Imagine the situation: two people are vigorously debating the best method for calculating tides. The first person argues that the rule of twelfths gives a superior approximation. The second argues that the calculation method condensed in Tables 5 and 5A of the Fisheries and Oceans Canada Tide Tables provides a better approximation. This debate, which actually took place, was reported to me by a colleague.

Who is right?

On a geekness scale of 0 to 10, this debate level is about 120. But if you want an incentive to keep reading, I’ll start by summarizing the punch line: both approaches do pretty much the same thing. If one method has a large forecast error, then the other method will also have an error. It’s the same approximation logic.

## The Rule of Twelfths

The rule of twelfths breaks down the half-cycle of the tide into six intervals of time. Variations in tidal height are then approximated in twelfths (hence the name). A visual presentation of the approach is given in the top image, illustrated over a twelve-hour cycle.

The table below also summarizes the approach. The first three columns present the generic rule, while the last three present the idea for a tidal range of three meters, with a rising tide from 1800.

Time elapsed | Variation | Tidal Height | Time | Variation (m) | Tidal Height (m) |

+0100 | +1/12 | 1/12 of the range | 1900 | 0,25 | 0,25 |

+0200 | +2/12 | 3/12 of the range | 2000 | 0,50 | 0,75 |

+0300 | +3/12 | 6/12 of the range | 2100 | 0,75 | 1,50 |

+0400 | +3/12 | 9/12 of the range | 2200 | 0,75 | 2,25 |

+0500 | +2/12 | 11/12 of the range | 2300 | 0,50 | 2,75 |

+0600 | +1/12 | 12/12 of the range | 2400 | 0,25 | 3,00 |

In principle, all you need to remember is that you add up the twelfths in the sequence “1, 2, 3, 3, 2, 1”, i.e. “1, 2, 3” and “1, 2, 3 in reverse”. Then it’s a matter of a few calculations to find the tides at the given times.

## The Twentieths Rule

The twentieths rule corresponds to Tables 5 and 5A of the Tide Tables published by Fisheries and Oceans Canada. In terms of presentation, there are three important differences with the sixths rule.

Firstly, the tables divide the tidal half-cycle into twenty periods instead of six (hence the name!). Secondly, the time intervals for each period are not equal. Rather, the intervals are chosen so that the variation in tidal height is equal during each interval. In short, in the rule of sixths, the time intervals are constant, whereas in the rule of twentieths, it’s the tidal variations that are constant.

Table 5 of the Tide Table is reproduced below. The table shows the first ten time intervals, arbitrarily labelled from A to I. Thus column A gives the elapsed time in the first time period, column B gives the elapsed time in the second time period, and so on. Each row shows the duration of each interval for a tidal cycle duration specified in the first column.

For example, a tidal cycle of 6 hours (row where the first column shows “6 00”) will give the time elapsed between each interval. After one interval, 52 minutes have elapsed. After five intervals, two hours and after 10 intervals, three hours have elapsed. (The other ten periods are approximated in the same way, starting from the other end of the tidal half-cycle, so they are not shown in the table: the table must be used in reverse).

Table 5A (shown below) directly calculates the tidal height for the given time interval (columns A to I) as a function of the tidal range identified in the first column. Thus, for a tidal range of 3 meters (line “3.0”), the tidal height after two hours of a six-hour half-cycle corresponds to 0.75 meters (column E). Note that the tidal intervals are divided equally between each period.

If you haven’t already noticed, note that the 0.75 metre tide corresponds to exactly the same answer as that obtained by the rule of sixths in the previous section. In fact, for each identical time interval (and the same tidal cycle length and tidal range), both approaches will provide the same answer. For example, after three hours have elapsed (column I), the variation in tidal height will be one and a half metres, i.e. the same answer as that given by the rule of sixths.

## The Same Thing?

It is hard to be more convincing without introducing a bit of high-school mathematics. Both approaches to tidal calculations are based on the same mathematical model, whose foundation is the sine curve (\sin(t)). A graphical representation of the curve is shown below.

For the purpose of calculating tidal heights, the following expression is what counts:

z = \frac{\Delta z}{2}\left[1+\sin\left(\pi\left(\frac{t-t_0}{\Delta t}-\frac{1}{2}\right)\right)\right],

where z is the tidal height, \Delta z is the tidal difference \Delta t is the tidal half-cycle in minutes and t is the elapsed time since the beginning of the tidal cycle (in minutes). Of course, \pi is the circumference to diameter ratio of a circle. (Note: the expression assumes the argument of the sine curve is expressed in radians.)

The advantage of the formula developed above is that it works at any moment of the tidal cycle. It is also compact, requiring only the tidal difference and the duration of the tidal half-cycle. If the tide is falling, however, you need to change a sign in the expression (which one?). It is a compact form that is a good alternative to the on-board computer in case of breakage. On the other hand, it requires a minimum understanding of “hard math”.

### An Example

If we take the same example as above, i.e. a tidal range of three metres and a rising tide duration of six hours, the expression becomes:

z = \frac{3}{2}\left[1+\sin\left(\pi\left(\frac{t}{360}-\frac{1}{2}\right)\right)\right].

This expression will produce exactly the same answers as tables 5 and 5A and roughly the same answers as the rule of sixths. I illustrate it below, both in a table and in a graph:

Time (h m) | Time (m) | Sixths Rule | Twentieths Rule | Formula | Approximation Error |

52 | 52 | | 0,15 | 0,15 | 0 |

1 00 | 60 | 0,25 | | 0,20 | 0,05 |

1 14 | 74 | | 0,30 | 0,30 | 0 |

1 31 | 91 | | 0,45 | 0,45 | 0 |

1 46 | 106 | | 0,60 | 0,60 | 0 |

2 00 | 120 | 0,75 | 0,75 | 0,75 | 0 |

2 13 | 133 | | 0,90 | 0,90 | 0 |

2 25 | 145 | | 1,05 | 1,05 | 0 |

2 37 | 157 | | 1,20 | 1,20 | 0 |

2 49 | 169 | | 1,35 | 1,36 | -0,01 |

3 00 | 180 | 1,5 | 1,50 | 1,50 | 0 |

3 11 | 191 | | 1,65 | 1,64 | 0,01 |

3 23 | 203 | | 1,80 | 1,80 | 0 |

3 35 | 215 | | 1,95 | 1,95 | 0 |

3 47 | 227 | | 2,10 | 2,10 | 0 |

4 00 | 240 | 2,25 | 2,25 | 2,25 | 0 |

4 14 | 254 | | 2,40 | 2,40 | 0 |

4 28 | 268 | | 2,55 | 2,54 | 0,01 |

4 46 | 286 | | 2,70 | 2,70 | 0 |

5 00 | 300 | 2,75 | | 2,80 | -0,05 |

5 08 | 308 | | 2,85 | 2,85 | 0 |

6 00 | 360 | 3,00 | 3,00 | 3,00 | 0 |

Note from the table that the approximation error from the rule are at most five centimetres! These deviations are deliberately introduced into the sixths rule to make it easier to use. It’s much easier to remember the “1, 2, 3” rule than to remember a hypothetical rule where the first twelfth is replaced by the fraction 67/1000, which would eliminate the error!

## Are These Rules Useful?

At a time when tide levels can be obtained from your on-board computer or cell phone, you might well ask what is the point in knowing these methods. Old-school advocates will certainly say that it is for the sake of safety.

Manual approaches provide a form of redundancy. If, for any reason, the on-board electronics should fail, the prudent navigator should be able to make an alternative calculation using on-board resources.

The rule of sixths is simple to remember and requires very little information to make an assessment of tidal conditions. Tables 5 and 5A provide the same information, but at finer intervals. In this sense, it better reproduces the sinusoidal curve at the beginning and end of the tidal cycle.

However, it requires the simultaneous examination of two tables, which introduces a greater potential for error at the time of evaluation. In short, the first manual tool is simple but less accurate, while the second tool is a little less easy to use, but produces a better approximation. No “hard math” is required.

Personally, I prefer to remember the sine equation explained above and bring along a solar-powered calculator. If the on-board computer, cell phone, calculator AND VHF radio were to fail (that would be a really bad sailing trip!), I’d probably resort to the rule of sixths. But in such a failure scenario, I would have my mind set on more urgent priorities than tidal calculations.

## What is A Good Approximation?

The examination of the two calculation approaches, basically two ways of approximating a sinusoidal curve, prompts a broader reflection on what a good approximation is. It would not be difficult, for example, to make a “rule of fiftieths” that would subdivide the tidal half-cycle into fifty time intervals. The only conceptual difference is that tables 5 and 5A would be larger.

But what if the sine curve misrepresents the actual tidal curves? This is not an insignificant question, as tides are very different in different parts of the world. A good place to start is by looking at how Fisheries and Oceans makes its own tide level predictions. Their prediction program is public (here), as is its user manual (here). They even share data to test the program (how about that!).

A reading of their approach shows that they employ two statistical forecasting paradigms. The first is a least-squares forecast based on the position of the planets (a “statistical-physical” model). The second is a time-series forecasting paradigm, using past tidal values to predict future values (time series).

All this to say that Oceans and Fisheries’ forecasts are certainly more robust for the purposes of calculating tides… present or future, not least because they are agnostic regarding the shape a tide can take as a function of time. Another redundancy paradigm would be to bring along the estimated coefficients of their forecasting model. It would then be possible to calculate, as required, each of the entries in each tide table. Not far from what an on-board computer can do…

That said, it’s the SOLAS convention that perhaps best informs us of another way to protect against system failure. Commercial vessels subject to this international convention bring on board a second electronic navigation system with an independent power supply. On a sailboat, it may be more convenient to bring a second cell phone, or a second tablet, with a tidal application.

## References

Oceans and Fisheries Canada (s.d-a). *L’application de niveaux d’eau optimisée pour les appareils mobiles*, webpage retrieved online in november 2023 at this address.

_____________________ (s.d.-b). *Application de suivi des niveaux de marée*, webpage retrieved online in november 2023 at this address.

_____________________ (s.d.-c). *IOS Tidal Package*, webpage retrieved online in november 2023 at this address.

Foreman, M.G.G (1996). *Manuel for Tidal Heights Analysis and Prediction*, document prepared for Fisheries and Oceans Canada, retrieved online in november 2023 at this address.

Oceans and Fisheries Canada (2023). *Table de marées et des courants du Canada*,* Tome 2*, document retrieved online in november 2023 this address.

Wikipedia (s.d.). *Rule of Twelfths*, image reproduced from this address.