An Ode to Music: Chaos Theory, Meridians and Melakarthas
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An Ode to Music: Chaos Theory, Meridians and Melakarthas Sai Venkatesh Balasubramanian and Gomathi Balasubramanian Sree Sai Vidhya Mandhir, Mallasandra, Bengaluru-560109, Karnataka, India. Correspondence: [email protected] Abstract: It is well known that the variations in frequencies (Swaras) and timbre (tone) form melody (Raga), the key aspect leading to all kinds of music experience. The pinnacle of exploring the frequencies within an octave is the Melakartha system of Carnatic Music, forming the motivation of the present work. A time domain waveform of the Swara is used to form an iterative map explaining the evolution and behavior of the Raga, and thus music experience. The presence of sensitivity, or chaotic behavior is seen and studied using phase portraits, maximal Lyapunov Exponents and Distance Plots. Following this, the basics of Meridian based healing in Traditional Chinese Medicine are explored, and by formulating a procedure to compute the “Deviance” of a Raga, the 72 Melakartha Ragas are mapped to the 72 Meridians in the Human Body. It is opined that this article takes the first step towards a series of research developments culminating in an era of “Holistic Healthcare and Personal/Personnel Management using Music”. Keywords: Music, Melakartha Ragas, Chaos Theory, Iterative Map, Meridians, TCM 1. Introduction: The history of music has been for the most part as rich as music itself. Around the world, music has been used for various purposes including but not limited to entertainment, spiritual progress, holy communion, recitation, lullabies, courtship and healthcare [1-14]. From a physics perspective, this is only appropriate, since music pertains to one of the primordial forms of energy – sound [15-18]. It is a well-known and established fact that most of the experience from music comes by virtue of its melody, composed of scales and notes, where notes, also called Swaras are frequencies or bands of frequencies [19-23]. Fundamental to this idea is the concept of octave, which is seen as the frequency space between a frequency and its double. That is, the frequencies f of a Swara S, in successive octaves n and n-1 are related by f(Sn) = 2f(Sn-1). Across varied cultures, it is established over time that within an octave, the number of discernible frequency bands is 7. This is the genesis of the concept of notes, or Swaras. However, frequency bands have been resolved into smaller bands, or as individual frequencies as well, which gives rise to the types of Swaras such as Major (Theevra) and Minor (Komal) [19-23]. While most of the western music is based on the Chromatic Scale, containing 12 notes as “tones” and “semitones” in an octave, or any of its subsets, Arabic Music in particular uses 24 notes, including “quartertones” as well [23-26]. Arguably though, no system of music has explored the frequencies of an octave as much as Indian Music, for reasons as explained below [27]. 1. Though on first glance, the Indian music has settled for 12 Swarasthanas in an octave, this is far different from the Western Music, as the 12 notes are not absolute frequencies. It is a frequency ratio based system where the 12 notes are chosen in relation to the fundamental C (Shadjam or “Sa”), while the absolute frequency of Sa itself is left to the choice of the performer. 2. The concept of Gamakas, or „oscillations‟ enable a form of ornamentation to notes, in the sense that notes are allowed to oscillate in the vicinity of the corresponding ratio. Ranging from simple ascent and descent to more complex forms, atleast ten varieties of Gamakas have been enumerated in treatises (Dasha Gamaka), and the use of these depends heavily on the context, such as the Raga being performed, its characteristics and so on. 3. Since the absolute frequencies of Swaras are not fixed in Indian music, it is often seen that different combinations of Swaras effect a slight shift on the frequencies of neighbouring Swaras. Thus, if a Komal Swara is replaced with its Theevra counterpart in a Raga, not just that Swara, but most other Swaras will undergo a frequency shift, albeit very slight. Using these concepts, the pinnacle of frequency exploration in Indian Music has to be the Melakartha Raga system of Carnatic Music. Conceived as a simple exercise in mathematical permutations and combinations between various varieties of Swaras, the fundamental premise is that these Ragas arise from scales that have exactly seven notes in their ascent and seven in their descent, each seven containing exactly one type (Komal or Theevra) of the 7 Swaras, and the type being the same in both ascent and descent. An interesting concept introduced in the Melakartha System is the concept of Vivaadhis, where both varieties of a Swara are used, one of which takes the role of a neighboring Swara. Though the possibilities of such permutations and combinations easily cross a hundred, a preliminary requirement that the frequency progression through Sa, Ri, Ga, Ma, Pa, Dha and Ni in the ascent as well as its reverse in the descent be unidirectional has ensured that the number of Melakartha Ragas is an all inclusive 72, of which 40 have atleast one Vivaadhi Swara [27]. In this article, the system of Melakartha Ragas are explored rom two perspectives. First, the interaction and interplay between the frequencies constituting a Melakartha Raga is seen as the embodiment of a well-known physics concept – Chaos Theory, and this is done by obtaining a Difference Equation describing the evolution of Music, as well as plotting Distance Maps. The second perspective stems from the mention in ancient treatises that the 72 Melakartha Ragas map to and influence the 72 Nadis, or energy channels, known in Traditional Chinese Medicine as Meridians. We begin with the fundamental premise that Kharaharapriya, the 22nd Mela Raga is the Raga of Vedic music, attributed to the highest of the three Gunas or “behaviors” (Sattva). From this concept, we compute how much each Mela Raga deviates from this configuration, termed “Deviance”, and using some fundamental mappings between music and Chinese Terminology, provide and list a mapping of the 72 Melakartha Ragas with the 72 Meridians. It is opined that the studies elaborated upon in this paper, both from a physical and astronomical perspective will usher in a music-dominated era, characterized by the almost no side-effect music based healthcare and personal/personnel management using music. 2. Deriving the Waveform of a Raga in Time Domain It is a well-known fact that the fundamental properties of sound are dynamics (loudness), pitch (frequency) and timbre (tone). While the frequencies indeed correspond to the Swaras, the choice of musical instrument (including vocal) affect the timbre [15]. From a signal processing perspective, Swaras merely determine the central frequencies of each of the 7 bands, whereas the timbre determines the amount of harmonic content around the central frequencies, and differs from instrument to instrument. Various attempts have been made at studying the spectral and timbre characteristics of various musical instruments and formulating mathematical expressions to synthesize such sounds, of which the most interesting result yields empirically the following expression as the output of a flute, the instrument with least number of harmonics, for a Swara S with centre frequency fs [28-34]: ( ) ∑ ( ) ( ) where A1=2.54, A2=0.245, A3=0.009 and A4=0.00001. The above equation describes the time domain waveform obtained for the flute output of a single Swara. However, a typical Raga has 12 possible Swaras, with Sa and Pa having one variety each, and Ri, Ga, Ma, Da and Ni having two varieties each. Of these 12, any Raga only allows a certain subset, and certain Ragas also have a proportion distribution among Swaras. For instance, Shankarabharanam, the 29th Melakartha Raga often associated with the Major Scale of Western Music has Sa, R2, G2, M1, Pa, D2 and N2, all in equal proportions, while Mohanam, a “child” Raga has Sa, R2, G2, Pa and D2. A Raga such as Kedhaaram, a “child” of Shankarabharanam, has this structure – Sa M1 G2 M1 Pa N2 – Sa N2 Pa M1 G2 R2. A convenient method to express these details is the Composition Matrix M. This matrix is formed by the computing the number of non-zero Swara types the Raga has, and expressing each Swara as a relative proportion of that sum. The composition matrix is a 1x12 1 Dimensional Row Vector with the following structure: [ ] ( ) The Composition Matrices of Shankarabharanam, Mohanam and Kedhaaram are respectively as follows: [ ] ( ) [ ] ( ) [ ] ( ) The Swara value for each Swara in the Composition Matrix is taken as a coefficient (weight) for equation 1 of that corresponding Swara. By forming 12 such equations for all the Swaras, and adding them up, the “experience equation” of a “Raga”, yielding the Raga Waveform R(t) is obtained as follows: ( ) ( ) ∑ ∑ ( ( ) ) ( ) In this equation, the term fs(j) refers to the central frequencies of the 12 Swaras, represented as a 1D vector, similar to M, as follows: [ ] ( ) The term f0 is the fundamental frequency of Sa, left to the choice of the performer. All calculations in the present article use a value of 100Hz for f0, a mathematically convenient value corresponding to the voice frequency of a typical adult male. However, as mentioned earlier, it is the ratio and not absolute frequency that contributes to the experience of Indian Music. 3. Chaotic Characterization of the Raga From the time domain equation of the Raga given in Equation 3, one obtains the time derivative dR(t)/dt as follows: ( ) ( ) ( ) ∑ ∑ ( ( ) ) ( ) Using this, we rewrite the derivative dR(t)/dt as a difference term Rn – Rn-1, and discretize the entire equation to obtain the following iterative function, termed the “Music Iterative Map”.