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I am not sure how these two sounds are used in Japanese. Which one is more common, in terms of frequency? Under what cases is the other one used?

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Did you look at Wikipedia? "Of the allophones of /z/, the affricate [dz] is most common, especially at the beginning of utterances and after /ɴ/ (or /n/, depending on the analysis), while fricative [z] may occur between vowels. Both sounds, however, are in free variation." –  Zhen Lin Aug 2 at 19:57

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From The Sounds of Japanese (Vance 2008), p.85:

We'll transcribe [dz] phonemically as /z/ because there's no contrast between [dz] and the voiced lamino-alveolar fricative [z]. Typically, though not consistently, [dz] occurs at the beginning of a word or in the middle of a word immediately following a syllable-final consonant (§5.1, §5.6), and [z] occurs in the middle of a word immediately following a vowel. In short, [dz] and [z] are allophones of this /z/ phoneme. Most native speakers of Japanese are quite surprised to discover there's actually a phonetic difference to worry about, but you'll hear it if you listen carefully to pronunciations of zu [dzɯ] 図 'diagram' and chizu [cɕizɯ] 地図 'map'.

In the above, "a syllable-final consonant" means either /N/ or /Q/, which Vance explains in sections §5.1 and §5.6 respectively. In the latter section (p.108), he goes on to write:

As we saw in §4.3, /z/ has both [z] and [dz] as careful-pronunciation allophones, but following /Q/, /z/ is always [dz].

Although note that voiced geminates like this appear almost exclusively in loanwords, and even there under certain conditions are commonly devoiced—see e.g. A corpus-based study of geminate devoicing in Japanese (Kawahara and Sano 2013) or their other recent work for discussion.

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Note that this lack of contrast described by Vance does not apply to all Japanese varieties. Speakers in the south and west tend to differentiate between [dz] and [z]. More at en.wikipedia.org/wiki/Yotsugana. –  Eiríkr Útlendi Aug 3 at 7:16
@EiríkrÚtlendi Yes, there are a few dialects that retain the distinction, around Shikoku and Kyushu according to Wikipedia. Thank you for pointing that out. But this answer is about standard Japanese. (I would be careful with the phrase "in the south and west", since that might be taken as describing a much larger region than actually makes the distinction. They're merged for most speakers.) –  snailboat Aug 3 at 7:25

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