I am using a spectrogram to find the frequency contours in Hz of the different tones that are present in Chinese, and trying to compare this to musical intervals to see if there is actually a musical element to tone. Is anyone aware of similar research that has been done?
五度标记法 （five level tone mark, labeled 1, 2, 3, 4, 5, roughly do re mi fa so）is similar to intervals. It was invented by Zhao Yuanren in the 1920s. It's considered inaccurate, but almost always, at least in Chinese-written books, used to describe 调值 （the0 tone values） of the Chinese languages, and other languages in China that has a tone, like Yi, Hmong, Lhasa Tibetan, Zhuang and etc. In fact, it's the main tool that Chinese scholars used to describe 调值 for any tonal language if they do not want to go into much detail. Since 五度标记法 is not accurate but used as a standard, if it's a book dedicating to the phonetics and phonology of a language, the common practice is to use 五度标记法 to describe it first, then raise issues, and describe it using some other tools. One tool I particularly like is 分域四度制 （《语音学》朱晓农）. Another commonly used tool is polynomials. For other purposes, 五度标记法 usually suffices.
Again, as 五度标记法 is not accurate, there are not many research dedicated to it. But rather it's widely used, so you can find chapters of it in many books, like 《语音学教程》林焘 王理嘉. Here are some applicable examples.
普通话 （Standard Mandarin） has 4 tones. Tone 1 is called 阴平, with 调值 55. It's often written as superscript, as [ma55]. Tone 2, 阳平, 35. Tone 3, 上声, 214. Tone 4, 去声, 51. 《现代汉语》北京大学中文系现代汉语教研室.
香港话 （Hong Kong Cantonese） has 9 tones. 阴平 53 or 55, 阳平 21 or 11, 阴上 35, 阳上 13, 阴去 33, 阳去 22, 阴入 5, 中入 3, 阳入 2. 《粤语（香港话）教程》
彝语诺苏话 (Nuosu Yi) has 4 tones. Tone1 55, Tone2 34, Tone3 33, Tone4 21. 《凉山彝语语音概论》
To use 五度标记法 correctly, one needs to know that 1）the intervals between levels are equal, with no step, half-step distinction. Thus the intervals 2-1=3-2=4-3=5-4. 2）it describes the frequencies of ONE person speaking ONE language. 1 is a relative lowest level for ONE person speaking ONE language. So my 55 is different to yours when speaking Standard Mandarin, and my 55's are different when speaking Standard Mandarin and Hong Kong Cantonese.
I once calculated my tone values for my dialect of Chinese. (To calculate the tone values for a language/dialect, you need to take averages on many people) The highest frequency is 374Hz, and lowest is 164Hz. Then take logs on frequencies, we have 2.573 and 2.215. Then divide the region into five equal bands: 2.215-2.286, 2.286-2.358, 2.358-2.430, 2.430-2.501, 2.501-2.573, labeled 1、2、3、4、5. For borderline values, we need to look case by case according to what it feels like and the values of neighboring tones. My testing results are 阴平 23, 阳平, 21, 上声 41, 去声, 212, for a Sub-dialect under Chengyu Region of Southwestern Mandarin.
Traditionally, Chinese music is 依字行腔, that is we sing along the tonal contour. However, modern Mandarin pop music doesn't completely follow this. We sometimes see violations. But at least, composers try to avoid obvious differences, like a rising melody with Tone 4. This is to avoid confusion for listeners. Confusion is obvious for translated pieces, especial in musicals. If you're interested, there's an excellent video, in Chinese, discussion the Chinese version of French musical Romeo and Juliet https://www.bilibili.com/video/BV1G44y1u7HW. They also published a paper: Guo. F., Zhang, C., Zhang, Z., He, Q., Zhang, K., Xie, J., Boyd-Graber, J., 2022. Automatic Song Translation for Tonal Languages. https://doi.org/10.48550/arXiv.2203.13420.
Cantonese pop requires a stricter match of melodies and tones. For example, the lines 3 0624700 of 《数字人生》reads the same as it sings.
If you search 依字行腔 in cnki, there are tons of papers. For example， 京剧唱腔中的腔词关系——对“依字行腔”原则的剖析, 从一字一音到依字行腔以及汤、沈之争,浅析民族声乐作品《孟姜女》的演唱技巧.
There's also a book chapter Tone and music processing in Chinese in The Routledge Handbook of Chinese Applied Linguistics https://www.taylorfrancis.com/chapters/edit/10.4324/9781315625157-43/tone-music-processing-chinese-caicai-zhang. I haven't read it, but the title suggests high relevance.
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The answer is no.
In music, we divide an octave into 12 half-steps, and each amounts to 100 cents. This makes the relative intervals fixed. But in Chinese tone values, we divide the full used pitch range into 5 bands (taking logs converts frequencies to cents https://en.wikipedia.org/wiki/Cent_(music). The base doesn't matter for a relative relationship). Here are two issues,
The used pitch range for a person in a language is not fixed, which makes the difference in levels 1 and 2 not 200 cents, but rather arbitrary. Take my frequencies for my dialect. The highest is 374Hz, or roughly F#4 (A4=440Hz). The lowest is 164Hz, or roughly E3. This is more than an octave, but still we label it from 1 to 5. My level one corresponds roughly to E3~G3.
The music note is a point, fluctuating for more than 1 cent is detectable by human ears. But the level for tone values is a band. If we migrate this to music, it would be that any frequency between C4 and D4 is considered as a do, which is not the practice in music.
These being said, most people who are learning a new dialect by textbook take do re me fa so as approximation. This usually suffices.