In the early 15th century, a major change took place in the methods of creation of calendars in Tibet. Within the space of one decade – they all give dates within the 1440s – three new calendrical traditions were established, all following one and the same basic principle: the development of siddhānta calculations (grub rtsis). These systems were the Phugpa, Tsurphu and Error Correction ('khrul sel) systems. They all assumed that they had managed to recreate a lost accurate system of calculating the calendar, that had presumably been described in the lost Kālacakra Mūlatantra. Why this idea suddenly popped up is not yet clear from available texts, but an intriguing possibility is that these ideas were introduced by the last great Kālacakra teacher to visit Tibet, Vanaratna (nags kyi rin chen). He visited at about the right time, and we know that he taught Zhonnu Pal (gzhun nu dpal), the creator of the Error Correction system. One piece of evidence is given in the Phugpa text, the "pad dkar zhal lung". It states there that Vanaratna aligned the 60-year Chinese and Indian cycles differently from normal. It is usual that Fire-Rabbit is equated with the Indian Prabhava, but we are told that for Vanaratna this was Wood-Mouse, a difference of three years. Zhonnu Pal has exactly the same alignment – see below – and may very well have come to this conclusion based on chronology as explained by Vanaratna (one possible explanation could be years expunged from the Indian cycle used by Vanaratna since the Kālacakra was introduced to Tibet). There are many texts by students of Vanaratna and complete copies of Zhonnu Pal's work that have been discovered in the Drepung library finds. Maybe all this will be clarified when these become available.
Whatever the source, these three traditions based their calendars on a very simple idea: that, apart from the weekdays, the calendar cycle of intercalary months – governed by the solar longitude – repeats itself exactly every 65 years. They built tables of mean values that would occur in the calendar, and constructed their calculations on the basis of matching observations with the values in those tables. Given that the calendar was now said to be accurate and would repeat itself indefinately, further observations would not be necessary (Zhonnu Pal said that it would "cycle repeatedly through the aeon, without even the slightest variation (de nyid bskal pa'i bar du yang dang yang 'khor ba ma gtogs rim pa gzhan cung zad tsam yang yod pa ma yin no). No more observation – this must have been very compelling.
Unfortunately, in creating these siddhānta calculation systems, they actually made their calendars less accurate – in particular, the monthly mean motion of the Sun was much less accurate than the Kālacakra figures they had used previously. They based their new calculations on data given in two verses of the Kālacakra Tantra; the one from which they derived the mean motion of the Sun is in fact an approximation for determining intercalary months. Some further information on these values and how they were derived is available here.
The Phugpa and Tsurphu systems compounded the error. Existing implementations of calendars in Tibet had been criticised as early as the 13th century CE by people such as the translator Shongton; he complained that the Tibetans misaligned the Indian names of the months by one month. The great Kasmiri Pandit, Vimalaśrī, also complained that the months were named one month earlier in Kashmir. For further information on the reason for this misalignment, see here.
Presumably in order to justify this alignment of the months, combined with a misunderstanding of the nature of precession and the difference between sidereal and tropical zodiacs, the founders of the Phugpa and Tsurphu traditions established the view that the winter solstice – and other seasonal phenomena – was observed in Tibet two to three weeks (Phugpa 23 days, Tsurphu 14 days) before it is observed in India. They therefore, when observing the winter solstice in order to correct the longitude of the Sun, instead of setting the longitude at the correct value of 0° Capricorn, they set the value at 8° Sagittarius (Phugpa) and 16° Sagittarius (Tsurphu). These errors, combined with the increasing error due to the inaccurate mean motion of the Sun, mean that currently, the error in the Phugpa mean Sun is 36° and in the Tsurphu mean Sun 31° – in terms of days, the Phugpa is out by 37 days and Tsurphu by 32 days. These are huge errors in any calendar.
Zhonnu Pal did not make the same errors. He followed the Kālacakra siddhānta method for correcting the longitude of the Sun, without assuming that the solstice was observed at a different time from India. As a result, he also named the Indian months correctly, aligning them properly with the Chinese months. He defined and combined the months of the Indian and Chinese traditions in a way that would certainly have met with the approval of both Indian and Chinese experts. Unfortunately, he made other mistakes.
Like many Tibetans, he took certain chronology given in the Kālacakra Tantra as literal, and tried to get his calendar to fit both these data and other factors, such as events involving the life of the Buddha, based on texts such as the Lalitavistara. He got this all hopelessly wrong, and placed the Kālacakra astronomical epoch in 340 CE rather than 806 CE, and the passing of the Buddha in 1352 BCE! He also misaligned the Indian and Chinese 60-year cycles. As Zhonnu Pal is better known as the author of the famous historical work, The Blue Annals, translated by George Roerich, his rather weak grasp of chronology suggests that dates in that work should perhaps be considered with care.
Regarding other astronomical errors, there was an unfortunate one in the weekday definitions, and his planetary calculations were hopelessly inaccurate: he took the mean positions of the planets at the Kālacakra epoch, as given in the tantra, and applied them to his epoch, 466 years earlier! The planets move a long way in that time. Quite how he justified the results of his calculations, which were effectively random, with the positions of the planets in the sky is not recorded.
His calendar failed to be adopted, partly presumably because of politics, but also because of the errors – his errors were much easier to spot than those made in the other systems. In fairness to Zhonnu Pal, having taken his work apart in this way, I should say that his text (rtsis la 'khrul pa sel ba) on his Error Correction system shows three things: a genuine attempt properly to implement the intentions of the Kālacakra system, the best description in the Tibetan language of how the siddhānta calculation systems were developed, and, a mathematical clarity and accuracy unfortunately lacking in some of his contempories. For anybody wishing to study the siddhānta calculation systems in the Tibetan language, his text would make the best starting point. Software implementing Zhonnu Pal's calendar is available on this page.