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This question started when I learned that "hannya haramita" (般若波羅蜜多) comes from Sanskrit "prajñā pāramitā" (प्रज्ञापारमिता). It is not hard to see that what was /p/ in Sanskrit became /h/ in modern Japanese.

I have a guess: there was a [p]-like sound found in an early stage of Japanese (about the time when "prajñā pāramitā" was borrowed) and a sound change affected the sound turning it into modern /h/ after a while, and a new /p/ was created from some other source. Another possibility is that something in a chinese language affected it, since the Japanese terms were borrowed from China, which borrowed it from the indian sources.

Is one of these hypotheses right? If it is how was the phonological system of this early stage of Japanese and whence comes the modern phoneme /p/? What chinese language acted as an intermediate language and what was its phonological inventory (if it had any effect at all)?

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    Your first guess is absolutely right: Old Japanese /p/ first merged with /w/ intervocalically in Early Middle Japanese; later on, in Late Middle Japanese, most /p/’s that didn’t immediately follow a consonant became /f ~ φ/, which eventually became /h/ in the Modern Japanese period. Loan words, analogy and mimetics have provided new /p/’s so that they are now distinct phonemes, but even in modern-day Japanese, /p/ and /h/ are clearly related; e.g., in rendaku, /h/ becomes /b/ (← voiced /p/), as in 人人 hito-bito ‘people’. Jul 4 at 16:47

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This is due to sound changes in both Japanese and Chinese.

In Japanese, f & h are largely allophones of each other, reflecting earlier *f which became h except before rounded vowels. Loanwords are starting to introduce a phonemic split here though.

Late Middle Chinese f descends from Early Middle Chinese p when there is a following glide (which is then lost). As all Chinese varieties other than Min (which often then acquired readings based on this change as borrowings from non-Min varieties) underwent this change it does little to identify the source of this borrowing. That said, these characters have a b in most modern Chinese varieties, rather than an f. This may be due to their medial glide being w rather than j. It's possible that the source variety had slightly more extensive labiodentalisation than the ancestor of the modern varieties.

Luckily, different readings of characters in Japanese are categorised according to their source. The reading of 般 as han is a kan'on reading based on the pronunciation in Chang'an (modern Xi'an) during the late Tang and early Song dynasties. The reading of 波 as ha could be either kan'on or go'on (the latter corresponding to the earliest layer of Sino-Japanese readings, from the lower Yangzi). Generaly go'on are common in Buddhist terminology, but the likelihood is this entire term was borrowed at the same time, so this particular use likely reflects a kan'on reading.

So, the p's here were borrowed into Chinese, and developed a medial glide. This medial glide triggered labiodentalisation, so the word was borrowed into Japanese as *f which then debuccalised to h as the following vowel was not rounded.

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    I haven’t looked at Sanskrit loans in Chinese for a long time, but if memory serves, it’s fairly regular that Skt. <p> /p/ → Cn <b> /p/, while Skt. <ph> /pʰ/ → Cn <p> /pʰ/. 般 bān and 波 never had palatal glides, only labial ones, so we wouldn’t expect them to develop an initial /f/. But they wouldn’t need to, since plain <b> /p/ was generally borrowed into Japanese as <p> /p/, which subsequently became /f/ some time in LMJ and eventually /h/ in NJ. (I’m not sure if it’s relevant that the ModCn pronunciation of 般若 is irregular – not bānruò as expected, but bōrě.) Jul 4 at 16:39

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