Monday, November 29, 2021

A Metric System for Faerun

In many tabletop roleplaying games, the round of 6 seconds, and the space or square of 5 feet serve as basic units of measurement for game mechanics. Just for fun, let’s try to actually construct a system of measurements using these as our base units for time and distance.

The self-imposed rules of the exercise are as follows:

  • Derived units are expressed simply in terms of base units. (Coherence)
  • Larger or smaller units use a consistent exponential power (Though I’m not sticking to decimal).
  • And if possible, the resulting system should feel psuedo-medieval, be intuitive to a 21st century American, and be useful for rules of thumb in a tabletop rpg.


Units of Time

One of the starting points is that our base unit of time is the round, which is equal to to 6 seconds.

Here’s an interesting observation: A 24-hour day is equal to 14400 = 120 times 120 rounds. This suggests the use of a base-120 system of units, which is handy for that psuedo-medieval feel, because “hundred” used to refer to units of 120.

Whenever you see “hundred” in the rest of this post, you should interpret it as a “long hundred”, meaning 120. For example:

  • The base unit of time is the round (6 seconds).
  • There are a hundred rounds in a turn (12 minutes),
  • a hundred turns in a day,
  • a hundred days in a season (120 days),
  • and roughly a hundred seasons in a career (40 years).

And the nice thing about 120 is that it has lots of devisors, so all the other units of time we’re used to slot right in 1.

And a year of 365 days is a bit more than 3 seasons, but eh. This is a fantasy world, so if that bothers you, then you can totally just make the year a bit shorter. Nobody will even notice. I’ve also seen “watch” (sixth of a day, 4 hours, 20 turns) and “march” (third of a day, 8 hours, 40 turns) used as convenient rpg units.

Also, I don’t see such a tiny unit being useful in ttrpg time accounting, but we can also go one level smaller. 1/120th of a round is 50 milliseconds. Just for fun, let’s call this a tick2.

Units of Length

The size of a square on a tabletop rpg grid is 5 feet. Calling the unit of length a “square” would be rather confusing. Classically, the Roman pace was 5 Roman feet, so let’s call our base unit of length the “pace” (even though it’s slightly longer than a Roman pace). And then in-universe, “space” could be a pormanteau of “square pace”, making the battle-grid diagetic.

  • There are a hundred fingerwidths (half an inch)3 in a pace.
  • The base unit of distance is the pace (5 feet).
  • There are a hundred paces in a stadium (600 feet),
  • and a hundred stadiums in a hex (about 13.6 miles or 11.85 nautical miles).

It’s more traditional to have a 12 mile hex in rpg maps, but we can comfort ourselves by pretending that all those old rpg modules meant nautical miles.

Units of Volume

To get units of volume, we simply cube our unit of length, and then divide this cubic pace by 120.
  • There are a hundred cups4 in a pony keg or firkin (about 7.8 gallons),
  • and there are a hundred firkins in a cubic pace, also called a case or cord (125 cubic feet).

And if you want the in-betwen units, it’s quite simple to remember: There are two cups in a pint, two pints in a quart, two quarts in a pottle, fifteen pottles in a firkin, two firkins in a kilderkin, two kilderkins in a barrel, two barrels in a hogshead, and fifteen hogsheads in a cubic pace. I didn't make any of these units up.

Units of Speed

The unit of speed is paces per round. It’s coherent, and that’s the unit of speed which is actually used in game mechanical terms. Easy peasy.

A side of effect of the consistent exponents is that paces per round, fields per turn, and hexes per day are all equivalent units. That’s great for reckoning distances and travel times. Take a look at how many spaces a character can move in a round on the battlemap. That’s the same number of hexes they can move in a day on the worldmap, if they travel all night. Divide by three to get a more reasonable 8 hour travelling day, and double it to get a speed which will drive them to the brink of exhaustion.

Here is a table of speeds for comparison:

Description Paces/Round Hex/8hr mi/hr km/hr
Sloth Top Speed 1/3
0.2 0.3
Tortoise Speed 1
0.6 0.9
Crawling 3 1 1.7 2.7
Brisk Walk 6 2 3.4 5.5
Typical Run 12 4 6.8 11.0
Sailing Ship5 12
6.8 11.0
Athletic Run 18 6 10.2 16.5
Bike Ride 24 8 13.6 22.0
Clipper Ship 30
17.1 27.4
Cheetah Sprint 120 40 68.2 109.7

Horses can gallop at up to 60 paces per round, but this speed isn’t sustainable. Over a long distance, horses can travel at about the same average speed as a person on foot: 2-3 hexes per day.6

And a little calculator:

Sidenote: Units of Acceleration

The unit of acceleration is paces per round per round, or pprpr for short.

Acceleration from gravity at the Earth’s surface is about 232 pprpr. That’s somewhat close to 240 = 2 * 120, and it’s a fantasy world, so if that’s the kind of thing that bothers you, you can crank up gravity about four percent to make g = 240 pprpr = 2 fields per round per round = 2 fingerwidths per tick per tick exactly.

It’s worth noting that it takes about 1 round for a human to reach terminal velocity.

Also, it takes about 1 round for a typical sedan to reach highway speeds of 120 paces per round7.

Force, Mass, and all the rest.

All we’re missing to get a full system of measurement (for mechanics at least; we’re not getting into electrodynamics here) is a base unit for force or mass.

Ideally, it’ll give us some unit which can be thought of as the equivalent of one ‘inventory slot’ worth of stuff.

Units of Mass

In the real-world SI system, a liter of water has a mass of approximately 1 kilogram. But this is world of dragon hordes, so let’s base our units on gold instead of water. The base unit of mass is the draconic stone,8 equal to the mass of precisely one cup of of pure gold, (which is roughly 4.7 kg or 10.5 pounds).

  • There are a hundred draconic ounces in a draconic stone.
  • The base unit of mass is the (draconic) stone (about 10.5 pounds).
  • There are a hundred stones in a cartload (about 569 kilograms or 1255 pounds),
  • and a hundred cartloads in a hord (about 68 metric tons or 150 thousand pounds).

A hord is the mass of a cord of gold, which is the smallest amount that can be called a proper dragon’s hoard. It’s comparable to the mass of a bulldozer or large dinosaur.

A draconic ounce is about 40 grams, 1.4 avoirdupois ounces, or 1.3 troy ounces. And there are 12 draconic ounces in a draconic pound, which is about 1.05 avoirdupois pounds.

Also, there are 12 draconic stones in a draconic picul, which is the amount of weight a man can carry on a pole.

Derived Units of Force, Energy, and Work

Now that we have units for time, distance, and mass, we can use these to derive the remaining mechanical base units.

A newton is a force equal to one kilogram·meter per second per second.

Similarly, our unit of force is equal to one hord·pace per round per round.9 This works out to about 2892 Newtons, which is, according to Wolfram Alpha, comparable to the force of a punch from a professional boxer. Let’s call this unit a wallop.

A joule is 1 newton·meter.

Likewise, our base unit of energy is 1 wallop·pace, which works out to about 4407 joules, or about 1.05 Calories.

And the base unit of power is 1 wallop·pace per round, which is about 735 watts or 1 metric horsepower.10

Thus, after hundreds of words of explanation, and many hours of careful calculation, we have finally established why hp is the proper measurement of a fantasy creature’s power.11

  1. A minute is 10 rounds, an hour is 5 turns, a month is a quarter-season, etc.

  2. In some video games, the state of the world is updated at a rate seperate from the framerate. The length of time between each update of the game’s physics engine is called a “gametick” or “tick”. In the video game Minecraft, for example, the gameworld updates at 20 ticks per second.

  3. Annoyingly there is no folksy old name for a length of half an inch. A third of an inch is called a barleycorn, and a sixth of an inch is called a pica. There are also units called “digit” and “finger”, which are 3/4 and 7/8 of an inch, respectively. But I just measured my fingers, and with the exception of the thumbs, they’re all closer to half an inch wide. I guess the units were based on big meaty working-man fingers? Maybe “halfinch” isn’t such a bad name on its own though.

  4. In the US, a cup is legally defined as 240 mL. In Canada, a cup is 250 mL. It turns out that 1/120th of 1/120th of a cubic pace is 245.8 mL, right in between. Don’t even need to make up a weird fantasy unit. Cups slot right in.

  5. Based on the numbers for East India Company ships in the early 1800s in a fresh breeze. Kelly, M, and C Ó Gráda (2018), “Speed under Sail during the Early Industrial Revolution”, CEPR Discussion Paper 12576.

  6. Source: Something I vaguely remember a friend once telling me.

  7. I just skimmed through this list of car acceleration speeds. The sedan model names I recognized take around 4-6 seconds to go from 0mph to 60mph. Increase that number a bit, and you get 1 round to reach cheetah speed.

  8. I know the brits out there will say that a stone is 14 pounds, but I refuse to recognize the dictates of Edward the Third, and continue to insist that any unit of mass between 3 and 10 kg may, depending on context, be reasonably called a ‘stone’.

  9. You could use stone·paces per round per round, but then you’d get teeny tiny base unit for force and the rest.

  10. I actually started with horsepower as the unit of power, and worked backwards to figure out that I could use a cup of gold as the basis for the unit of mass, with the resulting discrepancy between fantasy horsepower and metric horsepower being less than one percent. What a happy coincidence.

  11. My apologies.

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