Time & Clocks

A Consideration of
Time Systems & Time Bases

Introduction

Most people in the world are familiar with the standard 24-hour Western time system. It goes something like this:

HH:MM:SS.DD…

where HH is a number of hours from 00 to 23, MM is a number of minutes from 00 to 59, and SS.DD… is a number of seconds, also from 00 to 59, followed by a decimal fraction where additional precision is required. Immediately, several issues jump out.

Problems

  1. It is inconsistent with normal Western numbers.

    The Western numbering system is decimal, base-10. For consistency, surely our time should also be base-10? Indeed, scientific disciplines tend to eschew the full Western time system altogether in favour of simply using multiples of seconds.

    However, decimalisation of time and its numerous proponents throughout the ages have always failed. Why? Perhaps because base-10 time systems do not divide well? It is very convenient to be able to divide time units ‘smoothly’ and thence keep the denominators small: we often speak of ‘three quarters of an hour’, but how often have you heard someone talk about ‘ten twelfths of an hour’, ‘fifty minutes’, or even ‘a fifth of an hour’? It is not usually necessary because we have the ‘quarter of an hour’ measurement, which is close enough for almost all everyday purposes. I propose, without any particular evidence, that fractions in general are difficult to intuit, or perhaps even impossible: learning each new denominator requires an extra cognitive fact. You can probably picture a quarter or maybe even a third of a cake quite easily, and thence three quarters or two thirds, but how much extra effort does it take you to picture four fifteenths of a cake?

    Base-10 time limits us to the denominators 2 and 5 (and multiples thereof). Not only is 5 a denominator people do not tend to use, it is quite small: the base is not ‘smooth’. There is a clear and large gap between a half and a fifth that is not easily filled by any simple base-10 fraction, and this issue is compounded by the fact that the decimal hour, minute, and second are all rather much larger than their mixed-base counterparts. To wit, the difference between half a decimal hour and a fifth of a decimal hour is $\frac1{20} - \frac1{50} = \frac3{100}$ of a day, or almost three quarters of a base-24 hour — far too large a time difference to be negligible in day-to-day life.

  2. It is inconsistent within itself.

    The numbers of hours and minutes are not the same — it doesn't even read like a number. It is what is known as a mixed-base positional system: each position has a value, but those values are not the same, nor even regularly predictable. In terms of numbers of seconds, it looks something like this:

    <36000><03600>:<00600><00060>:<00010><00001>.<0.1000><0.0100>…

    Like the similar irregularities with the Imperial system of measurements, now largely obsoleted by the more regular metric system, this makes arithmetic using times very difficult, requiring consideration of various compound constants. For example, 100000₁₀ seconds does not yield 100₁₀ days, and nor does 1000₆₀ seconds yield one day: to make the conversion to days, there is no simple procedure. Instead, one must break down the value, first into 24ths, then into 60ths, and then into 60ths again (and then into 10ths if one also needs to consider fractions of seconds). The same issues apply when converting to time intervals smaller than a second, since…

  3. It is limited in range.

    We have hours, minutes, and seconds — but beyond that, there is no further appropriate measurement, and people must resort to decimal fractions of seconds, introducing a new base. Why does it not continue down past the second in a regular base? I suspect the answer is that in ancient times people had no reliable way of measuring times smaller than a second. That is not the case today, of course, so why should we continue to support the legacy limitations of an old system?

  4. It is wasteful of space.

    The use of a sub-base aids arithmetic by allowing one to relate to the native decimal base, but introduces considerable redundancy due to the use of multiple characters to represent one value: the first digits of the minutes and seconds will only ever go up to 5, and of the hour, only 2! Furthermore, a separator must be inserted to make it clear when the mixed-base digits begin and end.

On closer inspection, a more subtle problem surfaces:

  1. The lengths of the units are not well-suited to everyday usage.

    There are two scales on which time is specified in everyday life: the larger scale, used for arranging synchronised times of events such as meetings, in which leeway of perhaps a dozen minutes is permissible; and the smaller scale, used for precision tasks such as cooking, in which permissible leeway is perhaps a dozen seconds either way. These very specifications reveal the intrinsic unsuitability of our units for these tasks: the unit boundaries should be such that one unit is sufficiently precise for the task at hand. As it is, we often have to mix our units in specifications, such as ‘meet me at 15:30’ or ‘cook the noodles for 3:15’: the hour is insufficiently precise but the minute excessively precise for the former, and the minute too vague but the second too specific for the latter. With a well-considered time unit, this can be avoided.

A Solution?

Sexagesimal (base-60) time solves problems 2, 3, and 5, though problem 1 can only be solved by using a sexagesimal base for native arithmetic (however, it is mitigated considerably by using a similar base, such as dozenal). The sexagesimal ‘hour’ and ‘minute’ are equivalent to twenty-four mixed-base minutes and seconds, respectively. This gives a single unit a precision of precisely ±12₁₀ of the next equivalent mixed-base unit on either side, which, it turns out, is roughly the sort of precision appropriate for the two everyday scales mentioned above. The system, being consistent, can be logically extended indefinitely in pure powers of 60, to any precision required for sciences and other extremely precise measurements.

The use of an intentionally-designed pure sexagesimal digit set, rather than a sub-base, allows times to be specified much more concisely: almost all everyday times can be represented with only two characters, rather than the four to eight required by the mixed-base system. This I intend to cover in a separate post.

The main problem of large number bases, the difficulty of multiplication and division, does not apply with times, since it makes no sense to multiply together two times. Considering this and the existing sexagesimal nature of most of our current mixed-base system, I would argue that sexagesimal is the ideal base for our time, and the units that arise from a sexagesimal division of the day are the ideal units for the specification of time as we use it in our everyday lives, as well as being less cumbersome in a technical context.

I have created a simple example of a software sexagesimal clock using Javascript and the <canvas> element; it may be found here. Insofar as I am aware, the full clock does not currently work in Internet Explorer.

Problems that remain with this implementation are digit design and vocal specification of times: I hope to cover these topics in later posts.