The recent additions to my series on Messiaen (here and here) remind me that he was a major influence on Toru Takemitsu (1930–96). Here I’ll just feature some of his works directly inspired by the traditional Japanese soundscape—though of course there’s much more to explore in his ouevre (wiki; see also e.g. Tom Service’s succinct general introduction).
Having spent his early years until 1938 with his family in Dalian in occupied northeast China, where his father worked as a businessman, military conscription in 1944 further alienated him from Japanese militarism and nationalism; coming to associate these—not incorrectly—with the musical traditions of Japan (see e.g. this article on gagaku), he was drawn instead to new Western Art Music. He extended his initial aversion to Japanese music to other traditional forms:
There may be folk music with strength and beauty, but I cannot be completely honest in this kind of music. I want a more active relationship to the present. (Folk music in a “contemporary style” is nothing but a deception). [Hah! Discuss!]
Indeed, by contrast with the value-free ears of outsiders, some younger urban native listeners often hear their own traditions as tainted by association with a repressive or stultifying establishment—such as Chinese and Moravian folk, English choral music, or Russian Orthodox liturgy.
So it was only from the early 1960s, partly through John Cage—another important inspiration for him—that Takemitsu came to value the Japanese concept of ma 間 “empty space” (exemplified by Noh drama) and began consciously to borrow from Japanese music. As he recalled:
One day I chanced to see a performance of the Bunraku puppet theatre and was very surprised by it. It was in the tone quality, the timbre, of the futozao shamisen, the wide-necked shamisen used in Bunraku, that I first recognized the splendour of traditional Japanese music. I was very moved by it and I wondered why my attention had never been captured before by this Japanese music.
Here’s the second story, “Woman of the snow”, from the soundtrack for Kwaidan (Masaki Kobayashi, 1964):
Best known among his Japanese-inspired works is November steps (1967), for shakuhachi, biwa, and orchestra, commissioned by the New York Phil at the behest of Bernstein, premiered under Seiji Ozawa:
For the same combination is Autumn: into the fall after a little while (1973):
Traditional Japanese music, notably the courtly tradition of gagaku, deriving from Tang China, had long inspired Japanese and Western composers. Henry Eichheim‘s visits to east and southeast Asia (for his trips to China, see here) led to works such as Oriental impressions (1919–22), including the gagaku-derived E[n]tenraku (cf. Japanese nocturne); in 1931 Hidemaro Konoye (who the previous year conducted the very first recording of Mahler 4!) made a more faithful orchestral arrangement of Etenraku. Both works were soon taken up by Leopold Stokowski in his programmes with the Philadelphia orchestra.
After the war, Etenraku was again the basis for Yoritsune Matsudaira‘s Theme and variations for piano [hmm] and orchestra (1951); he followed it in 1961 with the orchestral piece Bugaku. Also channelling gagaku were Henry Cowell in Ongaku (1957), and Olivier Messiaen in Sept haïkaï (1963— to which I devoted a separate post. 
The Reigakusha ensemble (site, largely in Japanese).
But now some composers actually began writing for the gagaku ensemble itself, as innovation became a significant subsidiary theme in the gagaku world. Takemitsu wrote Shūteiga for the gagaku ensemble of the Imperial Household (hichiriki oboe, shō mouth-organ, ryūteki flute, biwa lute, gakusō zither, and percussion), later incorporating it into In an autumn garden (1973), one of the most enthralling essays in the genre. Besides the Imperial Household’s own version, the sonorities of this recording, by the Reigakusha ensemble that grew out of it, are even more mesmerising (cf. this live performance):
Garden rain (1974), for brass ensemble, evokes the cluster-chords of the shō mouth-organ (so very different from the anhemitonic pentatonic organum of its Chinese ancestor the sheng!):
See also this interview with the enterprising shō performer Mayumi Miyata.
Just in case you suppose all these to be avant-garde creations far distant from their model, here’s some “traditional” gagaku:
 For the Japanese explorations of Eichheim, Cowell, and others, see W. Anthony Sheppard, Extreme exoticism: Japan in the American musical imagination (2019), ch.3. For other Western works inspired by gagaku, click here. For Western devotees of Zen, see The great Gary Snyder, and More East-West gurus; see also under Some posts on Japanese culture.