## Afterthoughts on Dozenal Clock

I have written a clock to display in dozenal time. Having using it for only two days, I’ve already discovered that it is so much more efficient than our standard clocks.

## Telling Time on Dozenal Clock

My dozenal clock is all based on power of twelve. The clock shows four digits, each digit holding a value of zero to eleven. The base unit of time is the day, so each digit on this clock divides the day by successive powers of twelve. That is, the first digit on the far left divides the day by twelve. Then the next digit to the right of that further divides this one-twelfth day by another twelve, and so on for the last two digits on the clock. Such that each digit going to the right represents a smaller and smaller unit of time.

To convert this dozenal time back into our standard time is quite simple. The first digit on the left is one-twelfth of a day, which is about two hours. So multiply by two and you get the military hour, or hour in a 24-hour clock. For instance, a 3 as the first digit on a dozenal clock means 6 in the morning on our standard time.

The next digit is equivalent to ten minutes. Six of these units constitute one hour, and twelve equals two hours. Six would advance the military time by one hour, so add one hour to the previous digit if the second digit is six or higher. A three or nine means half past the respective hour. For instance, 39 for the first and second digits means 07:30 in the morning on our standard time.

The third digit on the dozenal clock is one-twelfth of ten minutes, or 50 seconds. This is just short of the minute which is 60 seconds. Twelve of these units is ten minutes and advances the second digit. Six is half of that, so worth five minutes. The conversion can be done simply in the head by multiplying the third digit on the dozenal clock by 0.8 (=50/60) to get the ones value of the minute in our standard time. For instance, 9 as the third digit on the dozenal clock is worth about 9*0.8 = 7 minutes approximately. So 399 as the first three digits on the dozenal clock is about 07:37 in the morning on our standard time.

The fourth and last digit on my dozenal clock is one-twelfth of 50 seconds, worth 4-and-1/6 seconds. For the most part, one can ignore this digit, just as most clocks don’t show the seconds, but only the hours and minutes. However, what I have discovered as interesting is that counting from one to twelve in a brisk pace takes about the same time for this digit to advance. So instead of counting with “one one-thousand, two one-thousand”, we can keep it simpler with “one, two, three..” up to “ten, eleven, twelve.” This bodes very well for the next digit if one would devise a clock or timing system based on this fifth digit, or 12^(-5) of a day.

## Advantages of the Dozenal Clock

Compared to the current standard method of telling time, the pure dozenal clock can be proven to be much more efficient, in terms of units per digit, time arithmetic, and easy learning curve.

Efficiency of time systems can be compared by the ratio of the number of smallest units divided by the amount of digits used. The higher result means more efficient in keeping time. Our standard time with four-digits holds the hours and minutes. Each day is 24 hours and each hour has 60 minutes. So a four-digit clock we usually see has efficiency of 24*60/4 = 360 units per digit. For the dozenal clock, each digit is simply a successively greater power of twelve. So four-digit dozenal clock has efficiency of 12^4/4 = 5184 units per digit. That is a magnitude 14.4 times more efficient than our current standard clock system.

If we include the seconds to the hours-minutes, then we get 24*60*60/6 = 14400. That looks a lot better, but consider that seconds is not applicable for general time telling. Most of the time, we just need the approximate minute, where a second is simply too tiny to consider. Nevertheless, let’s see efficiency of dozenal clock with six digits. Six digits means twelve to the sixth power, so 12^6/6 = 497664 units per digit. That is 34.56 times more efficient than our current hour-minute-second clock system. Actually, we only need five digits on the dozenal clock because the fifth unit is already smaller than a second–it’s 0.3472~ seconds. In this case, 12^5/5 = 49766.4, which is still over three times more efficient than our HMS system.

Beyond that, the dozenal system will further the gap in efficiency as the HMS second is fractionally divided into powers of ten, while the dozenal time, naturally, continues with powers of twelve. The powers of twelve obviously grow at a faster quadratic rate than powers of ten.

One may surmise that using two digits to represent base 60 is not efficient at all. We could maximise efficiency if every digit of a clock exploited all the digits of a single base. In this case, we’d be better off if we used base 8 to tell time. In four digits that’s 8^4/4 = 1024 units per digit. That’s almost three times more efficient than our current system. (Hmm, an octal clock doesn’t seem that bad of an idea. The fact that eight is a power of two, the binary base, would be very appealing for the masses.) Of course, if we used an even higher base, we’d have even more efficient time system. Like base twelve–which we should anyway because twelve is overall better base than eight and ten in general measurements and math, not only for time.

As for time arithmetic and easy learning curve, since dozenal clock uses the same base for each digit, we can treat any time value as an ordinary number when performing math with time. So any arithmetic operations, such as addition, subtraction, multiplication, and division is very simple because of the base twelve. On the other hand, even simple adding and subtracting time with our current system is cumbersome. Forget multiplying and dividing. That’s more complicated than algebra.