# Dozenal Time: Proposal for a duodecimal method of measuring time

The duodecimal system (also known as base 12 or dozenal) is a positional notation numeral system using twelve as its base. In this system, the number ten may be written as “A”, “T”, or “Ϫ”, and the number eleven as “B” or “Ɛ”. Another common notation, introduced by Sir Isaac Pitman, is to use a rotated “2” (2) for ten and a reversed “3” (3) for eleven. The number twelve (that is, the number written as “12” in the base ten numerical system) is instead written as “10” in duodecimal (meaning “1 dozen and 0 units”, instead of “1 ten and 0 units”), whereas the digit string “12” means “1 dozen and 2 units” (i.e. the same number that in decimal is written as “14”). Similarly, in duodecimal “100” means “1 gross”, “1000” means “1 great gross”, and “0.1” means “1 twelfth” (instead of their decimal meanings “1 hundred”, “1 thousand”, and “1 tenth”).

The number twelve, a superior highly composite number, is the smallest number with four non-trivial factors (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing range. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, with neither 3 nor 4), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system. Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers (such as 2, 3, 4, 6, 8, 9...) have a terminating representation in duodecimal. In particular, the five most elementary fractions (1⁄2, 1⁄3, 2⁄3, 1⁄4 and 3⁄4) all have a short terminating representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9, respectively), and twelve is the smallest radix with this feature (because it is the least common multiple of 3 and 4). This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary, octal and hexadecimal systems, although the sexagesimal system (where the reciprocals of all 5-smooth numbers terminate) does better in this respect (but at the cost of an unwieldy multiplication table and a much larger number of symbols to memorize).

If we were to switch to a dozenal counting system as well as to a dozenal measuring system, our clocks would have to change; it’s just that simple. We were able to keep the 60-divisions circle for our clocks in spite of counting in decimal because we use twenty-four hours, each of which divides into twelve neat little packets of 5 divisions (minutes/seconds) each. Some duodecimal advocates, when discussing time, think no further than slapping two new digits on the clock face in place of 11 & 12; but it is actually a much more complex issue.

If we changed to a base twelve counting system, our perception of “5” would change. It would no longer be that convenient binary of ten, our current base number. Instead, it would be this inconvenient number that kids don’t like reciting multiplications for (imagine the multiplication table for seven; that’s what five would be like in a dozenal). In any discussion of dozenal counting, you have to remember that one dozen would be the new 10 and six, by consequence, would be the new 5. Since it is important that calculating time be easy for us, it is important that we concentrate on groupings of 6, 12, 72, and 144 (in dozenal: 6, 10, 60, 100 — remember, 2 x 6 = 10 in duodecimal multiplication). What does this mean for time? It means that the old 60-divisions circle is going to have to be replaced; or it must have ten divisions — these two options are mutually exclusive.

Dozenal will call for a clock that has 72 divisions. The day will still be 24 hours, but each hour will have 72 minutes, and each minute 72 seconds. What does this mean for you? Hours will still be the same and timekeeping will not be constructively different than it is now. The only major difference is that the minute and the second will be slightly shorter than what we’re used to (1 dozenal minute = 50 standard seconds and 1 dozenal second = 25/36 standard seconds).

Value in Dozenal Seconds |
Dozenal Time Units |
Equivalent in Current Time Units |
Equivalent in Current Time Seconds |

1 sec | 1 sec | 25/36 sec | 25/36 sec |

72 sec | 1 min | 5/6 min | 50 sec |

5 184 sec | 1 hour (72 min) | 1 hour (60 min) | 3 600 sec |

124 416 sec | 1 day (24 hours) | 1 day (24 hours) | 86 400 sec |

#### Pronunciation

In terms of pronunciation, Donald P. Goodman, president of the Dozenal Society of America, says that Ϫ should be called “ten”, Ɛ called “elv” and 10 pronounced “unqua”. So, when counting, we’d say, “...eight, nine, ten elv, unqua”.

Interestingly, in the 1973 episode “Little Twelvetoes” of Schoolhouse Rock! television series, an alien child uses a base-12 system and pronounces the last three numbers “dek”, “el” and “doh”. “Dek” was derived from the prefix “deca”, while “el” was short for “eleven”, and “doh” a shortening of “dozen”. Many dozenalists have adopted this particular pronunciation system.