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In all Cantonese romanization systems I could find, tone 4 is labelled the falling tone. However according to http://www.freehongkong.net/cantonese-pronunciation-cantonese-tones/ Cantonese does not have a falling tone. The author of that page has renumbered the tones so his tone 6 corresponds to tone 4. He describes this tone as "bottom flat".
When listening to people pronounce words in tone 4, I can clearly hear it as falling. Is there any truth to the author's claim?
Standard Cantonese's 陽平 tone is definitely pronounced with a falling contour (21). Modern Cantonese Phonology by Robert S. Bauer, p. 144 appears to acknowledge, but did not find, a low-level contour for this tone though:
For the Mid-Low Falling tone both Yuan (1983:181) and Zhan (1985:168) also recognized a variant low level contour of
˩11in addition to the mid-low to low falling contour of˨˩21. However, analysis of the Mid-Low Falling tone contours produced by speakers in this study has not found such a low level variant contour.
"Perceptual learning of Cantonese lexical tones by tone and non-tone language speakers" by Francis et al. (2008) also measured the contours of the fundamental frequencies of each tone, and a definite falling contour was seen. This figure from Wikipedia reproduces the contours found in that paper (refer to the line for tone 4, 時):
I would try to avoid using the romanization scheme advocated in your link. It appears to reorganize the tones to make them appear more systematic than how they are actually pronounced. While that may help with learning the tones, it may introduce more confusion later, especially since it goes against the numbering scheme that all of the other Cantonese romanization schemes use.
Adding to @Claw's answer, Standard Cantonese used to have (and I heard in some dialects still has) TWO falling tones (and thus a total of 7 tones). This is still acknowledged in modern-ish dictionaries like the one published by CIHK. The former tone #2 was a high-mid falling tone, akeen to Hakka's fourth tone.
