Voder-Vocoder

A galactic system of measures used be several species to communicate with each other in a SF universe in my head. This is an explicitly designed system of units. Base 16 is used because of the importance of binary. Aliens might find caesium-133 as easy to use as humans do. I don’t use multiples of the plank length—which might seem like the natural thing to do—because measuring the constant of gravitation accurately is harder than measuring the elementary charge, speed of light in vacuum, Planck’s constant, or Boltzmann’s constant. But my unit of electrical charge is √(2)×16^15 times the Planck charge.

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L = length unit = wavelength of radiation from the transition between hyperfine ground states of caesium-133 atoms at absolute zero, as that radiation passes through a perfect vacuum = 3.26122557 centimeters = 0.0326122557 meters. (The length unit is exactly 299792458/9192631770 meters.)

T = time unit = 16^8 × (Length unit / speed of light in a vacuum) = 0.467218464 seconds. Or the duration of 16^8 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom at rest at a temperature of 0 K. (The time unit is exactly 4294967296/9192631770 seconds.)

M = unit of mass = 16^32 × Planck’s constant / (speed of light × length unit) = 0.0230618313 kg, which makes plank’s constant 16^-24. [I'm not sure of a nice way to state this definition.] (This unit is only as good as our measurement of Planck’s constant, currently with an uncertainty of around 5×10^-8. But we’re still arguing about how to define the kilogram.) [In the galaxy, the simplest way to realize this definition is to ask the most scientifically advanced species you can find to make a measurement of Planck's constant with minimal uncertainty and manufacture a few 1-unit-of-mass artifacts.]

C = unit of current = 52.3615416 amperes. The Current Unit is a constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 Length Unit apart in a vacuum, would produce between these conductors a force per length equal to 1/(2π) M×L/T²/L.

E = unit of energy = M×L²/T² = 0.000112360968 joules

P = unit of temPerature = 7.05881935 Kelvins, which makes Boltzmann’s constant 16^-16. One defines temperature in terms of the ideal-gas-law behavior of an ideal gas: the temperature of an ideal gas is defined to equal (16^16)(pressure)(volume)/(number of particles).

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Therefore,

speed of light = 16^8 L/T (by definition)

Planck’s constant 16^-24 = M×L²/T (by definition)

elementary charge = 16^-16 × √(2 × the fine-structure constant) C×T (measured)

Newtonian constant of gravitation = 9.6853×10^-9 L³/M/T² (measured)

electric constant = vacuum permittivity = 16^-16 T⁴×C²/L³/M (by definition)

magnetic constant = vacuum permeability = 1 M×L/T²/C² (by definition)

Boltzmann’s constant = 16^-16 M×L²/T²/P (by definition)

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Well, if we were really starting from scratch, I would go with a base 16 numbering system (already in use by programmers everywhere).

One would have to throw out SI units, which would cease to make sense. It would also give us an opportunity to throw out the base 360/60/60 system (degrees/minutes/seconds) and 24/60/60 (hours/minutes/seconds) that the Babylonians saddled us with.

The proposal: The sixteen digits 0123456789abcdef. Integers would be written

dd,dddd,dddd

with four digit groups (16 bits each). Mixed numbers would be witten

dd,dddd,dddd.ddd

and a decimal separating fractions. Floating-point numbers (or numbers in scientific notation) would always be written

d.dddPdd

or

d.dddP-dd

where “xPy” means “x × 16^y” (P for exPonent.)

The new minute would be 1/256 th of an average day. (5.625 old-minutes) The new second would be 1/65536 th of a day (1.318 old-seconds).

The new standard length would be around 9.202 centimeters. that would make the speed of light equal to an even 16^8 (new standard lengths/new seconds)

The new degree would simply be 1/256th of a circle.

Translating to and from binary would of course be trivial.

One would have to come up with new prefixes which mean “256 times”, “1/256 times”, “1/65536 times”, and “65536 times”—like “kilo” means “1000 times”.

A proposal for a better calendar. (If we didn’t have the Gregorian calendar already and wanted to start from scratch.)

No      Name            Days
--      ----            ----
01      January         30
02      February        31
03      March           30
04      April           31
05      May             30
06      June            31
07      July            30
08      August          31
09      September       30
10      October         31
11      November        30
12      December        30 or 31

This would space out the months more equitably. Almost all two-month periods would be exactly 61 days, the exception being December in non-leap years.

I would define January 1 to be the day after the Winter Solstice (at a fixed zero meridian). So all dates would be around ten days off from the Gregorain calendar.

As a result of this rule there would be around 8 leap days every 33 years, similar to how the Iranian/Persian/Jalālī Calendar works.

* * *

Of course the final calendar we will all come to accept millennia from now will probably simply be TAI with picosecond precision, expressed as a number of seconds since some fixed instant. Then calculating intervals become trivially simple, and we will just ignore where the sun is.

* * *

Additional, optional rules to make this a perpetual calendar: January 1 is always a Saturday. Then on 365-day years, you would get two Saturdays in a row:

12-29 12-30 | 01-01 01-02 01-03
 Fri   Sat  |  Sat   Sun   Mon

Or a double weekend on leap years:

12-29 12-30 12-31 | 01-01 01-02 01-03
 Fri   Sat   Sun  |  Sat   Sun   Mon

* * *

Note that one could make the existing Gregorian Calendar into a perpetual calendar by defining March 1 as always a Saturday.

02-27 02-28 | 03-01 03-02 03-03
 Fri   Sat  |  Sat   Sun   Mon

02-27 02-28 02-29 | 03-01 03-02 03-03
 Fri   Sat   Sun  |  Sat   Sun   Mon

* * *

UPDATE 2008-10-25: Here’s very simple python code to give the month, day, and week, given the day of year.

# returns (month, day-of-month, weekday)
# given day-of-year
def dayandmonth(yday):
    assert (yday > 0) and (yday < 367)
    da = ((yday - 1) % 61) + 1
    mo = (((yday - 1) // 61 ) * 2) + 1
    if (da > 30):
        da = da - 30
        mo = mo + 1
    wd = (yday - 1) % 7 +1
    return (mo, da, wd)

And a leap year function:

# offset is determined by solar
# observations at the reference
# longitude
def isleapyear(y):
   offset = 2
   y = (y + offset) % 33
   if (y != 32) and ((y % 4) == 0):
        return True
   else:
        return False

Every 10,000 years, the offset may need to be changed. After 100,000 years, a new algorithm will likely be necessary