Of Tuning and Temperament

The temperament and tuning systems in Western music is based upon the theoretical properties of physics such as wavelengths and the harmonic series; whereas the temperament and tuning systems in Chinese music is based on the physical properties of materials used such as the length of the pipes or strings being used.

However, even though both cultures look upon tuning and temperament from a slightly different approach, they are both based upon sound physical properties and hence are both scientific and could find a confluence in the systems of tuning that are developed.

Based upon the writings of the ancient Chinese in the Book of History [尚書], sounds from music are based upon language and the various tones are used to coordinate the relationships between the pitches of the sounds. In a way, to the Chinese, the spoken word and the voice has always been of the utmost priority and has perhaps influenced the aesthetics of Chinese music. Many phrasings and inflections are dependent heavily upon how a sentence is being spoken.

Similar to Western music, each octave in the Chinese musical system is being divided into 12 tones. Each of these pitches is known as a lü [律]. These 12 lüs are divided into yin and yang, with the odd numbered ones being yang and called lü [律] and the even numbered ones called lü [呂]. Combined together, they are known as the 12 lü-lü [12 律呂] or simply known as the 12 lü [12 律].

The lüs are the absolute pitch names, different from gong [宮], shang [商], jiao [角], qing jiao [清角], zhi [徵], yu [羽], and bian gong [變宮] - the names of the relative pitches in the diatonic scale. Each of these lüs have their own name and in order from the first, they are known as huang zhong [黃鐘], da lü [大呂], tai cu [太簇], jia zhong [夾鐘], gu xi [姑洗], zhong lü [仲呂], rui bin [蕤賓], lin zhong [林鐘], yi ze [夷則], nan lü [南呂], wu she [無射], and ying zhong [應鐘]. Each of these lüs are based upon an absolute pitch, similar to the letter names of the notes in Western music. Huang zhong is the pitch of the first note and is also the starting pitch that is being considered, from which other pitches are calculated. The absolute pitch of each lü is obtained from a standard tuning object, which sometimes might be a pipe of a certain length, a string of a certain length and so on. Throughout history, the absolute pitch of these notes varied, for example, in the pre-Qin dynasty, the pitch of huang zhong is close to a Ab, during the Western Zhou dynasty, close to a F, in the Tang dynasty similar to a E and during the Yuan and Ming dynasties, a D. Now for convenience' sake, scholars have been using the C as the pitch of huang zhong.

The Chinese have always looked upon music as being a part of a greater whole within the entire cosmos, and hence have related the pitches to the different seasons and months as well, each of the 12 pitches referring to each of the 12 months.

The Chinese also have different tuning systems for these 12 pitches.

Relative Fifth Tuning System

The most commonly used form of tuning that is being adopted is one that is similar to the Pythagorean tuning system, known as the Relative Fifth tuning system (五度相聲律). This kind of tuning system could be traced as far back as the Spring Autumn Warring States (722 B.C.). This system of tuning is a kind of Just tuning, in which the frequency of the pitches are in the ratio of 3:2. With this kind of tuning, the fifths sound perfectly harmonious but unfortunately, it is impossible to obtain a perfect octave. In the Pythagorean tuning system, what is done is to adjust some of the ratios within the octave to create two wolf intervals. The thirds are also impure but with this, the octaves are always in the ratio of 2:1.

In the Chinese Relative Fifth tuning system, all the intervals within an octave are in the ratio of 3:2. With this however, the octaves can never attain the ratio of 2:1, a problem already attested by the ancient Chinese people, because the starting pitch, huang zhong and the perfect octave is 1200 cents apart while the pure perfect fifth, lin zhong, is 702 cents apart, instead of 700 cents if the octave were to be divided into 12 equal intervals.

Tuning in this way also, with each interval being in the ratio 3:2, the pitches will go higher and higher, the third fifth already out of an octave range. Hence intervals that falls out of the octave range will have 1200 cents, the interval of an octave, deducted from it, keeping all the pitches that are being derived within the range of an octave.

Based upon this, we can obtain the following table for the pitches derived from the Relative Fifth tuning system:

degree of scale	1	5	2	6	3	7
pitch name	huang zhong	lin zhong	tai cu	nan lü	gu xi	ying zhong
cents	0	702	1404 - 1200 = 204	906	1608 - 1200 = 408	1110

4#	1#	5#	2#	6#	3#	7#
rui bin	da lü	yi ze	jia zhong	wu she	zhong lü	qing huang zhong
1812 - 1200 = 612	1314 - 1200 = 114	816	1518 - 1200 = 318	1020	1722 - 1200 = 522	1224

As can be seen, the qing huang zhong [清黃鐘] ("qing" referring to pitches an octave higher) is 1224 cents instead of the 1200 cents for a pure perfect octave.

Equal Temperament Tuning System

The equal temperament tuning system, the most commonly used tuning system in Western music in the present, had been proposed by music theorist and mathematician Zhu Zai Yu [朱載堉] (1536 - 1611) in the Ming Dynasty.

With the problem inherent in the Relative Fifth tuning system, scholars tried different ways and means to derive another tuning system.

One of the methods was to carry on deriving more pitches through the relative fifth tuning after the 12 pitches, until it finally resolves onto the pure perfect octave. In this way, a note series comprising more than 12 different tones was obtained. In one instance, Western Han music theorist Jing Fang [京房] (77 - 27 B.C.) derived 60 pitches to obtain a huang zhong which approaches the pure perfect octave closely. In the Southern Song dynasty, another theorist Qian Le Zhi [錢樂之] went further and derived 360 tones to approach the pure perfect huang zhong even more closely, but because the number of pitches was too many to be practical, this was never adopted. Southern Song dynasty theoretician Cai Yuan Ding [蔡元定] (1135 - 1198) took the 12 pitches derived through relative fifth tuning and added 6 more modified pitches to create a total of 18 different pitches. These were in fact, the first 18 pitches calculated by Jing Fang. Although these 18 pitches solved the problem of modulation within the 12 pitches, the problem of the pure perfect octave was still unresolved.

Following a different path from this group of scholars who attempted to resolve the problem of the pure perfect octave by adding more pitches to the tuning system, there is another school of theorists who decided not to add more notes to the system, but rather, to modify the way of calculating the pitches. He Cheng Tian [何承天] (370 - 447), a philosopher from the Northern and Southern dynasties came up with an innovation in tuning. With the deviation from the pure perfect octave through relative fifth tuning, he divided it equally into 12 and added this deviation in pitch to each of the pitches derived through relative fifth tuning. In this way, although still entirely not equal temperamental, the problem of modulation is being resolved and this could be seen as one of the pioneers to equal temperament tuning.

In 1581, Ming dynasty scholar Zhu Zai Yu published a work on tuning and temperament and in the forward, suggested a new method of tuning. Instead of using the ratio 3:2 for deriving the fifth, he used the number 749153538. 500000000 divided by this will give an approximation to 0.667, giving the new pitch of the fifth (lin zhong), and 1000000000 divided by this number will give an approximation to 1.334, giving the new pitch of the octave below the second fifth (tai cu). He further elaborated on this method of tuning in his subsequent work published in 1596, thereby establishing the method of equal temperament. However, the ruler of that time did not recognise the importance of this work and sadly, this method of tuning was never utilised. It was only in 1890 that Belgium musicologist Victor Mahillon (1841 - 1924) studied Zhu Zai Yu's calculations and found them to be perfectly accurate and in 1991, musicologist Liu Yong tested the pitches of the pitch pipes constructed out of Zhu Zai Yu's calculations and proved it to be exactly that of the equal temperament tuning that we know of now.

Just intonation

Just intonation is another important tuning system used in ancient China. This system of tuning is based upon the intervals of the pure perfect fifths, and the pure major thirds. Starting from qing jiao [清角], a pure perfect third above gives the yu [羽], a pure perfect fifth above gives a gong [宮]; from gong, a pure major third above gives jiao [角] and a pure perfect fifth above gives zhi [徵]; from zhi, a pure major third above produces the bian gong [變宮] and a pure perfect fifth above produces shang [商]. In this way, all the seven tones of the diatonic scale is obtained. In this method of tuning however, the semitone that are produced are wider than a usual semitone obtained from pure relative fifth tuning or equal tempered tuning.

In these 3 main tuning systems, the just intonation and the equal tempered tuning systems produces octaves that are pure. The relative fifth tuning system however, has pure perfect fifths within, but has trouble justifying the perfect octave. Each of these tuning systems are being used at some time or other, and in some instances, are being adopted together, for example in the tuning of the guqin whereby both the relative fifth tuning and the just intonation systems are being utilised. In music where modulation is plentiful, the equal tempered tuning system may be used.

References:

Du, Ya Xiong and Qin, De Xiang (2007). 中國樂理 [Zhong Guo Yue Li]. Shanghai: Shanghai Yin Yue Xue Yuan Chu Ban She.