Why do Chinese and Japanese have a similar alphabet, but not Korean?

Korean used to be written in Chinese characters until the 12th century, when a Korean ruler (I forget his name) wanted to increase the literacy rate among his people, so he simplified the writing system by creating the Korean alphabet (yes it is a TRUE alphabet, and not ideograms like Chinese or Japanese Kanji, nor is it a syllabary, like Japanese Hiragana or Katakana). It definitely worked, because Korea enjoys one of the highest literacy rates in the world, much higher than the US.