29 EDO is the smallest number of equal divisions of the octave, resulting in a perfect fifth better than 12 EDO. Its principal third is about as imprecise as 12-TFW; However, it is flattened by 14 cents and not net by 14 cents. It also tuned the 7th, 11th and 13th harmonics flat, about the same amount. This means that intervals such as 7:5, 11:7, 13:11, etc. are very well suited in 29-TET. There is exactly one family of equal temperaments that sets the semitone to any fraction of a whole tone while keeping the notes in the correct order (which means, for example, that C, D, E, FA and F♯ are in ascending order if they maintain their usual relationship with C). That is, fixing q to an eigenshare in the relation qt = s also defines a single family of the same temperament and its multiples that satisfy this relation. The twelve-tone temperament prevailed for various reasons. It fit comfortably into the design of the existing keyboard and allowed complete harmonic freedom at the expense of a little contamination at each interval.
He also built a 12-string tuning instrument, with a series of height pipes hidden in its lower cavity. In 1890, Victor-Charles Mahillon, curator of the Brussels Conservatory, duplicated a series of pitch pipes according to Zhu Zaiyu`s specifications. He said China`s sound theory knows more about the length of height pipes than its Western counterpart, and the amount of pipes duplicated according to Zaiyu data proves the accuracy of this theory. [22] The following table compares the quantities of different equitable intervals with their equal equivalents, expressed in both ratios and cents. The first mention of the same temperament in relation to the twelfth root of two in the West appeared in Simon Stevin`s manuscript, Van De Spiegheling der singconst (c. 1605), published posthumously nearly three centuries later, in 1884. [31] However, due to insufficient precision in his calculation, many of the string-length numbers he received deviated by one or two units from the correct values. [32] As a result, the frequency ratios of Simon Stevin`s chords do not have a uniform ratio, but a ratio per tone, which Gene Cho calls false. [33] The two characters to whom the exact calculation of the same temperament is often attributed are Zhu Zaiyu (also romanized as Chu-Tsaiyu. Chinese: 朱載堉) in 1584 and Simon Stevin in 1585. According to Fritz A. Kuttner, a critic of the theory,[5] is known that “Chu-Tsaiyu presented in 1584 a very precise, simple and ingenious method for the arithmetic calculation of monoak chords of equal temperament” and that “Simon Stevin in 1585 or later offered a mathematical definition of the same temperament plus a slightly less precise calculation of the corresponding numerical values”.
The developments took place independently of each other. [6] Kenneth Robinson attributes the invention of the same temperament to Zhu Zaiyu[7] and provides verbatim quotations as evidence. [8] Zhu Zaiyu is quoted as saying that in a 1584 text, he said, “I have established a new system. I put a foot as a number from which to extract the others, and with the help of proportions I extract them. One of the earliest discussions of equal temperament is found in the writings of Aristoxenus in the 4th century BC. [ref. Vincenzo Galilei (father of Galileo Galilei) was one of the first practical defenders of twelve-tone temperament.