Ornamentation: Polishing Gems by the Numbers

Hope Diamond, By David Bjorgen

Hope Diamond, By David Bjorgen

Back in June I wrote some ideas of new rules for gems. I started with the Pathfinder Roleplaying Game rules for gems, which provide standard values and some variation.

I wanted a little more detail, though, so I drafted a table that separated ‘size’ and ‘quality’ so they could each be rolled with weighted frequencies (that is, ‘medium’ is more common than ‘tiny’ or ‘huge’, and ‘average’ is more common than ‘flawed’ or ‘excellent’).

This got me closer to where I wanted to be, but there was one thing still bothering me. The base values of the various gem grades grow unevenly.

Grade Name Standard Value
1 Least Semi-Precious 10 gp
2 Lesser Semi-Precious 50 gp
3 Semi-Precious 100 gp
4 Greater Semi-Precious 500 gp
5 Lesser Precious 1,000 gp
6 Greater Precious 5,000 gp

The progression “10, 50, 100, 500, 1,000, 5,000” is not a smooth curve, and the size and quality adjustments (or the random values from the original table) leave gaps in the values. I prefer a smoother progression (similar thoughts led to the smoother damage progression I posted about this time last year) that transitions between the different grades. I’ve got a couple of progressions in mind.

The first has the value of stone multiply by five every six steps, which means that a lower stone increased by three steps or a higher stone decreased by three steps will have the same value. For instance, a semi-precious stone (base 100 gp value) that has “+3 value” and a greater semi-precious stone (base 500 gp value) would both have the same value: 223 gp.

The second has a shallower curve and spans two steps on the base table, and the value of a stone is multiplied by ten every twelve steps. This would span two grades in the original table (semi-precious, 100 gp, to lesser precious, 1,000 gp), with greater semi-precious at the midpoint between them (316 gp). Each grade is still six steps from the one above or below it, with three steps being the midpoint between those.

Adjustment Exact 5 Exact 10 Approx  5 Approx 10 Ratio 5 Ratio 10
-12 4.00 10.00 4 10 0.04 0.10
-11 5.23 12.12 5 12 0.05 0.12
-10 6.84 14.68 7 14 0.07 0.14
-9 8.94 17.78 9 18 0.09 0.18
-8 11.70 21.54 11 22 0.11 0.22
-7 15.29 26.10 15 26 0.15 0.26
-6 20.00 31.62 20 31 0.20 0.31
-5 26.15 38.31 25 38 0.25 0.38
-4 34.20 46.42 35 46 0.35 0.46
-3 44.72 56.23 45 56 0.45 0.56
-2 58.48 68.13 60 68 0.60 0.68
-1 76.47 82.54 75 82 0.75 0.82
0 100.00 100.00 100 100 1.00 1.00
1 130.77 121.15 130 120 1.30 1.20
2 171.00 146.78 170 145 1.70 1.45
3 223.61 177.83 225 175 2.25 1.75
4 292.40 215.44 290 215 2.90 2.15
5 382.36 261.02 380 260 3.80 2.60
6 500.00 316.23 500 315 5.00 3.15
7 653.83 383.12 650 385 6.50 3.85
8 854.99 464.16 850 465 8.50 4.65
9 1118.03 562.34 1150 560 11.50 5.60
10 1462.01 681.29 1450 680 14.50 6.80
11 1911.81 825.40 1900 825 19.00 8.25
12 2500.00 1000.00 2500 1000 25.00 10.00

These have nice curves, very smooth and consistent, and they scale across all base values the same way. That is, regardless of the value of the stone I put in ‘position 0’, the curve will have the same shape. There is no difference, when using the ‘Exact 5’ method, between starting with a 100 gp stone and adding 6 and 12 to the value (100 gp, 500 gp, and 2,500 gp respectively) and starting with a 500 gp stone and subtracting 6 value, using the base, and adding 6 value (also 100 gp, 500 gp, and 2,500 gp respectively).

Given that I usually prepare things at my computer, it’s no big deal to have a spreadsheet open… but I don’t care for systems that depend on spreadsheets. Looking it over I see some approximations that might work, usually rounded to two or three significant figures. I’d probably still want a calculator, though. Can it be simplified more?

Probably. Given that I’m usually going to be within five points of where I started, I’m going to drop the second option (multiplying or dividing by ten for every twelve steps)

Adjustment Exact 5 Approx  5 Ratio 5 Fraction
-6 20.00 20 0.20 1/5
-5 26.15 25 0.25 1/4
-4 34.20 35 0.35 1/3
-3 44.72 45 0.45 4/9
-2 58.48 60 0.60 6/10
-1 76.47 75 0.75 3/4
0 100.00 100 1.00 1
1 130.77 130 1.30 4/3
2 171.00 170 1.70 10/6
3 223.61 225 2.25 9/4
4 292.40 290 2.90 3
5 382.36 380 3.80 4
6 500.00 500 5.00 5

This gets me down to a total of seven numbers to remember: 1, 4/3, 10/6, 9/4, 3, 4, 5. If I’m reducing quality I simply invert the fractions… and these are fractions I can more or less do in my head (my Grade 9 Maths teacher spent a lot of time drilling us on this sort of thing, thanks Ray!)

Closing Comments

To be completely honest, I don’t know if I’d actually use this in practice. Mathematically I like it, arithmetically it looks like it could be annoying. The approach I described in June (which amounts to “plus or minus ten percent per change”) is simpler, and ‘simpler’ means a lot to me. On the other hand, the fractions above aren’t that difficult for me to work with.

I need to think about this some more, I suppose.

Postscript

I have considered another model that’s even simpler. Each grade of stone is worth three times as much as the previous, and each point of value difference changes the value by ten percent. Thus, each grade at +5 (150% of base) is equal in value to the next grade at -5 (50% of 300% of current base = 150%). It makes for a continuous curve, and relatively simple arithmetic. I’m not as enamored of it, though.

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