訓読み: Japanese reading
音読み: Chinese read
Kunyomi is like the Japanese way to pronounce kanji. Onyomi is the Chinese way to pronounce them, basically. Since all kanji originated from China, the Japanese like had the actual Chinese way to say them and then also had their own way of saying them. (If you notice, sometimes the onyomi for certain kanji sound almost identical to the Chinese pronunciation!)
Kunyomi is used mostly for standalone kanji words, such as verbs, adjectives, and like single kanji nouns. But when you pair two kanji together, pronunciation switches from kunyomi to onyomi to say the word.
Example: 木。Means tree, pronounced き。When used to say thursday, the pronunciation changes from き to もく。木曜日・もくようび。
There are two exceptions to the rule above, however.
1. Some kanji don't even have a kunyomi. Most have both, but every kanji has an onyomi. If this is the case then obviously, only the onyomi will be used when pronouncing the kanji.
Example: 院 has no kunyomi, so every time you read this kanji it is pronounced いん。
2. Some kanji use the onyomi as their primary reading. I only know one example of this, which is 気。Some sources say that 気 doesn't even have a kunyomi, but this website says it does. The point being that technically 気 does have a kunyomi but it is so obsolete that some sources say it doesn't. So, き, which is the onyomi for 気, clearly is the primary reading for when it is both alone and in a compound.
Hopefully this explanation helped you out!
August 3, 2014 at 12:49am