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In my new language, I use tones, such as those in Mandarin and Cantonese, to diversify my words. When I try to make a dictionary, I do not know how to describe these tones. I have tried using parts different English words to describe what a tone sounds like, but that is not very efficient. Is there a simpler way to describe these tones?

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I think vowel + shift in tone is key. If you'd just have different tones (as in notes) for vowels, that would conflict with expressions of pain, surprise, etc. In Asian Philology, when we talk about "tones", we actually mean a change in pitch for a vowel. Chinese has 4: Same, Up, Down and Down+Up. Thai for example has more.

Here's a grid of the most used tonal notation systems for Chinese:

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The most common in use today is Pinyin

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You can use standard musical notation to describe the tones of your conlang. Level tones are just represented by different notes, and tonal glides can be represented by groups of notes joined with a slur. The notes can also hint the duration of a syllable.

An alternative to that are Tone Letters introduced by the Chinese linguist Yuen Ren Chao providing a nice visual cue to tone levels and contour tones.

These means are for the explanation of the tones to the aspiring learner of your conlang, their representation in a writing system is another thing worth a separate question.

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    And you can of course notate tone in IPA. – curiousdannii Oct 13 '18 at 12:21
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    Musical notation for languages is problematic, because while you could certainly map e.g. “high tone” (a phonemic concept) to a certain note purely for writing purposes, but this would entirely obfuscate the reality wherein intonation, stress etc change the actual realization of tone. Similarly, if you were to write out the actual realizations, you’d only get a crude approximation. At that point, it would be better to just write a continuous line over your text indicating the pitch relative to some baseline. – Adarain Oct 22 '18 at 16:46
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Vietnamese is a tonal language that is written with a Latin character set, using accents and other "punctuation" to indicate the tones. Take a look at it because at the Wikipedia entry on the Vietnamese alphabet for some ideas about how it is implemented.

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