As far as I understand it’s a mouth position thing. If you mime the “sh” sound, and then you mime the “k” sound in your mouth, you’ll notice your jaw moves ever-so-slightly forward. Chinese speakers tend to have that mouth position when making the “sh” sound and this is an attempt to capture that in the transliteration. It isn’t perfect, but transliterating it as “sh” wouldn’t be either.
So the Pinyin system is not the original phonic symbols, it was actually ChuYing aka BoPoMoFo, PinYin is a legacy of the Romanization of Chinese movement by the Chinese communist party after PRC is formed, where people were trying to replace Chinese characters with Latin alphabet spelled “words.”. Obviously this was a failure because of how the Chinese words are but PinYin which is the foundation of making up the sound, became the phonic standard in China.
In essence, PinYin borrows Latin characters to map to BoPoMoFo, so X maps to the ㄒ sound which is sh- but Sh maps to the ㄕ Shi sound, so on and so forth.
Short answer: romanization is an arbitrary way to map Chinese sounds to a roman alphabet characters, so they decided to use X for SH. For convenience, efforts were made to keep as many sounds the same or close to the English sounds those letters represent, but some letters were repurposed.
There have been other romanizations in the past, for example “Wade-Giles” which renders 北京 as “Peking” instead of “Beijing” and “Wade-Giles” as “Wei Chai”. It’s not so much that Messrs Wade and Giles didn’t hear well (although there is some of that) but they used a dialect of Mandarin that had a “k” sound instead of the “j” of the modern standard Mandarin. Also that they were perhaps favoring sounds that English speakers could more easily make. It’s no longer used much (if at all) except in historic things like older restaurant names in the U.S.
In Japanese romanization, there are similar oddities: “si” is used in some texts to represent the し/シ sound, but it’s more commonly written as “shi” these days, which is how it sounds to an English speaker. Any word that’s romanized with an “r” could be pronounced almost as a “l”, because the actual sound spoken in Japanese is in between: ra, ri, ru, re, ro can sound like la, li, lu, le, lo. English speakers have trouble hearing and saying this sound correctly, and Japanese speakers have trouble hearing and saying the difference between “glass” and “grass” because in Japanese there is no difference between those consonants.
To give a broader answer:
Not all languages have the same sounds, and indeed Chinese has several sounds that aren’t really in English.
Languages like Chinese aren’t originally written with the Latin alphabet (which is the alphabet English, Spanish, French etc. uses), so people decided on certain rules for how to do the conversion. It’s primarily for convenience, to facilitate communication between speakers of the different languages, but it’s only an approximation. Because as mentioned earlier, languages simply don’t have the same sounds as each other.
There is no such thing as a “Chinese X” so to speak. The “x” symbol is just used as an approximation to write Chinese words using a system that was not designed for it.
And also the short answer is “sh” had already been assigned to a different sound. The sounds represented by “sh” and “x” in the pinyin system are distinct, so they wanted to use different symbols to represent them.
Chinese has a systematic romanization standard called Pinyin to write Chinese words using Latin characters.
Of course, the Latin alphabet was invented to write Latin, not Chinese, and Chinese has some sounds that don’t have a precise equivalent in the Latin alphabet and vice-versa. So basically they took the Latin letters that Chinese doesn’t need and used them to describe sounds that other letters can’t. Chinese has two sounds that sound close to “sh”, so one of them is written “sh” and the other uses “x.”
Other folks have explained the principles behind pinyin, correctly identifying the repurposing of the character <x> for the sound /ʂ/ (which is fairly similar to English /ʃ/, which we write as <sh>).
However, no one has yet answered why they chose that sound-symbol correspondence in particular. Why should the <x> denote /ʂ/ and not another sound?
I poked around online and couldn’t find much, especially because I don’t speak or read Chinese. However, one person gave an answer which matched my suspicion (and cited a source in Chinese which I can’t read): it’s based on the Iberian usage of <x> for /ʃ/. A number of languages of the Iberian peninsula use or have used <x> that way in the past, including Portuguese, Basque, Old Spanish, Catalan, and Galician. This usage was extended to unrelated languages under colonial influence by Iberian powers, such as Nahuatl. Indeed, in the word “Mexica,” the people for whom Mexico is named, the <x> represents that /ʃ/.
It seems within reason that the choice to use <x> for /ʂ/ in pinyin was made by analogy.
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