Chapter Fundamentals of Indian Classical Music.1 Symbols, Notes and Octave Indian Classical Music (ICM) has its origin from Sama Veda, one of the four Vedas. Bharata Muni has illustrated ICM in his book N atyasastra [Muni, 2]. In Indian Classical Music (ICM) there are seven symbols: [Sa, Ri, Ga, Ma, Pa, Da and Ni], which are same as the seven symbols in Western Classical Music, [Do, Re, Me, Pha, So, La and Te]. Further, each of the five notes [Ri, Ga, Ma, Da and Ni] are sub-divided to form the basic group of twelve notes in ICM as: {Sa, Ri 1, Ri 2 (or Ga 1 ), Ga 2 (or Ri 3 ), Ga 3, Ma 1, Ma 2, P a, Da 1, Da 2 (or Ni 1 ), Ni 2 (or Da 3 ) and Ni 3 }; these are also called as Semi-tones 1. Note that Sa and Pa have only one note each and these are referred to as Sthayi Swara. It has been established that these semi-tones, if plotted in one dimension with logarithm of frequency to base two (log 2 f) as axis, will be points which are equi-distant as indicated in Fig..1 [Moorer, 1976]. The note Sa has twice the frequency as that of Sa and the space between these frequencies is called an octave. Rendering of a typical music (instrumental or vocal) spans more than one octave; usually the vocal music spans from two to two and half octaves, while instruments (specifically string instruments) can have larger span of octaves. Ga1 Ri 3 Ni 1 Da 3 Sa Ri 1 Ri 2 Ga 2 Ga 3 Ma 1 Ma 2 P a Da 1 Da 2 Ni 2 Ni 3 Sa f 2f log 2 f Fig..1: Twelve notes placed equidistantly in a single octave 1 It is worth noting that Ri 2 and Ga 1 are same notes, Ga 2 and Ri 3 are same notes, Da 2 and Ni 1 are same notes and Ni 2 and Da 3 are same notes) 1
Fig..2: Representation of semi-tones in a key-board The semi-tones are also represented as the twelve keys in a single octave of a keyboard, as shown in Fig..2. Using these twelve flexible notes and a set of inflexions anchored on these notes, it is possible to model and synthesize virtually any Indian musical phrase..2 Swara Right from Vedic period, music has been rendered as a combination of notes of frequencies. In the context of Carnatic Music 2, the twelve semi-tones do not exactly correspond to the frequencies mentioned in Section.1. The frequency of each note in ICM is slightly above or below the semi-tone frequencies. Further, depending on raga (explained in Section.3) and style of rendering, each note will have inflexions anchored to them. These inflected notes are referred to as Swaras. This gives specific characteristics to each raga depending on the selection of Swara. Arvindh [Krishnaswamy, 23] has carried out extensive experiments and observations of musical digital patterns obtained from several accomplished musicians like Unnikrishnan, T.R. Mahalingham, U. Sreenivas, etc. One of the main observations noted by him is that : ratio of each of the twelve notes to the note Sa is a fraction of small integers as shown in Table.1. The difference in Cents 3 of these notes shown in the table are almost equal, qualifying them as pitch values of notes in ICM [Krishnaswamy, 23]. This kind of tuning system is called Just Intonation. In Western music, notes are considered to be the fundamental unit of music, while in Indian Music, swaras are considered as the fundamental unit. 2 also known as South Indian Classical Music 3 the value in Cents of a note with frequency, f is 12 log(f/f sa ), where f sa is the frequency of sa 2
Table.1: Swaras and their ratio to Sa Note Ratio to Sa Cents Sa 1 Ri1 16/15 111 Ri2 9/8 23 Ga2 6/5 315 Ga3 5/4 386 Ma1 4/3 498 Ma2 17/12 63 Pa 3/2 71 Da1 8/5 813 Da2 5/3 884 Ni2 9/5 117 Ni3 15/8 188.3 Raga Raga is an attractive combination of swaras. Associated with each raga are the characteristic phrases (of swaras), sequence of swaras, and the treatment given to each swara in terms of its timing, rendition, prominence or ornamentation. A raga is popularly defined as a specified combination, decorated with embellishments and graceful consonances of notes within a mode which has the power of evoking a unique feeling distinct from all other joys and sorrows and which possesses something of a transcendental element. [Pandey, 23] Further, a raga is characterized by the following attributes: The choice of swaras (from the 12 swaras) Ascending and descending sequences (arohana & avarohana) The nature of inflexion on different notes (gamaka) Characteristic phrases (groups of notes) (swara sanchara) For example, in a typical raga Kalyani, the ascending and descending orders are shown below: Sa Ri 2 Ga 2 Ma 2 P a Da 2 Ni 2 Sa (ascent) Sa Ni 2 Da 2 P a Ma 2 Ga 2 Ri 2 Sa (descent) Ga is usually held (prolonged) with inflexion. Ri, Da, and Ni can be held without inflexion and Pa Ma Pa is a characteristic phrase referred to as swara sanchara. 3
.3.1 Melakartha Raga (Janaka Raga) Any Raga which contains all the symbols in the ascending and descending order without change in order of symbol is called a Melakartha Raga or Janaka Raga [Sambamoorthy, 1963]. Since all the symbols should be there in a Melakartha raga, the number of possible Melakartha ragas is finite. As indicated earlier, both Sa and P a have one note each. Let us consider the other swaras. Ma can either be Ma 1 or Ma 2. If we exclude Ma, between Sa and P a we have four swaras (Ri 1 Ri 2 Ga 1 Ga 2 ) from which one can select Ri and Ga in 4 C 2 = 6 combinations. where, m C k = m! k!(m k)! (1) Similarly, if we consider the notes Da 1 Da 2 Ni 1 and Ni 2, again we can select two notes from them in 4 C 2 ways. So the total number of combinations possible for Melakartha Raga (each of which is made up of combination of seven swara-symbols) is 6 6 = 36 with Ma 1 and 6 6 = 36 with Ma 2 thus yielding 72 Melakartha Ragas as shown in Table.2. Further, the swaras present in each raga is shown in Table.3 and Table.4. An alternative representation which contain all these facts is shown in Fig..3..3.2 Janya Raga Many Ragas, called Janya ragas, can be generated from each Janaka raga making a combination of a fixed number of swaras. Depending on the number of swaras selected, the janya ragas are again sub-classified. For example, if five swaras are selected from the Janaka raga, the janya raga is called Oudava raga and if six swaras are selected from the janaka raka the janya raga is called Shadava raga. Further, if the arohana of janya raga contain five swaras and avarohana contain six ragas, the janya raga is called Oudava Shadava raga. There are plenty of such Janya ragas. A few popular Janya ragas along with the Janaka ragas from which it is generated are indicated in Table.5. In addition to the class of Janya ragas mentioned, many more ragas containing less than five swaras are also in vogue. Hoiber et. al. [Hoiberg, 2] have mentioned the use of two or three swaras in vedic chantings. 4
Table.2: 72 Melakartha Ragas No. Ragas (with Ma 1 ) Swaras No. Ragas (with Ma 2 ) Swaras 1 Kanakangi S R1 G1 M1 P D1 N1 37 Salagam S R1 G1 M2 P D1 N1 2 Ratnangi S R1 G1 M1 P D1 N2 38 Jalarnavam S R1 G1 M2 P D1 N2 3 Ganamurti S R1 G1 M1 P D1 N3 39 Jhalavarali S R1 G1 M2 P D1 N3 4 Vanaspati S R1 G1 M1 P D2 N2 4 Navaneetam S R1 G1 M2 P D2 N2 5 Manavati S R1 G1 M1 P D2 N3 41 Pavani S R1 G1 M2 P D2 N3 6 Tanarupi S R1 G1 M1 P D3 N3 42 Raghupriya S R1 G1 M2 P D3 N3 7 Senavati S R1 G2 M1 P D1 N1 43 Gavambhodi S R1 G2 M2 P D1 N1 8 Hanumatodi S R1 G2 M1 P D1 N2 44 Bhavapriya S R1 G2 M2 P D1 N2 9 Dhenuka S R1 G2 M1 P D1 N3 45 Shubhapantuvarali S R1 G2 M2 P D1 N3 1 Natakapriya S R1 G2 M1 P D2 N2 46 Shadvidamargini S R1 G2 M2 P D2 N2 11 Kokilapriya S R1 G2 M1 P D2 N3 47 Suvarnangi S R1 G2 M2 P D2 N3 12 Rupavati S R1 G2 M1 P D3 N3 48 Divyamani S R1 G2 M2 P D3 N3 13 Gayakapriya S R1 G3 M1 P D1 N1 49 Dhavalambari S R1 G3 M2 P D1 N1 14 Vakulabharanam S R1 G3 M1 P D1 N2 5 Namanarayani S R1 G3 M2 P D1 N2 15 Mayamalavagowla S R1 G3 M1 P D1 N3 51 Kamavardani S R1 G3 M2 P D1 N3 16 Chakravakam S R1 G3 M1 P D2 N2 52 Ramapriya S R1 G3 M2 P D2 N2 17 Suryakantam S R1 G3 M1 P D2 N3 53 Gamanashrama S R1 G3 M2 P D2 N3 18 Hatakambari S R1 G3 M1 P D3 N3 54 Vishwambari S R1 G3 M2 P D3 N3 19 Jhankaradhwani S R2 G2 M1 P D1 N1 55 Shamalangi S R2 G2 M2 P D1 N1 2 Natabhairavi S R2 G2 M1 P D1 N2 56 Shanmukhapriya S R2 G2 M2 P D1 N2 21 Keeravani S R2 G2 M1 P D1 N3 57 Simh. madhyamam S R2 G2 M2 P D1 N3 22 Kharaharapriya S R2 G2 M1 P D2 N2 58 Hemavati S R2 G2 M2 P D2 N2 23 Gourimanohari S R2 G2 M1 P D2 N3 59 Dharmavati S R2 G2 M2 P D2 N3 24 Varunapriya S R2 G2 M1 P D3 N3 6 Neetimati S R2 G2 M2 P D3 N3 25 Mararanjani S R2 G3 M1 P D1 N1 61 Kantamani S R2 G3 M2 P D1 N1 26 Charukesi S R2 G3 M1 P D1 N2 62 Rishabhapriya S R2 G3 M2 P D1 N2 27 Sarasangi S R2 G3 M1 P D1 N3 63 Latangi S R2 G3 M2 P D1 N3 28 Harikambhoji S R2 G3 M1 P D2 N2 64 Vachaspati S R2 G3 M2 P D2 N2 29 Dh. sankarabharanam S R2 G3 M1 P D2 N3 65 Mechakalyani S R2 G3 M2 P D2 N3 3 Naganandini S R2 G3 M1 P D3 N3 66 Chitrambari S R2 G3 M2 P D3 N3 31 Yagapriya S R3 G3 M1 P D1 N1 67 Sucharitra S R3 G3 M2 P D1 N1 32 Ragavardhini S R3 G3 M1 P D1 N2 68 Jyoti Swarupini S R3 G3 M2 P D1 N2 33 Gangeyabhushani S R3 G3 M1 P D1 N3 69 Dhatuvardani S R3 G3 M2 P D1 N3 34 Vagadheeswari S R3 G3 M1 P D2 N2 7 Nasikabhushani S R3 G3 M2 P D2 N2 35 Shulini S R3 G3 M1 P D2 N3 71 Kosalam S R3 G3 M2 P D2 N3 36 Chalanata S R3 G3 M1 P D3 N3 72 Rasikapriya S R3 G3 M2 P D3 N3 5
Table.3: (Supplimental Table): Swaras present in first 36 Melakartha raga (1 represents presence, represents absence) Raga No. Sa Ri1 Ri2 Ga1 Ga2 Ma1 Ma2 Pa Da1 Da2 Ni1 Ni2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 13 1 1 1 1 1 1 1 14 1 1 1 1 1 1 1 15 1 1 1 1 1 1 1 16 1 1 1 1 1 1 1 17 1 1 1 1 1 1 1 18 1 1 1 1 1 1 1 19 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 21 1 1 1 1 1 1 1 22 1 1 1 1 1 1 1 23 1 1 1 1 1 1 1 24 1 1 1 1 1 1 1 25 1 1 1 1 1 1 1 26 1 1 1 1 1 1 1 27 1 1 1 1 1 1 1 28 1 1 1 1 1 1 1 29 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 31 1 1 1 1 1 1 1 32 1 1 1 1 1 1 1 33 1 1 1 1 1 1 1 34 1 1 1 1 1 1 1 35 1 1 1 1 1 1 1 36 1 1 1 1 1 1 1 6
Table.4: (Supplimental Table): Swaras present in second 36 Melakartha raga (1 represents presence, represents absence) Raga No. Sa Ri1 Ri2 Ga1 Ga2 Ma1 Ma2 Pa Da1 Da2 Ni1 Ni2 37 1 1 1 1 1 1 1 38 1 1 1 1 1 1 1 39 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 41 1 1 1 1 1 1 1 42 1 1 1 1 1 1 1 43 1 1 1 1 1 1 1 44 1 1 1 1 1 1 1 45 1 1 1 1 1 1 1 46 1 1 1 1 1 1 1 47 1 1 1 1 1 1 1 48 1 1 1 1 1 1 1 49 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 51 1 1 1 1 1 1 1 52 1 1 1 1 1 1 1 53 1 1 1 1 1 1 1 54 1 1 1 1 1 1 1 55 1 1 1 1 1 1 1 56 1 1 1 1 1 1 1 57 1 1 1 1 1 1 1 58 1 1 1 1 1 1 1 59 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 61 1 1 1 1 1 1 1 62 1 1 1 1 1 1 1 63 1 1 1 1 1 1 1 64 1 1 1 1 1 1 1 65 1 1 1 1 1 1 1 66 1 1 1 1 1 1 1 67 1 1 1 1 1 1 1 68 1 1 1 1 1 1 1 69 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 71 1 1 1 1 1 1 1 72 1 1 1 1 1 1 1 7
Fig..3: An alternate representation of Melakartha Raga Source: https://en.wikipedia.org/wiki/melakarta#/media/file:melakarta.katapayadi.sankhya.72.png Table.5: A few Janya ragas, their ascending and descending orders and the corresponding Janaka Raga. Janya raga Ascending order Descending order Janaka raga Revati S R1 M1 P N2 S S N2 P M1 R1 S Rathnangi Bhoopalam S R1 G2 P D1 S S D1 P G2 R1 S Hanumatodi Bowli S R1 G3 P D1 S S N3 D1 P G3 R1 S Mayamalava Gowla Malayamarutam S R1 G3 P D2 N2 S S N2 D2 P G3 R1 S Chakravakam Mohanam S R2 G3 P D2 S S D2 P G3 R2 S Harikambhoji Hamsadhwani S R2 G3 P N3 S S N3 P G3 R2 S Dheera Shankarabharanam 8
.4 Musical Instruments Fig..4: Flute (Wind Instrument) Throughout history, various kinds of musical instruments have been in use and are classified based on various aspects. A typical classification is based on the way sound is produced. On this basis, musical instruments can be classified broadly as Wind Instrument, String Instrument and Percussion instrument. Wind instrument can further be divided into woodwind instrument and Brass Instrument. A model of typical wind instrument, Flute, is shown in Fig..4. Typical string instrument, Guitar and Violin are shown in Fig..5 and a typical percussion instrument, Tabla, is shown in Fig..6 (a) Guittar (String instrument) (b) Violin (Bowed string instrument) Fig..5: String instruments 9
.4.1 Wind Instrument The most common and ancient wind instrument is flute. Indian flutes can be broadly classified into Hindustani 4 flute and Carnatic flute. Hindustani flutes are commonly longer than flutes used in Carnatic Music and are called Bhansuri. Even though small flutes are used in Hindustani style, Bhansuri is the commonly used. Hindustani flute has six tone holes, while Carnatic flute has eight tone holes. The nomenclature of different holes in flute are given in Fig..7. Acoustic Aspect When air is blown with a pressure greater than the atmospheric pressure and is allowed to flow through the blow hole as shown in Fig..8, the air molecules inside get agitated and start propagating in all directions. Due to the resistance of the air, some of the blown air will be reflected back. Thus in effect a wave, V + e γz, propagates in one direction and another wave, V e γz, propagates in the opposite direction due to reflection, as shown in Fig..9, thus causing a standing wave of longitudinal nature. Here, V + represent wave traveling in positive z-direction and V represent wave traveling in negative z-direction and γ is the propagation constant. A few of the standing waves which can be found inside a hollow resonating cylinder while excited are shown in Fig..1. Similar waves and its variants occur in musical instrument flute also, depending on the pattern of open and closed tone holes. The points where pressure is zero are called nodes and the points where pressure is maximum are called anti-nodes. Usually at the end of the bore hole, there will be nodes. 4 also known as North Indian Classical Music Fig..6: Tabla (Percussion Instrument) 1
Fig..7: Nomenclature of different holes in flute Fig..8: Flow of air while blowing V + e γz V eγz Fig..9: Waves traveling inside flute Amplitudes Amplitudes Amplitudes 1.5 Standing waves with fundamental frequency.5 1 1.5 2 flute length Standing waves with first harmonics 1 1.5 1 1.5 2 flute length Standing waves with third harmonics 1 1.5 1 1.5 2 Flute length Fig..1: Standing waves in flute 11
.5 Plot of Single note of flute.5 Amplitude.1.15.2.25.3 2 4 6 8 1 12 Time sequence (n) Fig..11: A typical time-domain plot of flute note 3 25 2 Magnitude 15 1 5 5 1 15 2 25 frequency in Hz Fig..12: Magnitude spectrum of the flute note in Figure.11 Signal Processing Aspect A single note played by any wind instrument is periodic in nature. Depending on the note played, corresponding fundamental frequency and harmonics will be present in the spectrum. Plot of a portion of a typical note of flute is shown in Fig..11. The frequency domain representation of this note is shown in Fig..12. It can be readily noted from the figure that the fundamental frequency of the note for this particular flute is 6 Hz and there are two prominent harmonics, 12 Hz and 18 Hz. However the pattern remains the same for all flutes. 12
.4.2 String Instrument The commonly used string instruments in ICM are Veena, Sarod, Sitar, Violin, etc. Violin is an instrument adopted from western music and has become an integral part of Carnatic Music. Mandolin, an instrument similar to guitar, is another instrument adopted to ICM. Acoustic Aspect The nomenclature of different parts of a violin are shown in Fig..13. An excited string can be modeled as shown in Fig..14. As per Newton s second law of motion Force = Mass Acceleration. Hence, K 2 y x 2 = ɛ 2 y t 2 (2) where K is the String Tension and ɛ is the linear mass density and y is the string displacement. The wave equation is fully derived in [Morse, 1981]. Similarly, a bowed string instrument can be modeled as shown in Fig..15 by considering the bow force, bow velocity and bow position. For bowed strings, torsional waves should also be considered, since they affect the bow-string friction force and provide an important loss mechanism for transverse waves [McIntyre, 1979]. Equation (2) can be solved by any fixed string shape which travels to the left or right with speed K v =. If we express the wave traveling to right as y ɛ r(t x/v) and the wave traveling to left as y l (t + x/v), then the general solution for the wave equation is y(t, x) = y r (t x/v) + y l (t + x/v) (3) where y r and y l are arbitrary twice-differentiable function. Specifically, when a string is excited, two traveling waves are generated in both direction of excitation and finally a standing wave is formed depending on the length of the string, which resonates in a frequency corresponding to the length of the string. Signal Processing Aspect A single note played by string instrument is periodical in nature. If it is a plucked string instrument like guitar, the periodic wave will be decaying while bowed instrument will give sustained oscillation. The time domain plot of a typical note of violin is show in Fig..16. The periodic structure in the wave can be easily identified. The magnitude spectrum of violin note is plotted 13
CHAPTER. FUNDAMENTALS OF INDIAN CLASSICAL MUSIC Body Bow Bridge String Fig..13: Nomenclature of different parts of violin y(t,x) String Tension (K) Mass/length (ǫ) Position x Fig..14: The ideal vibrating string Fig..15: A schematic model of bowed string instrument 14
5 x 1 3 4 3 2 Amplitude 1 1 2 3 4 1 2 3 4 5 6 Time sequence (n) Fig..16: A typical time-domain plot of violin note 18 16 14 12 Magnitude 1 8 6 4 2 1 2 3 4 5 Frequency in Hz Fig..17: Magnitude spectrum of the violin note in Figure.16 in Fig..17. The spectrum of a single note of violin contains more harmonics compared to that of flute..4.3 Percussion Instrument There are many instruments in the percussion family and a number of ways in which they can be classified. One of the classification is to divide them into four groups: idiophones (xylophone, marimba, chimes, cymbals, gongs, etc.); membranophones (drums); aerophones (whistles, sirens); and chordophones (piano, harpsichord) [Rossing, 21]. As a sample from this, Tabla, which is a membranophone is selected for our study. Moreover, as opposed to common drums which are inharmonic in nature, the drums, which are in usage in India, produce 15
sounds which are harmonic in nature. This was pointed out by Raman et. al. [Raman, 192]. Mridanga and T abla are two popular harmonic drums in India. Acoustic Aspect The classical model of Indian drums proposed by Raman [Raman, 1934] is characterized by its spectrum representing five harmonics. The circular drum-head is loaded on a massive hollow body, mostly heavy wood, to suppress all the overtones above the ninth. The sound produced by these Indian drums has an analogy to the case of stretched string which give one or other of its five harmonics. The different modes of vibrations produced by the instrument are experimentaly shown by putting sand on the membrane just before striking. Different strokes gave different pattern of sands over the membrane. The patterns so obtained for different strokes are shown in Fig..18. In this thesis, musical instrument flute is used for the main experiments. Five musical instruments, flute, violin, saxophone, clarinet, guitar and piano are modelled using wavelets. 16
(a) (b) Fig..18: Different modes of vibration shown by sand put on Mridanga Indian Academy of Science 17