Inyo Gogyo Shiso (Yin-Yang Wu-Xing Thought) (陰陽五行思想)

Inyo (also pronounced Onmyo) Gogyo Shiso (Yin-Yang Wu-Xing Thought) is the idea that linked together the principles of Yin-Yang (positive and negative, light and shade) and Gogyo Shiso (Five Elements Theory) during the Spring and Autumn period and the Warring States period in China. It is also known as Inyo Gogyo Setsu (The Theory of Five Elements in Yin-Yang) or Inyo Gogyo Ron. A combination of the cosmic dual forces ideology and Gogyo Shiso made it possible to explain more complicated things.

Jikkan (The Ten Stems) and Junishi (the Twelve Signs of the Chinese Zodiac)
The basis of Onmyo Gogyo Setsu is that the Ten Heavenly Stems in yin and yang combine in pairs, each pair corresponding to each of the Five Elements; 木, 火, 土, 金, 水 (moku, ka, do, gon, read as 'gon' not as 'kin', sui). "甲, 乙, 丙, 丁, 戊, 己, 庚, 辛, 壬, 癸" are read as "ko, otsu, hei, tei, bo, ki, ko, shin, jin, ki" by using on-yomi (the Chinese reading of kanji). While on-yomi makes it hard to understand how the ten stems corresponds to yin-yang and the Five Elements, kun-yomi (the Japanese reading of kanji) makes it easy to do so because these readings refer to each element; kinoe, kinoto, hinoe, hinoto, tsuchinoe, tsuchinoto, kanoe, kanoto, mizunoe, mizunoto (wood [ki], fire [hi], earth [tsuchi], and metal [ka], water [mizu]). In terms of yin-yang, ending a word with 'e' means yang (positive) and ending with 'to' means yin (negative). Etymologically speaking, the letter 'e' refers to an elder brother and the letter 'to' refers to a younger brother.
Hence the name, the 'Eto.'
Eto' was originally the name of the Ten Stems or the Oriental zodiac.
Kinoe refers to 'yang of wood.'

The Wu Xing (The Five Elements) are also applied to the twelve signs of the Chinese zodiac. The basic principle of Wu Xing corresponding to the four seasons is that spring is moku, summer is ka, autumn is gon, and winter is sui. As for do (土), it corresponds to the last month of every season. The famous 'Doyo No Ushi No Hi' (Midsummer Day of the Ox), occurs in the last month of summer. The twelve signs of the Chinese zodiac are divided into seasons as follows.

For Spring, February is linked with tora (tiger), March with u (rabbit) and April with tatsu (dragon) (moku, moku, do in Wu Xing). For Summer, May is linked with mi (snake), June with uma (horse) and July with hitsuji (ram) (ka, ka, do in Wu Xing). For Autumn, August is linked with saru (monkey), September with tori (cockerel) and October with inu (dog) (gon, gon, do in Wu Xing). Finally, for Winter, November is linked with i (pig), December with ne (rat) and January with ushi (ox) (sui, sui, do in Wu Xing).

In yin and yang of the twelve signs of the Chinese zodiac, counting begins with ne (子), and odd-numbers are regarded as yang whilst even-numbers are regarded as yin. When the ten stems and the twelve signs of the Chinese zodiac are used together, yin and yang cannot be combined, however any combination of yin and yin or yang and yang is possible. That is why there are only sixty possible combinations, not 120 (10x12). Kinote-tora is possible but no kinoto-tora, and kinoto-u but no kinote-u.
Reki (the traditional calendar)

Risshun (the first day of spring) is considered as the beginning of the year in the traditional calendar of Inyo Gogyo Setsu and the Chinese astrology centered around it. In addition, the starting point of each month is not the first day but the 12th of 24 'setsu' (divisions of the old calendar) in the first half of the month. It depends on the year, but it is generally from the fifth to the eighth day on the month. It is called tsuki no setsuiri.

Risshun (the first day of spring) is around February 4. Keichitsu (awakening of insects) is around March 6. Seimei (literally, clear and bright) is around April 5. Rikka (the first day of summer) is around May 5. Boshu (grain in ear) is around June 6. Shosho (small heat) is around July 7. Risshu (the first day of autumn) is around August 7. Hakuro (literally, white dew) is around September 8. Kanro (late-autumn or early-winter dew) is around October 8. Ritto (the first day of winter) is around November 7. Taisetsu (great snow) is around December 7. Shokan (the second coldest day) is around January 5.

As a result, 'the animal symbol of the year' in January is the same as that of the previous year and 'the animal symbol of the month' on March 3 is the same as that of the previous month one as well.

The Logic of Inyo Gogyo Shiso (Yin-Yang Wu-Xing Thought)
Inyo Gogyo Shiso (Yin-Yang Wu-Xing Thought) is a combination of 'Yin-yang' and 'Gogyo shiso' as mentioned above.

Yin-yang' was originated by Emperor 'Fukki,' also known as Fukugi) who appears in ancient Chinese mythology. It is the idea that all the phenomena do not exist independently, but exist in one of two conflicting forms, 'yin' and 'yang' (for example, light and darkness, heaven and earth, men and women, good and evil, good fortune or bad), each one repeating rise and fall.

Meanwhile, 'Gogyo Shiso' was invented by 'U,' founder of the Ka Dynasty (also translated as the Xia Dynasty), and stated that all things consists of the five elements, 'moku, ka, do, gon, sui.'
Later, Suen, an Inyoka (a practitioner of the Yin and Yang school of philosophy) from Sei (State of Qi in ancient China), completed the Inyo Gogyo Shiso by combining Yin-yang ideology and Gogyo Setsu (the Five Elements Theory), making a connection between Gogyo, the five planets and various phenomena.

Gogyo Sosho (The Harmonious Relationship between the Five Elements) and Gogyo Sokoku (Rivalry between the Five Elements)
A distinctive feature of Gogyo shiso is the idea that the five elements influence each other.
Element which complement and have a strengthening influence on others are called 'Sosho' and elements which control and have a weakening influence on others are called 'Sokoku.'
However, attention should be paid to the fact that 'Sosho' is not necessarily good simply because of it strengthens just as 'Sokoku' is not always bad simply because it weakens.

Gogyo Sosho
"Gogyo Sosho" refers to relationships between elements such as 'wood makes fire, fire makes earth, earth makes metal, metal makes water, water makes wood.'

Gogyo Sosho' expresses the strengthening influence elements have on others in the following order: wood, fire, earth, metal, water, and wood. This is symbolized in the way that wood becomes fire after burning, which generates ash (or earth) that gathers to makes a mountain, where minerals (metal) are produced, metal returns to water after corrosion and finally water grows wood.

Gogyo Sokoku
"Gogyo Sokoku" refers to relationships between elements such as 'wood defeats earth, earth defeats water, water defeats fire, fire defeats metal, metal defeats wood.'

Gogyo Sokoku' expresses the weakening influence elements have on each other, in that water influences and weakens fire, as fire does metal, metal does wood, and wood does earth. It is symbolized in the fact that water puts out fire, fire melts metal, from which cutting tools are made, metal cuts wood, wood shoots through earth, and earth holds streams of water.

Inyo Gogyo Shiso in Japan
Inyo Gogyo Shiso came to Japan in the 5th-6th Centuries along with traditional Chinese culture such as rekiho (method of making calendars), in the same way as Buddhism and Confucianism. Thus a public office called Onmyoryo (a government office that had jurisdiction over calendar preparation, astronomy, divination, etc.) was built under the ritsuryo system. From then on it developed in a way unique to Japan into Onmyodo (the way of Yin and Yang), with Taoist methodology.