The Sixty Cycle: How Stems and Branches Pair Up

The East Asian calendar counts time with sixty named pairs, each made of one of ten heavenly stems and one of twelve earthly branches. This page explains why there are sixty and not 120, and shows how to find the pair for any year with two remainders.

Two wheels that turn together

Picture two wheels side by side. The first carries the ten heavenly stems, 甲 乙 丙 丁 戊 己 庚 辛 壬 癸. The second carries the twelve earthly branches, 子 丑 寅 卯 辰 巳 午 未 申 酉 戌 亥. Each step of the count moves both wheels forward by one notch, and the two characters that line up form one pair.

The count starts with 甲子, pair number 1. Then come 乙丑, 丙寅, 丁卯, and so on. At pair 10, 癸酉, the stem wheel has used all ten stems and starts again at 甲, but the branch wheel still has two branches left, so pair 11 is 甲戌, not 甲子. Because the wheels have different sizes, they keep slipping against each other, and that slip is what produces so many different combinations.

Why sixty, and not 120

If every stem could meet every branch, there would be 10 x 12 = 120 pairs. The cycle only ever produces 60 of them. The wheels come back to their starting position together after a number of steps that both 10 and 12 divide evenly. The smallest such number is the least common multiple of 10 and 12, which is 60. After 60 steps the stem wheel has turned exactly six times and the branch wheel exactly five times, and 甲子 comes round again.

There is a second way to see the same thing, through yin and yang. Stems alternate yang, yin, yang, yin, and so do branches. Both wheels have an even number of places and both start on a yang position, so a yang stem always lands on a yang branch and a yin stem on a yin branch. A pair such as 甲丑 (a yang stem with a yin branch) simply never occurs.

• 5 yang stems x 6 yang branches = 30 pairs
• 5 yin stems x 6 yin branches = 30 pairs
• 30 + 30 = 60, exactly half of the 120 conceivable pairs

Finding the pair for a year

Number the stems from 0 (甲) to 9 (癸) and the branches from 0 (子) to 11 (亥). The haru engine then works out a year's pair from two remainders: the stem is (year - 4) mod 10 and the branch is (year - 4) mod 12. Subtracting 4 simply lines the count up so that a remainder of 0 lands on 甲 and 子.

Worked example for 2026: 2026 - 4 = 2022. Dividing 2022 by 10 leaves 2, and counting 甲 = 0, 乙 = 1, 丙 = 2 gives 丙. Dividing 2022 by 12 leaves 6 (12 x 168 = 2016, and 2022 - 2016 = 6), and counting 子 丑 寅 卯 辰 巳 午 gives 午. So 2026 is a 丙午 year, the 43rd pair of the cycle, the year of the Horse.

The same method shows the cycle closing. For 1984, 1980 leaves 0 when divided by 10 and 0 when divided by 12, so 1984 is 甲子. Sixty years later, 2044 gives 2040, which again leaves 0 and 0, so 2044 is 甲子 once more. One small caution: in this tradition the year changes at the solar term 立春 in early February, not on 1 January, so a date in January belongs to the previous year's pair. The year pillar page explains that boundary.

One cycle for years, months, days and hours

The same sixty pairs label every layer of time in the Four Pillars. Years run through them one per year. Months run through them one per solar month, and since a year has twelve months, the month cycle repeats every 60 / 12 = 5 years. Days run through them one per day without any break, so the day cycle repeats every 60 days. Hours are counted in twelve double hours per day, so the hour cycle repeats every 60 / 12 = 5 days.

This is why a birth moment can be written as four pairs, eight characters in all: one pair each for the year, month, day and hour. The tradition reads each pair through the element of its stem and its branch, which is why the table below also shows the stem's element and its yin or yang side.

The full sixty-pair table

Read the table from top to bottom. After 癸亥, number 60, the count returns to 甲子.

Questions

Why does 甲丑 never appear?

甲 is a yang stem and 丑 is a yin branch. Because both wheels alternate yang and yin and start together on yang, a yang stem only ever meets a yang branch, so that combination cannot come up.

Is the sixty cycle the same as the twelve animals?

The animals belong to the twelve branches only. The sixty cycle adds the ten stems on top, so each animal appears five times in sixty years, each time with a different stem, for example 甲子, 丙子, 戊子, 庚子 and 壬子.

Can I use (year - 4) mod 10 for any year?

Yes, for the stem and branch of the year as a whole. Just remember that the pillar year starts at 立春 in early February, so a birthday in January or the first days of February may belong to the previous year's pair.

Where does 2026 sit in the cycle?

2026 is 丙午, pair number 43. The next 甲子 year after 1984 is 2044.

Read next

Traditional Four Pillars (myeongni) and yin-yang five-element interpretation. Not a medical diagnosis or advice.