the smaller scale e.g. to avoid having 1 1/8 feet.
Why Logarithmic Not Exponential?
The values in the scale increase exponentially, so why do we call the scale "Logarithmic"? It's because we're saying Dwarves are height -4 compared to a Human. So the scale height in feet => height rank is the act of taking the logarithm of the height. This makes it like logarithmic scales in science.
Does this all matter?
Probably not!
Anyway I can now go back to talking about proper stuff. Next up is how much can you lift and how fast can you run - both of which incidentally use these scales - but which are also of some actual practical use!
Why Logarithmic Not Exponential?
The values in the scale increase exponentially, so why do we call the scale "Logarithmic"? It's because we're saying Dwarves are height -4 compared to a Human. So the scale height in feet => height rank is the act of taking the logarithm of the height. This makes it like logarithmic scales in science.
Does this all matter?
Probably not!
Anyway I can now go back to talking about proper stuff. Next up is how much can you lift and how fast can you run - both of which incidentally use these scales - but which are also of some actual practical use!
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