THE CONTAINED SOUNDS OF CHINESE MUSICAL NOTATION

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1 THE CONTAINED SOUNDS OF CHINESE MUSICAL NOTATION Derek Walters ABSTRACT From Pythagoras to Copernicus, even up to the present day, there has been tradition of astronomers supplementing their studies with musical pursuits, and the Chinese sages of the early dynasties were no different in pursuing the same dual interests. This article explores a few of the factors linking Chinese natural philosophy and musical theory. Scrutiny of the correlations between the notes of Chinese music and the theory of the Five Elements reveals some curious anomalies, leading to some surprising if not controversial conjectures. This article deals with the contributions made by musical theory to the evolution of specific aspects of Five Element philosophy. These aspects are specifically: the order in which the Five Elements are listed; the concept of repeated cycles of the Five elements; the association of the musical notes with the Five Elements; the concept of a generative sequence; and its corollaries; inconsistencies in the orders of the Five Elements with musical notes; the meaning of the Contained Sound (na 納音 ). In this article, the expression Western music is used to refer to the now international style of music which originated in late medieval Europe, and is distinguished by harmonic structure and tonality. KEY WORDS Contained sound, Chinese music, Five Elements, 1 HOW MUSICAL NOTES ARE NAMED: FIXED PITCH AND RELATIVE PITCH SYSTEMS The pitches of musical notes can be defined in one of two ways: either by a fixed system which defines the actual pitch of a sound, such as C, D, E, or by a relative system which merely identifies the pitch of a note relative to others in the same piece of music, for example the tonic sol-fa system uses the syllables Doh, Ray, Mi, Fah, Soh, Lah, Ti to represent the notes of any Western major scale at any pitch. When Western orchestral musicians tune their instruments to 1

2 concert A, this is a note of fixed pitch (440Hz). Singers, however, often prefer to learn their tunes by relative pitches, as best suits their voices. Over the past century or so, Chinese musicians commonly use the relative pitch notation, numbered musical notation (jian pu, 簡譜 ) originally proposed by the French philosopher Jean-Jacques Rousseau, which simply uses the numbers 1 to 7 to represent the degrees of the scale of any key (Chen et al 2011). 2 THE FIVE NOTES OF ANCIENT CHINESE MUSIC In common with much folk music throughout the world, ancient Chinese music (Xi et al 2011; Jin 2010) had a simple natural structure constructed from a tone bank of five notes (the pentatonic scale), the names of which are 宮 (Gong, Palace ); 商 (Shang Commerce ); 角 (Jue Angle ); 徴 (Zhi Fulfilment ); and 羽 (Yü Feathers ). Western references to Chinese music often transcribe the five notes as C, D, E, G and A, but this is misleading since the Five Notes can be either fixed or relative. As notes of relative pitch, the Five Notes respond to the notes doh, ray, me, soh, lah of the tonic solfa system. or 1, 2, 3, -, 5, 6, -, of the jian pu system, the notes fah and ti ( 4 and 7 ) being absent from the Chinese musical resource. This tone bank of five notes is vividly demonstrated by a piano keyboard, on which the black notes form the pentatonic scale, the gaps between the groups of two and three black notes corresponding to the missing notes fah and ti (jian pu, 4 and 7). 3 PREDICTIONS RELATED TO THE MUSICAL NOTES AND THE CORRESPONDING FIVE ELEMENTS In the Huai Nan Zi there are accounts of predictions made by listening to the buzz of the crowds gathering at the New Year. For example, If the sound (the overall noise made by the crowds) is the note Gong then the harvest will be good; if the note is Shang wars will break out; if the note is Jue there will be drought; if the note is Zhi the harvest will be bad; if the note is Yü it will be damp. The predictions are clearly drawn from the elements associated with the particular sound. The association of the note Shang with wars suggests the Metal element (swords), and Yü (wet weather) with Water. The predictions of the other three notes could be explained if it was assumed that each note of the pentatonic scale was already known to be associated with one of the Five Elements. Not only that, but the pitch of the sounds made by the crowd would have to be compared with the standard fixed pitches of the time. 4 REMARKS ON THE NAMES AND ORDER OF THE FIVE ELEMENTS The names of the Five Elements first appear in the Shu Jing, (Book 2, The Counsels of the Great Yu) where they are noted as Water, Fire, Metal, Wood, Earth. This is a closed list, not a repeated sequence, so cannot be compared with what is traditionally known as the mutually destructive order, which could only be the case if Earth were followed by Water. Theoretically there are 120 possible ways in which the names of any five things can be ordered. If the five things are arranged in a repeating sequence, the number of possibilities reduces to 24 because any one of the five terms can be the start of the group. 2

3 The most familiar series is the mutually generating (xiang sheng) order, in which the five elements follow the repeated sequence Water, Wood, Fire, Earth, Metal, Water, Wood each element generating its follower. The variant of this order, the mutually destructive is derived by taking the alternate terms from the foregoing repeated sequence; that is, Water, Fire, Metal, Wood. Earth, Water, Fire in which each element is said to destroy its follower. From the Han dynasty onwards the various properties such as season, colour, taste, planet, and so forth attributed to the Five Elements, were listed, and such lists sometimes included the musical notes associated with the elements, viz. gong Earth, shang Metal, jue Wood; zhi Fire and yü Water, which tally with the predictions mentioned above. It is remarkable that the order of the elements associated with the five notes does not relate to either the mutually generating or mutually destructive orders. Furthermore whereas the Five Elements are considered to have equal standing, this is not the case with the Five Notes. The musical sequence begins with gong (Earth) which, as will be shown below, generates the following four musical notes in turn, but does not return to gong. There is yet another musical sequence of the five notes, the order in which they are arranged according to rising pitch, which also differs from any of the conventional sequences of the Five Elements. 5 CALCULATING THE FIXED PITCHES Note: Two words, related but not identical, are both transcribed as lü in pin; 律 lù (falling tone) meaning a rule or regulation, was used when referring to the twelve notes of the chromatic scale, while 呂 lǔ (low rising) can mean either the actual musical pipes themselves or the notes which they produced. The fixed pitches of the Chinese musical notes were determined by lengths of the musical tubes which produced the sounds. The fundamental note, gong, (also called The Palace Note or Yellow Bell ) was produced by a musical flute-like tube with a length of 81 fen ( 分 ). A carpenter s rule from the Western Han dynasty reveals the length of 100 fen to be 23.1 centimetres, it follows that the length of a tube of 81 fen would be equal to 18.7 cm. The sound made by a musical pipe of this length would depend partly on the position of the hole where the air entered, and partly on the position of the edge of the blade, but a reliable estimate of the pitch which would be produced by a tube of 81 fen would be close to A. The reason why the fundamental pipe had to be 81 fen long is that the remaining four pipe lengths are each determined as being two thirds of the length of the previous pipe, 81 being the lowest number which can be so divided four times to result in a whole number. The theoretical resulting pipe lengths would be 81, 54, 36, 24, 16, this last figure not being divisible by 3. 3

4 Because the sounds produced by the shorter pipes at these lengths would be too shrill, in practice the theoretical lengths of the shorter pipes were doubled or even quadrupled to produce deeper sounds. This produced an effective and perfectly tuned set of pipes of lengths 81, 72, 64, 54, and 48 fen respectively. The pipe lengths were thus very close to 9 x 9, 8 x 9, 7 x 9, 6 x 9, and 5 x 9; ( that is 187 mm, 166 mm, 148 mm, 125 mm, 111 mm) The tones produced by the pitch pipes set the standard for other instruments such as metal bells, gongs, and stone chimes. Thus it is that from the pipe lengths, two series of notes are formed. The notes generated by division give the notes: Gong - Earth; Zhi - Fire; Shang - Metal; Yü - Water; Jue - Wood. This is an absolutely demonstrable order of a sequence wherein each element generates the next by a consistent mathematical formula. When the five notes are assembled in order of rising pitch another order is produced: Gong - Earth; Shang - Metal; Jue - Wood; Zhi - Fire; Yü - Water. The conclusion must be that the association of the musical notes with the five elements preceded the current concept of the productive and destructive sequences of the five elements. 6 THE FIVE PLANETS AND THEIR ASSOCIATED ELEMENTS Since the compilation of the astronomical chapters in the Book of History, by Sima Qian ( BC), the five planets known to the ancients have been named according to the Five Elements, Jupiter, Wood-star; Mars, Fire-star; Saturn, Earth-star; Venus, Metal-star, and Mercury, Water-star. These names are still in use today both by astronomers and the general public. It may be observed that if Saturn, the Earth-star, is replaced by the Earth itself, then the five planets (excluding Saturn but including the Earth) are listed, from furthest to nearest) in the order of their distances from the Sun. Whether or not the early Chinese astronomers knew this is of no consequence; they were certainly aware (and recorded the fact) that the planets all moved at different speeds: Saturn the slowest taking 28 years to make its circuit of the heavens, then Jupiter at twelve years, next Mars, two years, then our own Earth as the standard measure, followed by Venus next at a mere seven months and finally swift orbiting Mercury at 88 days. Thus the order of the Five Elements reflected the speeds of their corresponding planets. It can be assumed that early philosophers would have considered an order of the five elements based on astronomical measurements would have been regarded as more celestial than an order based on the proportions of earthly sounds. 7 THE TWELVE LÜ 律 (CHROMATIC NOTES) The planetary sequence may have established the familiar order of the elements Wood, Fire, Earth, Metal, Water and so remained. On the other hand the musical sequence, instead of terminating at the fifth note was extended to produce a spectrum of twelve notes closely matching a complete chromatic scale. Liu An, in the Huai Nan Zi not only lists the appropriate 4

5 lengths for the five elemental notes, but continues with the two-thirds formula to calculate the pitches for another seven auxiliary tones until a note was reached which almost, but not quite, matched the fundamental sound of gong. These extra notes were not used in the musical composition of tunes, which used only five different notes, but the availability of a complete bank of twelve chromatic chimes meant that music could be transposed and played at any pitch. Indeed, the Zhou Li (Rites of Zhou) dictated that each transposition was to be associated with a particular month of the year. The Table 1 lists the measurements of the twelve pipes, the descriptive names of the twelve notes, and also shows the positions of the Five Element Notes of the pentatonic scale. Table 1: The Twelve Lü 律 (Chromatic Notes) Length of Pipe Name of Lü 律 Month Zhou Li 1 st 1 st 2 nd 2 nd 3 rd 3 rd 4 th 4 th 5 th 5 th 6 th 6 th 81 宮 Gong 黃鐘 Huang zhong Yellow Bell 大呂 Da lü Great Pipe 商 Shang 太簇 Tai cu Greatest Frame 夾鐘 Jia zhong Intervening Bell 角 Jue 姑洗 Gu xian Old and Pure 仲呂 Zhong lü Middle Pipe 蕤賓 Rui bin Luxuriant Vegetation 徴 Zhi 林鐘 Lin zhong Forest Bell 夷則 Yi ze Foreign Measure 羽 Yü 南呂 Nan lü Southern Pipe 無則 Wu yi Lacking Measure 應鐘 Ying zhong Responding Bell [ 宮 Gong] 5

6 Chimes and bells tuned to the twelve chromatic notes were suspended from a framework in two rows, in an order that a present-day percussionist would find baffling. The upper row of chimes (the notes) rose in pitch in whole tone steps from left to right, (in numbered musical notation, jian pu, 簡譜 ) 1, 2, 3, 4#, 5#, 6#, and the lower row (the notes) proceeding in reverse order, 7, 6, 5, 4, 2# 1#. Surviving examples of ancient stone chimes show that the first two chimes in each row were repeated at the end of the rows at higher octaves, probably to encompass all the notes of a set tune when transposed to a higher pitch as shown in Table 2. Chime Table 2: Chimes twelve chromatic notes Numbered Musical Notation ( 簡譜 ) Although the twelve chromatic notes were not a factor in the composition of Chinese music, they enabled ritual temple modes to be transposed and played at any pitch. There were strict ritual observations regarding the particular pitch appropriate to calendar factors and nature of the ceremony. Earlier, it was mentioned that the five notes could be either fixed pitch or variable pitch. So, paradoxically, although there are precise measurements for determining the pitch of gong (a musical tube 81 fen in length), gong could also be any one of the 12 transpositions as long as it was the first note of the pentatonic scale. To recapitulate, there are five notes each associated with a different element, and twelve transpositions, each associated with a month of the year. 8 EXAMPLE OF THE TWELVE TRANSPOSITIONS FROM THE ZHOU LI The Zhou Li (Rites of Zhou) describes in intriguing detail how the Twelve Transpositions were employed in a particular rite. The format described consisted of six sets of three actions, a signal, a chant, and a dance. Each signal was given in a tone and the chants in a tone, so covering the whole gamut of twelve tones. The tones of the chants however, were not always matched musically with the previous tone, but with the tone of the chime suspended directly above it. Thus the first set opens with the signal note in the 1 st tone gong followed by the 1 st tone, but for the second set, although the signal tone is the 2 nd, the music for the chant is not the 2 nd tone but the 6 th tone, which hangs beneath it. Each note for the chant is always the chime on the lower row which is found immediately below the corresponding note. What is not known is whether the notes of the chants were determined by the manner in which the twelve chimes were arrayed, or conversely, whether the chimes were arrayed to match the ritual requirements. 6

7 9 THE FIVE MODES In addition to the key (the transposition) of the music, it was also necessary for the mode of the music to be appropriate. The so-called pentatonic scale the five notes can be grouped in different patterns according to the principal note. The effect is equivalent in Western music to the change in mood between tunes in a major key from those in a minor key even though the actual vocabulary of notes used is the same. In the five Chinese modes the determining quality is revealed by the greater difference in pitch between particular pairs of notes. In effect, the five Chinese modes are distinguished by the following patterns, shown her in numbered musical notation. The distinguishing features lie in the positions of the gaps: Gong mode: Shang mode: Jue mode: Zhi mode: Yü mode: Even with a vocabulary of only five notes, the Chinese modes have emotional variance, some suitable for joyous occasions, others of a more solemn nature, and it was important that the construction of the music should harmonise with the correct temporal factors, in order that Heaven and Earth did not conflict. 10 THE ENIGMA OF THE NA YIN ( 納音 ) Four Pillars divination associates each of the sixty calendrical stem-and-branch binomials with a resultant element known as the na ( 納音 ) usually translated as the inner, contained or received sound. It features as one of the determining factors of a horoscope; and is often explained as being derived from the elemental qualities of the stem and branch pairs. This cannot be the case, however. To begin with, each binomial has the same na as the following binomial, but while the stem element is the same for both pairs, the branch element may not be. Indeed, this is demonstrated by the very first pair of binomials, where the respective elements for Stem1 and Branch I are Wood and Water, with Metal being the resultant na. But the second binomial, Stem 2, Branch II, also has the same Metal na, even though the element of the Branch, Earth, is different. Even more significantly, the converse also occurs. There are binomials which share the same identical pairings of stem-and-branch elements but have different na. As an example, the associated elements of the 7 th binomial, (Stem 7 Branch VII), Metal and Fire, are the same as those for the 18 th binomial, (Stem 8 Branch VI), yet the na for the seventh binomial is Earth while that for the latter pair is Metal. Clearly, the na had nothing to do with the elements of their stems or branches. It follows that the origin and purpose of the na must lie elsewhere. 7

8 11 THE FIFTEEN NA YIN 納音 Although there are sixty stem-and-branch pairs, the na sequence contains only fifteen elements. There are two reasons for this. Firstly, the na which governs a binomial also governs the following binomial, thus reducing the number of possibilities from 60 to 30. Secondly the na sequence for the binomials from 1 to 30 is repeated for the binomials 31 to 60. The entire sequence of fifteen is as follow: 金火木土金火水土金木水土火木水 12 CLARIFICATION OF THE TERM NA YIN 納音 Although in Four Pillar Fate Calculation the 納音 na is perceived as one of the Five Elements, the actual meaning of 音 () is sound. It might be supposed that the na represented one of the Five lü 呂. the musical notes 宮商角徴羽 ; but if the notes represented the elements, it must be remembered that the five notes have a different order to the usually accepted order of the elements. There is another puzzle; if the na represented one of the five lü 呂, it why were they called na and not na lü 納呂? The conclusion that na does not refer to a single sound, but may to a particular set of sounds, such as a chant, instrumental piece or some other musical feature of the ritual. When the sequence of fifteen na is divided into five groups of three, as shown in the table below, a remarkable symmetry is revealed. Each group of three notes commences with the key note of one of the five modes, followed by two notes which define the mode. Two of the component notes are missing from each mode set; one of these is the keynote (defining note) of the following mode, and the other is the final note of the preceding mode. The order is elegant and precise, a continuous chain which re-generates itself with repetition; if any group were erased, it could be reassembled simply by reference to the group which preceded or followed it. The table 3 shows the sequence of fifteen na, giving both the element and its associated musical note, also expressed in numbered musical notation (jian pu, 簡譜 ). Table 3: Sequence of Na Yin a b c d e f g h i j k l m n o 金火木土金火水土金木水土火木水商徴角宮商徴羽宮商角羽宮徴角羽

9 As an example, consider the second set, columns d e f beginning with gong 宮.. The five notes of the gong mode are 宮商角徴羽. Three of these notes, beginning with the key-note gong, appear in columns d e f. To complete the set, the final note of the gong mode, 羽 appears as the key-note of the next mode in column g, and the remaining note, 角 is the one which precedes the gong mode key-note in column c. The same pattern applies to each of the five modes. Each set of three notes begins with the key-note of the mode; each set is followed by the key-note of the next mode, and each set preceded by the final note of the preceding mode. 13 CONCLUSIONS The association of the Five Notes lǔ 呂 with the five elements ante-dates the concept of the mutual production sequence of the elements and related sequences. The manner in which the pitches of the fives notes was obtained by set mathematical formulas may have been the stimulus for the concept of the mutual generation of the elements. The expression na, rather than na lǔ, suggests that na is a reference to a group of notes rather than a single note. The fifteen elements of the na sequence are shown to be the elements of the five modes, with the first and last notes of each mode overlapping the adjacent ones. Descriptions of ancient ritual music, including those fragments which have been restored* show structural similarities to the stanzas obtained by division of the sequence of na into five modes. It is entirely possible that the sequence na, before it became a factor in the calculation of Chinese Four Pillars horoscopes was originally a pattern for the composition of ceremonial music for the rites of the Shang and Zhou dynasties. *See J.A. van Aalst, Chinese Music Shanghai 1884 Walter Kaufmann, Musical References in the Chinese Classics Detroit Corresponding Author: Derek Walters derekwalters205@gmail.com 9

Asian Social Science August, 2009

Asian Social Science August, 2009 Study on the Logical Ideas in Chinese Ancient Mathematics from Liu Hui s Commentary of the Chiu Chang Suan Shu (Research of the Relations between Calculation and Proof, Arithmetic and Logic) Qi Zhou School