2025-02-01
The article describes one way of setting up a lunisolar calendar. It largely follows the current practice in China.
It is important to define the terms, such as day, month, etc. to avoid confusion. I don’t think I can define the term time. Time is very difficult to talk about, when you need to reckon with the theory of relativity. Alright, so I fail at the very beginning.
Most of the terminologies below are taken from Wikipedia.
A second is a length of time defined in terms of the caesium frequency1.
A day is defined to be 86400 seconds2. A solar day is the time it takes for the sun to return to its upper culmination point (its highest point in the sky). We point that that a longitude on earth must be chosen before hand, so indeed a lunisolar calendar constructed using the exact same rules described below can be very different when we use a different longitude. Here we refrain from going into the issue of timezones, which can be sidestepped by choosing a longitude that match with the timezone. The average of the length of the solar days, taken over a long enough period of time, is called the mean solar day length. In recent decades, the mean solar day length is approximately 86400.002 seconds. We will ignore the discrepancy with a day (86400 seconds) and will ignore the issue of leap seconds. From a certain epoch, which I don’t know exactly when, we cut up the time into blocks of 86400 seconds. Each block is called a calendar day. Each calendar day starts approximately when the sun is at its lower culmination point in the sky.
A year is defined to be 365.25 days long. We probably will not use this concept much. For this article, a tropical year is defined to be the time it takes for the sun to go from one winter solstice to the next winter solstice3. Some remarks are in order. Here winter solstice is defined in the ecliptic coordinates4 centred around the earth and it is the winter solstice with respect to the northern hemisphere, which means that it is the solstice that happens in December. It is the summer solstice in the southern hemisphere. For ease of exposition, we ditch the southern hemisphere. We do not use the tropical year from vernal equinox to the next vernal equinox. The reason for the choice is that I think the current practice uses winter solstices, as was that of the Qin state, to set up the lunisolar calendar. To describe the lunisolar calendar used in the Zhou dynasty, for example, one should use vernal equinoxes instead5. Due to the earth’s axial precession, the line that is the intersection of the ecliptic plane and the equatorial plane keeps changing direction in the sidereal frame, resulting in changes in the equinox points and the solstice points. Thus the length of a tropical year is different from that of a sidereal year, which we will not define. One can think of sidereal objects to be essentially objects defined with respect to an absolute reference frame. The mean length of a tropical year is 365.259636 days at the epoch J2011.0. (I do not know how exactly one takes the mean at the epoch J2011.0.)
We divide the ecliptic longitude into 24 equal parts. The points that start at the winter solstice and are 15 degrees apart are the solar terms6. Those with index congruent to that of the winter solstice modulo 2 are called major terms and the rest are called minor terms. I personally think of these as lying on a standing wave with the minor terms being the nodes and the major terms being where the amplitude is the largest.
We introduce the concept of the calendar sui. We define it to be the time from the calendar day (included) that contains the winter solstice to the next one (excluded). I am unsure if we should use true solar day or not. A calendar sui is approximately one tropical year long. We see that a calendar sui contains either 365 or 366 calendar days. When there are 366 calendar days, one of days is designated to be special.
So far only the sun and the earth play a role. Next we divide up the calendar sui based on information from the moon.
A synodic month7 is the time from the new moon to the next. The new moon is the phase of the moon when the moon and the sun have the same ecliptic longitude. At the J2000.0 epoch, the average length of a synodic month is 29.53059 days. A calendar yue is the time from the beginning of the calendar day that contains a new moon to the beginning of the calendar day that contains the next new moon. Again I am unsure if we should use true solar day or not. A calendar yue consists of 29 or 30 calendar days. We introduce the concept of a calendar yue-aligned sui. A calendar yue-aligned sui starts from the beginning of the calendar yue that contains a winter solstice and ends at the beginning of the calendar yue that contains the next winter solstice. We do some basic approximation \begin{aligned} \label{eq:month} \frac{\text{length}(\text{mean tropical year})}{\text{length}{(\text{mean synodic month})}} = \frac{365.259636}{29.53059} \approx 12.3689. \end{aligned} It can be shown that a calendar yue-aligned sui contains 12 or 13 calendar yue. When there are 13 calendar yue, one of them is designated to be special and determining which is special makes use of the major terms.
Let us adopt the current convention of 定正, which determines when the new year starts, in China. The calendar yue that contains a winter solstice is labelled as yue 11. If a calendar yue-aligned sui has 12 calendar yue, then we label them from 11 onward and when the number becomes bigger than 12, we do modulo 12, so, starting from the calendar yue containing the winter solstice, the labels are \begin{aligned} 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. \end{aligned} Now assume that a calendar yue-aligned sui has 13 calendar yue. By Schubfachschluss, in a calendar yue-aligned sui, there is at least one calendar yue which does not contain a major term. The ellipse orbit of the earth around the sun probably does not permit there being more than one. The first such calendar yue is designated as the intercalary yue and it will use the same the number as the calendar yue before it. The other 12 calendar yue will be labelled from 11 onward and when the number becomes bigger than 12, we do modulo 12. For example, we can have, as the yue numbers \begin{aligned} & 11, 11 \text{(intercalary)}, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10; \text{ or}\\ & 11, 12, 1, 2, 3, 4, 5, 6, 6 \text{(intercalary)}, 7, 8, 9, 10. \end{aligned} Some sequences are less likely to occur due to the earth’s ellipse orbit. We note that yue 11 always contains a major term since, by definition, it contains a winter solstice. The first calendar day of yue 1 is then the the lunisolar new year’s day. Just to be more precise, if we have both yue 1 and yue 1 (intercalary), the latter does not produce a second new year’s day.
In ancient China, other conventions of 定正 were used and there are other ways for labelling which calendar yue is the intercalary yue. We will not go into it.
Consider 0.3689 which is the non-integral part of [eq:month]. With bounded denominator, we can ask for the best approximation using fractions. Since 0.3689 is already an approximate number, we should not let the denominator be too big; there is no point to approximate beyond the precision that 0.3689 has. We should limit the denominator to at most 100. Besides considering human longevity and the longevity of past dynasties, it is best not to strain over 100.
One way for finding the best fraction approximation is via continued fractions89. [TODO: I need to find a better source for this.] It is easy to get the continued fraction for 0.3689: \begin{aligned} [0; 2, 1, 2, 2, 5, 2, 1, 3, 7, 1, 1759218603, 1, 1, 3, 2] \end{aligned} and for each truncation, we get a fraction and we list them below: \begin{aligned} 1/2, 1/3, 3/8, 7/19, 38/103, 83/225... \end{aligned} with error given by \begin{aligned} 0.13110, -0.035567, 0.0061000, -0.00047895, 0.000032039, -0.000011111, 0.0000024390... \end{aligned} We see that having 7 intercalary calendar yue in every cycle of 19 calendar yue-aligned sui is a good enough approximation. Going up to 38/103 stretches the precision too much.