Numbers in Asian Music
TRÂN QUANG HAI
The ancients saw music as an application of cosmic algebra. The Chinese, in their science, considered only the qualitative aspect of the Numbers that they manipulated as signs and symbols. Among the three functions of numbers, the distinction between cardinal and ordinal use is less essential than the distributive function. Thanks to this quality, numbers provide the function of uniting a set, of grouping.
The ratios expressing relationships among musical sounds have a correspondence in all other aspects of an event. The governance of these relations allows comparisons between musical harmony and all other harmonic classifications: colours, shapes or planets.
The number 5 represents the 5 elements, the 5 movements. “Five” evokes the 5 senses, the 5 organs that are a coagulation of breaths (as wind instruments).
The pentatonic range has a centre (GONG) surrounded by four notes assimilated to the four directions in space (SHANG-West, JIAO-East, ZHI-South, YI-North).
It is therefore the law of numbers that governs the proportions of the musical edifice. Sounds, just like numbers, obey the stimuli of attraction and repulsion. Their successive or simultaneous order – i.e. melodic and harmonic movements of the chords respectively – allows for an apparent structure of musical form. Sounds are ordered, paired up and make up structures that evoke real and imaginary worlds.
The root note (also known as tonic or keynote) constitutes the basis and the centre: it is the reference point that allows the construction of the musical edifice.
In the example where the tonic is the FA, the DO note will maintain a fifth interval with its tonic. The very note DO will play a role of fourth in a range of SOL. Our usual DO (Ut) is a tonic in the construction of the classical scale model.
The first and basic tube (Huang Zhong) reproduces the FA (Gong tonic). This FA is close to the FA sharp of the physics scale with its 708.76 vibrations per second. This generator tube represents the Central Palace around which the other elements gather. Yellow is the emblematic colour of the centre: it evokes the Sun, centre of the Sky or heart of the flower. It is reserved to the Emperor, central individual on Earth.
The 12 LÜ or Musical Tubes
We will not insist on the well-known legend of the discovery of the LÜ scale at the hands of Ling Lu (Linh Luân), a music master at the time of the famous Emperor Huangdi (Hùynh Dê, 2697-2597 before our era). In the solitary valley of the mount Kouen Louen (Côn Lôn), at the western borders of the empire, he found bamboos of the same thickness and obtained the fundamental sound, the HUANGZHONG (Hoàng Chung, the Yellow Bell) by blowing in one of the canes after cutting it in between two knots. He obtained the complete LÜ scale thanks to the sound made by a male and a female phoenix.
The LÜ scale would basically correspond to the modern chromatic scale. The absolute pitch of the fundamental sound, the HUANGZHONG, changes according to the dynasty. We will choose the FA, just like Louis Laloy, a French missionary specialist in Chinese music at the beginning of the nineteenth century.
The LÜ scale will then be:
HUANGZHONG (Hoàng Chung, the Yellow Bell): FA
TALÜ (Dai Lu, The Great Lyu): FA#
TAIZU (Thai Thô’c, The Great Iron of Arrow): SOL
JIAZHONG (Gia’p Chung: The Narrow or Still Bell?): SOL#
GUXIAN (Cô Tây, Ancient Purification): LA
ZHONGLÜ (Trong Lu, cadet Lyu) : LA#
RUIBIN (Nhuy Tân, Beneficial Fertility) SI
LINZHONG (Lâm Chung, The Bell of the Woods): DO
YIZE (Di Tac, The Same Rule): DO#
NANLÜ (Nam Lu, The Lyu of the South): RE
WUHI (Vô Xa, the Imperfect): RE#
YINGZHONG (Ung Chung, The Bell of the Eco): MI
But, when composing their melodies, the Chinese did not use the scale of the 12 sounds thus obtained – starting from the tonic note, the HUANGZHONG, through a succession of fifths. Instead, they were content with the five degrees that form the pentatonic scale GONG, SHANG, JIAO, ZHI, YI (Cung, Thuong, Giôc, Chuy, Vu). In the SHI JI (Su Ky: Historical Memoirs), Xi Ma Tian (Tu Ma Thiên), has given the dimensions of the tubes that create the notes of the Chinese pentatonic scale:
9 x 9 = 81 (placing 81 millet grains one next to the other gives a length corresponding to that of a bamboo cane giving the fundamental sound of the Yellow Bell HUANGZHONG).
In their cosmogonic system, the Chinese determined relations based on the law of numbers. These relations are also valid in the music world. The 12 musical tubes, or LÜLÜ, are the basis of this music theory. They generate in turn, in a rhythmic proportion, both by diminishing by a third and by increasing by a third: generation therefore happens through the action of the Three.
According to the type of operation carried out, two generations of tubes are created. The inferior generation provides a shorter tube, whose sound is more acute than the other one and length is reduced by a third. The superior generation provides a longer and deeper sound than the previous one, with respect to which it has been increased by a third.
The inferior generation is the result of a multiplication by two thirds, that is the inverse of a fifth. So, the first tube measuring 81 produces the second shortest tube: 81 x 2/3 = 54. Therefore, from FA (81) and DO (54) there exists a fifth.
The superior generation is the result of a fourth, since a 4/3 ratio is what characterises it. Therefore, the second tube (54) produces the third tube: 54 x 4/3 = 72. From DO (54) to SOL (72, longer and therefore deeper), there exists a fourth.
FA first tube 81
DO second tube 54 81 x 2/3
SOL third tube 72 54 x 4/3
RE fourth tube 48 72 x 2/3
LA fifth tube 64 48 x 4/3
MI sixth tube 42 64 x 2/3
SI seventh tube 57 42 x 4/3
FA# eighth tube 76 57 x 4/3 inversion
DO# ninth tube 51 76 x 2/3 inversion
SOL# tenth tube 68 51 x 4/3 inversion
RE# eleventh tube 45 68 x 2/3 inversion
LA# twelfth tube 60 45 x 4/3 inversion
The GONG (Cung) note = FA.
2/3 of 81 give 81 x 2/3 = 54; this note is known as ZHI (Chùy): DO
4/3 of 54 give 54 x 4/3 = 72; this is the note known as SHANG (Thuong): SOL
2/3 of 72 give 72 x 2/3 = 48; this note is known as YI (Vu): RE
4/3 of 48 give 48 x 4/3 = 64; note known as JIAO (Giôc): LA
Edouard Chavannes quoted and commented Lyus’s passage in appendix II, p. 636, of the third volume of his Historical Memoirs: the HUANGZHONG produces the LINZHONG; the LINGZHONG produces the TAIZU; the TAIZU produces the NANLÜ; the NANLÜ produces the GUXIAN and so on. Another part is added to the three parts of the generator to create a superior generation; one part is taken from the three parts of the generator to create an inferior generation. The HUANGZHONG, the TAIZU; the JIAZHONG, the GUXIAN, the ZHONGLÜ, the RUIBIN belong to the superior generation; the LINZHONG, the YIZE, the NANLÜ, the WUYI, the YINGZHONG belong to the inferior generation.
According to LIU PUWEI, the tube whose length is 4/3 of the generator tube belong to the superior generation and provides the fourth inferior, that is the low octave of the fifth of the sound of the generator tube. The tube whose length is 2/3 of the generator tube belongs to the inferior generation and provides the fifth of the sound of the generator tube.
These five notes correspond – according to SIMA QIAN, quoted by Maurice Courant – to “the 5 LÜ” HUANGZHONG, TAIZU, GUXIAN, LINZHONG and NANLÜ, the only ones whose measure is expressed in whole numbers starting from the basic 81:
GONG (81): Huangzhong: FA
SHANG (72): Taizu: SOL
JIAO (64): Guxian: LA
ZHI (54): Linzhong: DO
YI (48): Nanlü RE
This scale would have been used under the Yin dynasty (1776-1154 B.C.). According to Maurice Courant, the heptatonic scale – obtained by adding the two complementary or auxiliary notes BIEN GONG (Biên Cung) and BIEN ZHI (Biên Chùy) to the pentatonic scale – “existed at least a dozen centuries before the Christian era”. These auxiliary degrees are obtained by pushing the succession of fifths up to the seventh, starting from the fundamental sound. If we give the GONG (Hoàng Chung) degree the pitch of the FA, the BIEN GONG will have the pitch of a MI and the BIEN ZHI that of a TI.
Therefore, the cycles obtained starting from the fifths present analogies with the movements of the sun (seasons), of the moon (moon months), but also with planets mirroring in the organs. The Chinese associate them with a symbolic colour that has a therapeutic value.
Symbolism in Musical Instruments
In Chinese mythology, there are eight instruments destined to let resound the eight forces of the compass rose. Each instrument has a sonorous body made of a different material, which determines its peculiar character.
1. The sound of SKIN corresponds to the North
2. The sound of CALABASH-GOURD corresponds to the N-E
3. The sound of BAMBOO corresponds to the East
4. The sound of WOOD corresponds to the S-E
5. The sound of SILK corresponds to the South
6. The sound of EARTH corresponds to the S-W
7. The sound of METAL corresponds to the West
8. The sound of STONE corresponds to the N-W
In the North, and at the SKIN, the drums are eight. The GOURD (or CALABASH), in the North-East, has the peculiarity of consisting of a series of 12 Liu, some YIN, some YANG. The instrument allowed making four different sounds at the same time.
The sound of BAMBOO, in the East, was produced by sonorous tubes (Koan Tse). There were three types of these, all of which had 12 tubes: deep, medium and acute sounds, corresponding to the Earth-Man-Sky triad). Later, they evolved towards a separation between yin and yang tubes that constituted two distinct and complementary instruments.
The sound of WOOD was represented by various instruments of which the Tchou – shaped like a bushel and named after Ursa Major – would initiate the concert in the same way as the Ursa indicates the beginning of the day or of the year with its position.
The sound of SILK was produced by stringed instruments known as Qin. These were five-string zithers that originally had a rounded top part representing the Sky and a flat front part representing the Earth. They had five strings to represent the five planets or the five elements.
The sound of EARTH was produced by instruments made of clay that gave a GONG – that is the tonic FA – as deep sound as well as four more tones (SHANG; JIAO; ZHI; Yi). In the West, the sound of METAL was rendered by twelve copper and tin bells that gave the twelve semitones of the LÜ. The sonorous STONES situated in the North-West were assigned to ceremonies that evoked the Sky, establishing a spiritual link thanks to the pure quality of their sounds.
In Yoga, there exist 7 chakras corresponding to 7 vowels, 7 sounds or pitches, 7 overtones and 7 points of the human body. The author has carried out experimental research in the presence of overtones in Yoga. The result of his three-year study was presented at the International Congress of Yoga in France in 2002 .
According to his research, the fundamental of voice should be at 150Hz .
Number Name of Chakras Location Overtones Vowels Number of Hz
1 Mulâdhâra coccyx H n° 4 U 600Hz
2 Svâdhishthâna genitals H n° 5 O 750Hz
3 Manipûra navel H n° 6 Ö 900Hz
4 Anâhata heart H n° 8 A 1200Hz
5 Vishuddha throat H n° 9 E 1350Hz
6 Ajnâ between eyebrows H n°10 AE 1500Hz
7 Sahasrâra top of head H n°12 I 1800Hz
The lute in the shape of a Vietnamese moon – DAN NGUYET or DAN KIM – was conceived in a harmonious style and in a totally empirical way. Every part of this instrument can be divided into three parts: the two deep strings (0.96 mm) and the high string (0.72 mm) have a vibrating length of 72 cm.
The instrument measures 108 cm overall; the sound board 36 cm; the thickness of the resonance chamber is 6 cm. The bridge is 9 cm long and 3 cm high. The decorative part of the instrument, located opposite the bridge, measures 12 cm in length. The wooden pins measure 12 cm and there are 9 ring nuts measuring 3cm in width.
The Vietnamese monochord DAN DOC HUYEN (unique stringed instrument) or DAN BAU (gourd instrument) is the only musical instrument in the world using Pythagorean theory to create overtones by the division on harmonic knots of the unique steel string of the instrument into 2, 3, 4, 5, 6, 7, 8 equal parts corresponding to the series of overtones. (In fact, if the open string is tuned in C, the overtone 2 will be C an octave higher than the pitch of open string, 3 = G, 4=C 2nd octave higher, 5= E, 6=G, 7=Bb, 8= C 3rd octave higher.)
Chinese tradition has it that – over 5000 years ago – Emperor DU XI asked his lute maker to produce a zither. This imperial instrument was to be based on the relationship between the Sky and the Earth. It therefore had the length of three thuocs, six tâcs and one phân, so as to match the number 361 representing the 360 degrees of the circle plus the centre, that is unity and multitude. The height of the zither was eight tâcs and its bottom four tâcs, to match the eight half seasons and the four cardinal points or four seasons, that is space and time. Its thickness, two tâcs, bore the emblem of Sky-Earth. The twelve strings vibrated similarly to the twelve months of the year, while a thirteenth string represented the centre.
This story shows the domination of the Number and its application in Chinese thought.
Around 1911, a musician playing the Chinese fiddle, called LIU Thien Hoa, adopted the cipher notation for writing musical scores proposed by Jean Jacques Rousseau in 1746, perfected by Pierre Galin (1786-1821) and later made popular by Aimé Paris and Emile Chevé (1804-1864). The number 1 corresponds to the root note – whatever the tonic (DO, RE, FA, SOL, LA) – and the five main degrees correspond to the five number (1,2,3,4,5). The 6 and 7 represent intervals.
The cipher notation also appears in Indonesian music: it is found in the musical scores used in the GAMELAN.
Numbers are found in the notation used for Tuvan and Mongolian throat singing. The harmonics are numbered according to frequencies starting from the root note. A cipher notation was proposed to write musical scores for split-tone singing. Ted Levin and the author have used cipher notation to transcribe Tuvin and Mongolian folksongs.
Frequency Analysis Using a Sonagraph
The sonagraph, an instrument for measuring spectra, allows pushing split-tone singing beyond basic and experimental research. Since 1970, many people – including Emile Leipp, Gilles Léothaud, Trân Quang Hai, Hugo Zemp in France, Gunji Sumi in Japan, Ronald Walcott in the United States, Johan Sundberg in Sweden, Graziano Tisato in Italy, Werner Deutsch and Franz Födermayer in Austria – have utilised the sonagraph or similar types of instruments to increase the precision of the harmonics produced by split-tone singing thanks to hertzian spectroscopy.
An examination of sonagrams shows the diversity of styles in the split-tone singing of Tuvans, Mongolians, Tibetans and of the Xhosans in South Africa. It allows to establish a classification of styles, to identify the number of harmonics and better understand the how and why of vocal techniques – something that has so far been impossible. The sonagraph allowed Trân Quang Hai to carry out introspective experimental research in overtone singing (with one tonic and two independent partials or harmonics) as well as in “harmonic drone” with variation of the tonic – DO (harmonic 12), FA (harmonic 9), SOL (harmonic 8) and DO octave (harmonic 6 to create the same harmonic pitch). Other experiments by Trân Quang Hai that have produced interesting results are those on the realisation of different spectra starting from ascending and descending singing scale (normal voice, overtone voice with parallel harmonics, overtone singing with opposite moving harmonics between drone and harmonic melody). Basing himself on self-analysis, his research, whose originality lies in its experimental character, has led him to highlight the link between the harmonic drone and the tonic melody, which is the opposite of the initial principle of traditional overtone singing. Moreover, he has interwoven the two melodies (tonic and harmonic); explored overtone singing with one tonic and two independent partials or harmonics and highlighted the three harmonic zones on the basis of the same tonic sound.
Using the sonagraph and other medical systems of analysis (laryngoscopy, fibroscopy, stroboscopy, scanner), the author has been able to propose a new dimension of undertone study with which to develop undertones or sub-fundamentals by creating F-2 (an octave lower than the sung fundamental), F-3 (an octave + a fifth lower than the fundamental) or F-4 (2 octaves below the sung fundamental). This new aspect of research on undertones has attracted a number of researchers, namely Leonardo Fuks (Brazil), Johan Sunberg (Sweden), Masashi Yamada (Japan), Tran Quang Hai (France), Mark Van Tongeren (The Netherlands), and a few overtone singers from the Western world (Steve Sklar from the USA, Bernard Dubreuil from Canada).
This article is a brief introduction to the use of numbers in various areas of Asian music. It represents merely the beginning of a study that will continue in the future.
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