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Are there any particularly simple examples of circular tone sandhi known? For instance, is there an example of a rule in which two tones swap places but the rest are unaffected? Or an example of circular tone sandhi in a language with fewer tones overall?

The Wikipedia article on Taiwanese Hokkien in its section on tones shows an example of circular tone sandhi, reproduced here in an ascii diagram for convenience. This example is the simple tone sandhi case in the southern dialects (northern dialects have an arrow from 5 to 3 rather than from 5 to 7).

              ʔ
1 <----- 2 <----- 4
|        ^        ^
|   5    |        |
|  /     |        |ptk
v <      |    ʔ   v
7 -----> 3 <----- 8

Based on my understanding of the article, the different box diagrams are schematic representations of different sandhi rules that can be triggered under different circumstances, such as not being the final syllable in a phrase.

The contrast between the tones 1, 2, 3, 5, and 7 is phonemic since they can all appear on syllables that do not end in /ʔ/, /p/, /t/ or /k/. 4 and 8 are entering tones. I'm not sure why syllables ending in /ʔ/ can bear tones 2 and 3 in addition to the entering tones. The article is not explicit about whether the final /ʔ/ is deleted when a syllable changes from bearing an entering tone to bearing a non-entering tone.

This means that the given sandhi rule when restricted to tones 1, 7, 3, 2 permutes them cyclically.

Are there any simpler examples of circular tone sandhi?

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    Syllables ending in /ʔ/ do lose the /ʔ/ and change tone (4 becoming 2, and 8 becoming 3) when under sandhi. – Michaelyus Jan 16 at 12:01
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This paper analyzes Xiamen tone sandhi; this is the Southern Min dialect of Quanzhang, and is much simpler than what is given in that Wiki page. This is also the case most widely discussed in the phonological literature.

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    Please note that this paper, as it says in the 2nd footnote, "ignore[s] checked tones". Hence the resultant non-checked tone circle is exactly the same circle as Southern Taiwanese Hokkien apart from the pitch levels of each tone. – Michaelyus Jan 16 at 12:47

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