There have been certain claims that the Chinese musical scale is backward in nature, but is it really true?

The long-standing claim of lacking the true octave in the Chinese musical scales was repeatedly emphasized by Needham and Robinson in their comparison of Chinese and Greek works in musical acoustics, in that the Pythagorean acoustic system is basically different from the Chinese acoustic system. The primary difference lies in the proportions of the octave. A common belief held by sinologists and historians of musical acoustics is that there is no true concept of the octave in Chinese musical theory. For example, in the work of Needham and Robinson,we have the passage:
" The Chinese gamut of pitches, on the other hand, requires only the simplest mathematics and does not use the octave as a starting point. Indeed, it does not even include a true octave at all."

Both Needham and Robinson claim that there is the existence of a comma in the Chinese musical scale. A comma is the resulting difference between the twelve generated note of the scale and the expected thirteen note in the scale (which should be exactly an octave higher than the first note). (Refer to the diagram below) Put in words, say we start with a value of 1 for C, we end up with a value 0.7399 for F, the twelveth note of the generation procedure. Continuing with a down generation from this note in accordance with the Lu-Shi Chun-Qiu sequence, we would expect to get 0.5 (one octave lower than C) but instead we end up with 0.4932666..... This difference between the recurring decimal and 0.5 is the comma.

Because of this comma, the thirteen note would not be exactly an octave apart from the first note, and continual generation of the second octave of the scale from this "wrong" note would then lead to more "wrong" notes. The notes of the next octave would be all lesser than what they should have been due to the comma, and this will lead to the notes getting lower and lower than their expected values. Each time an octaval cycle of twelve notes is completed, a comma would be introduced. As such, a spiral is created. This is commonly referred to as the "Spiral of fifths". (Because recall that the Up-and-Down generation is also called "generating by the fifths" due to the use of the perfect fifth ratio 2/3) (Refer diagram below)

Needham and Robinson thus claim that because of this comma, the Chinese scale do not really have a "true" octave. If this claim were true, it would severely limit the music compositions using such a scale as they would have to be kept within an octave.

But Needham and Robinson may have overlooked, or misinterpreted the generation of the Chinese scales using the Lu-Shi Chun-Qiu method. They have understood it to be that the generation will proceed infinitely. As long as we want notes of higher octaves, we would carry on the sequence. However, the passage quoted in the Lu-Shi Chun-Qiu made no explicit mention of this. Either the Chinese stopped and were contented with the twelve notes thus generated, or a possible inference is that another method was adopted to generate notes of a different octave.

Recall that if you apply the up-and-down generation procedure on a same base note, the two resulting notes will be exactly an octave apart. Thus, rather than generate the thirteenth note from the twelveth note of the Lu-Shi Chun-Qiu sequence, we could have gotten the thirteenth note by up-generating backwards from the second note. (Refer to animation below)

To prove that the above holds, consider the value for Huang-Zhong (C) as ß. Then the value for Lin-Zhong (G) would have been (2/3)ß. We do an extra up-generation from Huang-Zhong to get Lin-Zhong at an octave higher. This would have the value (4/3)ß. Back generating from this Lin-Zhong the desired value for the Huang-Zhong would be ß'. and (4/3)ß = (2/3)ß' => ß' = 2ß => the new Huang-Zhong is exactly an octave above the original one.

If this method was used, then the Chinese scale would be free of the comma and thus would not degenerate into a spiral of fifths. Needham and Robinson's claim that the Chinese scale do not have a true octave would then not hold.