Concept of Probability in Sanskrit Texts on Classical Music

Invited Talk at ICPR Seminar on PRE-PRINT “Science & Technology in the Indic Tradition: Critical Perspectives and Current Relevance” 10-12-2017 I.I.Sc, Bangalore; February 4-5, 2017 Concept of Probability in Sanskrit Texts on Classical Music R.N. Iyengar Distinguished Professor Centre for Ancient History and Culture Jain University, Bangalore (Formerly KSIDC Chair Professor Dept. of Civil Engg., I.I.Sc; Bangalore) RN.Iyengar@jainuniversity.ac.in Introduction A classical music concert by a maestro, whether vocal or instrumental, either in the South Indian (Karnatic) style or in the northern (Hindustani) style turns out to be enjoyable and enchanting with the pleasant feeling remaining with a discerning listener for a long time. One wonders how transient groups of sounds without linguistic articulation, are made to convey some profound meaning, leading to satisfactory internal experience to a listener, even if it be called entertainment, so that this art form that literally builds castles in air has evolved as distinct from folk music and has sustained itself in this country over a period of two thousand years or more. It is usually pointed out that Indian classical music, known also as Rāga music in current parlance, has had an unbroken tradition of practitioners who have maintained the science of their art true to its original axioms but with many innovations and modifications over the centuries. Starting from Bharata’s Nāṭyaśāstra (BNS) dated variously, but perhaps originating around 100 BCE, Sanskrit texts numbering nearly a hundred exist on the theory of music1,2. While some texts may repeat what a previous author had already said, there are authors questioning and differing from their predecessors to arrive at new findings. The stated aim of many texts has been to reconcile theory and practice, where practice refers to the performance of professionals who by their theoretical knowledge, training and innate intelligence explore now vistas within the boundaries of the tradition. Almost all known treatises trace the origin of classical music to the chanting of the Sāmaveda. They also postulate one-to-one relation between the human body and the stringed instrument Vīṇā in sound production, a tenet adapted from the Aitareya Āraṇyaka (III.2.5) 1 V. Raghavan, Some Names in Early Sangita Literature, (in 3 parts), J of the Music Academy Madras, 1932. 2 M.S.Ramaswami Aiyar, Bibliography of Indian Music, J. Royal Asiatic Soc. of Great Britain & Ireland, No.3, pp.233-246, 1941. 1 belonging to the tradition of the Ṛgveda. These two are the foundational axioms PRE-or PRIN T principles 10-12-2017 accepted by all authors on Hindu music. The first tenet gradually led to the delineation of the seven svara (notes) and twenty-two śruti (ordered sound intervals) in an octave with special names and symbols. The second principle of similarity between the Vīṇā and the human body encouraged experiments with strings to clarify a variety of questions regarding human voice and vocal music, including theoretical understanding of svara and śruti. This tradition of experimenting with stringed instruments eventually culminated with the southern Sarasvatī Vīṇā and the northern Sitār that are too well known worldwide. The texts generally narrate esoteric concepts about sound and state the theory of svaras separated by certain śruti intervals. Attempts at quantification or fixing of svara positions are made, but the authors largely depend on verbal descriptions and similes to bring out ideas that are abstruse but essential to the music. For example discussion on the difference between svara and śruti occupies considerable space and in some texts the explanations are quite confusing. Interestingly enough, while many authors propound that the śrutis in an octave are finite in number, Kohala’s school holds that śrutis are infinite as noted in the text Bṛhaddeśī by Mataṅga (c 8th cent CE)3. It is easy to recognize that such diverse opinions are due to the fundamental question whether a line of finite length is to be described as a series of distinct points or is it adequate to take it as a sequence of sub-divisions. Several authors provide illustrative examples of their theory by providing rāga samples made out of svara symbols. These are actually short time series samples simulated by the authors with the full understanding that when sung the sound pattern will be continuous in time. How the internal composition of the various svaras relative to each other and to the whole, is to be handled by training, practice, intelligence and intuition to produce numerous pleasing rāga patterns with songs and accompaniments or extempore without any articulated song is the main subject matter of the texts. A large technical vocabulary is developed in the texts to delineate enchanting creation of melodies in human voice and on the Vīṇā. These technical terms are like parameters that one introduces in making a mathematical model for a physical process, except that they are too many, vague, neither fixed nor quantifiable but expected to be internalized by the musician to be able to visualize beauty in such sound patterns and be capable of conveying the same to the audience. In the midst of this palpable uncertainty there is one precise word dviguṇa to 3 AÉlÉlirÉÇ iÉÑ ´ÉÑiÉÏlÉÉÇ iÉÑ SzÉïrÉÎliÉ ÌuÉmÉͶÉiÉÈ| rÉjÉÉ kuÉÌlÉÌuÉzÉåwÉÉhÉÉqÉÉlÉlirÉÇ aÉaÉlÉÉåSUå || (Bṛhaddeśī. v. 29) 2 describe the doubling property of any svara from the lower to the upperPRE- PRIN register. T In the present paper, we first discuss this dviguṇa property of the notes in successive 10-12 -2017 registers. This is followed by a discussion on how uncertainty or unexpectedness is built into rāga representation through the often ignored probabilistic concept alpatva, bahutva in our ancient Sanskrit texts. The present study is an attempt at exploring scientific thought processes in the texts that helped our ancient musicologists to arrive at a theory for rāga music. Sapta-svara and Dviguṇa The seven svaras with their well-known names and vocalization symbols are: Ṣadja (Sa), Ṛṣabha (Ri), Gāndhāra (Ga), Madhyama (Ma), Pañcama (Pa), Dhaivata (Da), Niṣāda (Ni); in the increasing order of pitch. The number count of seven is likely to have continued from the oral tradition of the Sāmaveda that uses largely five but occasionally six and seven svaras in some chants. The Vedic names for the svara are not only different but also arranged in the descending order as Prathama, Dvitīya, Tṛtīya, Caturtha, Mandra, corresponding to Ma, Ga, Ri, Sa, Da with the infrequent Kruṣṭa and Atisvāra equated with Ni and Pa in some places. It is noted that the starting svara in the sacred music is Ma in contrast with the laukika (worldly) music of BNS and other texts starting with Sa. A simple description of sāmagāna practice including variant traditions and opinions is available in The Ragas of Karnatik Music by Ramachandran4. The monograph by Prajñānānanda5 and a recent article by Subhadrā Desai6 also provide some preliminary information on the relation between sāmagāna and classical music. Hymns of the Ṛgveda are chanted in three pitch levels, namely anudātta (low), svarita (middle) and udātta (high). This might have led to the five and later seven svaras of sāmagāna and also provided a model for the three registers of classical music. All traditional authors state that the same seven notes exist in the lower, middle and upper registers of human voice emanating from three levels corresponding with the chest, the throat and the head. The intriguing concept here is that of dviguṇa (double/twice) used to describe the relation between the svaras in the consecutive registers. The first clear statement of this 4 N.S.Ramachandran The Rāgas of Karnatik Music, University of Madras, 1938 5 Swami Prajnanananda, A History of Indian Music Vol.1 Ramakrishna Vedanta Math, Calcutta, 1963. 6 S. Desai, Vedic Chanting and its Relation to Indian Music, (published in 3 parts.) The Vedanta Kesari, July, August, September, 2014. Kolkata. 3 occurs in the Saṅgīta Ratnākara (SR) of Śārṅgadeva (12-13th cent)7 esteemed PRIN to this PRE- dayTas an authority on classical music. 10-12-2017 urÉuÉWûÉUå iuÉxÉÉæ §ÉåkÉÉ WØûÌSqÉlSìÉåÅÍpÉkÉÏrÉiÉå| MühPåû qÉkrÉÉå qÉÔÎklÉï iÉÉUÉå ̲aÉÑhɶÉÉå¨ÉUÉå¨ÉUÈ|| SR (1.7) The text has two famous commentaries namely, the Sudhākara of Simha-bhūpāla (14th cent CE)8 and the Kalānidhi of Kallinātha (15th cent CE)9. The first commentator explains the above verse as WØûÌS rÉ EimɱiÉå lÉÉSÈ xÉ qÉlSì CÌiÉ MüjrÉiÉå; rÉxiÉÑ MühPåû EimɱiÉå xÉ qÉkrÉÈ; rÉxiÉÑ qÉÔÎklÉï iÉÉUÈ | LwÉÉÇ qÉÉlÉÇ MüjÉrÉÌiÉ- ̲aÉÑhÉ CÌiÉ | rÉÉuÉÉlqÉlSìÈ iÉiÉÉå ̲aÉÑhÉÉå qÉkrÉÈ rÉÉuÉÉlqÉkrÉÈ iÉiÉÉå ̲aÉÑhÉxiÉÉUÈ || He says sound produced in the chest, throat and head is known as mandra, madhya and tāra respectively. Their measure (māna) is stated to be dviguṇa. It is double that of the mandra in the madhya; and double that of the madhya in the tāra sthāna. However Kallinātha interprets this as the relative effort needed by a person to produce the sounds in the three registers. According to him, .....̲aÉÑhÉÈ, aÉÑhÉÈ EŠÉUhÉ mÉërɦÉÈ.....ArÉqÉjÉïÈ....̲aÉÑhÉ-mÉërɦÉ-xÉÉkrÉiuÉÉSè ̲aÉÑhÉ CÌiÉ | There is no explanation on how the human effort required in a register could have been measured and found to be twice that needed in the lower register. Hence, we have to ignore this interpretation of dviguṇa as a passing opinion of the commentator. In any case he says that the mandra-ṣaḍja with twice the effort becomes the madhya-ṣaḍja and it is not a new note. He also further comments that when the octave is divided into 22-śruti intervals, the 23rd will be the dviguṇa10. Both the commentators and their predecessors have had the understanding that the same svara in the successive octave exhibits the property of doubling. Between the two commentators separated by nearly a century, the explanation of the former seems nearer to what the author of SR wants to convey. Somanātha in his Rāgavibodha (1609 CE) with auto commentary repeats previous authors on dviguṇa and emphasizes the sameness of the svara through an example that a person going from a lower level to an elevated place is 7 Śārṅgadeva son of Soddhala was the Chief Accountant in the court of King Singhaṇa, who ruled at Devagiri in Maharashtra during 1210-1247 CE. His grandfather Bhāskara had moved out of Kashmir and migrated towards South India. Simha Bhūpāla a king belonging to the Racherla dynasty ruled at Rācakonda, in Andhra Pradesh c 1330 CE. 8 Kallinātha was a scholar in the courts of King Vijaya I and of his successor King Devarāya II of the 9 Vijayanagara Empire in the middle of 15th century. 10 ´ÉÑÌiÉÃmÉlÉÉSÌuÉuɤÉÉrÉÉliÉÑ §ÉrÉÉåÌuÉÇzÉÉå ̲aÉÑhÉÈ || (Kallinātha’s commentary on SR I.7) 4 still recognized as the same individual11. The expression of frequency of vibration PRE-PRIN T in terms of cycles per unit time is modern and it is known that the frequency of a given10-12 -2017as note (taken a Sine wave) doubles in the immediate next octave. Hence the connotation dviguṇa to denote the interval measure of the octave, which is apt and precise, cannot be taken as a lucky coincidence. Beyond reasonable doubt, dviguṇa should have entered into the vocabulary of classical music through experiments with the stringed Vīṇā. The Vīṇā Almost all the texts have a section on the Vīṇā, representing stringed instruments. Starting from BNS it is taken as an accepted fact that the theoretical 22 śrutis in the three registers cannot be produced clearly by the human voice. Simha-bhūpāla in his commentary quotes a verse of Pārśvadeva (13th cent CE) to emphasize that a Vīṇā is necessary to demonstrate all the notes clearly12. Without going into all that is known and written about the evolution of string instruments from Vedic times we note that in the early period of BNS the single-string (eka-tantrī) and the bow shaped instrument with several strings must have been in wide use13. Bharata for his demonstration of the existence of 22-śrutis, invokes two identically tuned instruments one with fixed pitch positions and the other that can be changed further. How many strings these had? In the absence of direct evidence we have to infer that they must have had seven or more strings each. Abhinavagupta explaining the experiment of BNS says that the pañcama-tantrī has to be slackened14. This could mean either the fifth string or the one giving the Pa-svara had to be lowered by one śruti. Śārṅgadeva is quite clear that Bharata’s experiment has to be carried out on two instruments each with 22 strings with their pitch in ascending order when played in the downward direction15. There is mention in BNS of the matta-kokila vīṇā that had 21 strings like a harp, seven for each of the three registers. Here, even if the tuning were to be done by the ear, since the width of the bow shaped instrument increases upwards, the length of the string associated with any svara from tāra to madhya and then to mandra should double sequentially. Or conversely, as in the human body the pitch is said to double upwards, in the case of the above Vīṇā the doubling of the pitch happens in the opposite direction. In the later stringed instruments with or without frets also rÉjÉÉ SåuÉS¨ÉÉå lÉÏcÉxjÉsÉÉSÒŠxjÉÉlÉaÉiÉÉåÅÌmÉ xÉ LuÉåÌiÉ mÉëirÉÍpÉ¥ÉÉrÉiÉå iÉjÉåirÉÍpÉmÉëÉrÉÈ || 11 12 iÉå iÉÑ ²ÉÌuÉÇzÉÌiÉlÉÉïSÉ lÉ MühPåûlÉ mÉËUxTÑüOûÉÈ | zÉYrÉÉ SzÉïÌrÉiÉÑÇ iÉxqÉɲÏhÉÉrÉÉÇ iÉͳÉSzÉïlÉqÉç || (Saṅgīta-samaya-sāra) 13 A.K.Coomaraswamy, The Parts of a Vīṇā, J. of the American Oriental Society, Vol. 50. (1930), pp. 244-253. uÉÏhÉÉrÉÉqÉmÉUxrÉÉÇ mÉgcÉqÉiÉl§ÉÏ ´ÉÑÌiÉqÉɧÉÇ ÍzÉÍjÉsÉÏMüÉrÉÉï iÉSÉ qÉkrÉqÉaÉëÉqÉÉå eÉÉrÉiÉå|| (Abhinavabhāratī Commentary on BNS) 14 15 ²å uÉÏhÉå xÉSØzÉÉæ MüÉrÉåï rÉjÉÉ lÉÉSÈ xÉqÉÉå pÉuÉåiÉç | iÉrÉÉå²ÉïÌuÉÇzÉÌiÉxiÉl§rÉÈ mÉëirÉåMÇü iÉÉxÉÑ cÉÉÌSqÉÉ || AkÉUÉkÉUiÉÏuÉëÉÈ iÉÉxiÉ‹Éå lÉÉSÈ ´ÉÑÌiÉqÉïiÉÉÈ | (SR I. 3. 11 & 13a) 5 the same condition applies. This is mentioned clearly by Kallinātha16, Mahārāṇa PRE-PRIN T Kumbha (1433-1468 CE)17 and Dāmodara (1625 CE)18 among several others. Authors10-12 -2017 starting from Bharata onwards knew that as the length of the freely vibrating part of the string reduces the śruti increases. This viparyaya or reverse order must have led to the word dviguṇa as reciprocal of half the length of the open string for hearing the tāra-Sa. That the sound of the open string can be considered as Ṣaḍja, is an ancient concept traced to Dattila (c 3rd cent CE) by Simha-bhūpāla19. Śārṅgadeva famed to have designed a new instrument Niśśaṅka-vīṇā known by his honorific Niśśaṅka meaning Doubt-less, must have taken the middle octave to be equivalent to the interval [1, 2]. He says as much in the third chapter of SR while describing the process of ālāpana, where he introduces the technical term dvyardha which literally means one-and-half. Kallinātha comments that since, relative to the dviguṇa-svara this note is at the middle it is called dvyardha. The very form of the word indicates this to be based on the Vīṇā referring to Ma if the physical length of the octave on the string is taken. If the śruti value were to be meant then dvyardha would refer to Pa. Saṅgīta Pārijāta of Ahobala Paṇḍita (1665 CE) is the first known text to specifically state that at half the length of the open string the dviguṇa ṣaḍja arises. This work also provides the distances of the seven so called śuddha-svaras (pure notes) on the open string. Since theorization and experimental effort by Hindu musicologists is largely ignored by main stream historians of science in India, the relevant text is quoted here. xuÉU¶É WåûiÉÑpÉÔiÉÉrÉÉ uÉÏhÉÉrÉɶÉɤÉÑwÉiuÉiÉÈ | iÉ§É xuÉUÌuÉoÉÉåkÉÉjÉïÇ xjÉÉlÉsɤÉhÉqÉÑcrÉiÉå|| kuÉlrÉuÉÎcNû³ÉuÉÏhÉÉrÉÉÇ qÉkrÉå iÉÉUMüxÉÇÎxjÉiÉÈ| EpÉrÉÉåÈ wÉQèeÉrÉÉåqÉïkrÉå qÉkrÉqÉÇ xuÉUqÉÉcÉUåiÉç|| ̧ÉpÉÉaÉÉiqÉMüuÉÏhÉÉrÉÉÇ mÉgcÉqÉÈ xrÉÉiÉSÌaÉëqÉå| wÉQèeÉmÉgcÉqÉrÉÉåqÉïkrÉå aÉÉlkÉÉUxrÉÎxjÉÌiÉpÉïuÉåiÉç|| xÉmÉrÉÉåÈ mÉÔuÉïpÉÉaÉå cÉ xjÉÉmÉlÉÏrÉÉå ËUxuÉUÈ| xÉmÉrÉÉåqÉïkrÉSåzÉå iÉÑ kÉæuÉiÉÇ xuÉUqÉÉcÉUåiÉç || iɧÉÉÇzɲrÉxÉÇirÉÉaÉÉiÉç ÌlÉwÉÉSxrÉ ÎxjÉÌiÉpÉïuÉåiÉç|| (Sangīta Pārijāta v. 314-318) Ahobala remarks that on the Vīṅā the svara positions can be visually observed for better appreciation of their nature. The positions on the open string are given with the understanding that the free string produces the madhya-Sa. The positions are as follows: tāraka (upper-Sa) 16 rÉjÉÉ zÉUÏUå ´ÉÑiÉrÉÈ E¨ÉUÉå¨ÉUÉåŠÉ EimɱliÉå iÉjÉÉ uÉÏhÉÉrÉÉÇ AkÉUÉkÉUÉåŠÉ EimɱliÉå CÌiÉ oÉÉå®urÉqÉç || Kallinātha on SR (3.12-14). 17 Mahārāṇa Kumbha the Rājput King of Mewar is a great name in the political history of India. Besides waging wars, building forts and monuments he remains famous for his exhaustive encyclopaedic Sanskrit text of nearly 16000 verses on music, dance and Rasa titled Saṅgīta-Rāja and a commentary on the Gīta-govinda of the renowned poet Jayadeva. 18 ̲aÉÑhÉÈ mÉÔuÉïmÉÔuÉïxqÉÉSè ArÉÇxrÉÉSÒ¨ÉUÉå¨ÉUÈ| LuÉÇ zÉUÏUuÉÏhÉÉrÉÉÇ SÉUurÉÉliÉÑ ÌuÉmÉrÉïrÉÈ|| (Saṅgīta-darpaṇa 1.47) 19 S̨ÉsÉÉå ÌWû xuÉåcNûrÉÉ rÉxrÉÉÇ MüxrÉÉqÉÌmÉ ´ÉÑiÉÉæ wÉQèeÉÇ xjÉÉmÉrÉå¨ÉSmÉå¤ÉrÉÉ cÉ ´ÉÑÌiÉÌlÉrÉqÉålÉ AlrÉxuÉUÉlxjÉÉmÉrÉåÌSirÉÑ£üuÉÉlÉç | rÉSÉWû- wÉQèeÉiuÉålÉ aÉ×WûÏiÉÉå rÉÈ wÉQèeÉaÉëÉqÉå kuÉÌlÉpÉïuÉåiÉç | iÉiÉ FkuÉïÇ iÉ×iÉÏrÉÈ xrÉÉSè GwÉpÉÉå lÉÉ§É xÉÇzÉrÉÈ|| (Comm. on SR I.4.15-16) 6 is at the centre of the string; Ma is at the midpoint between the middle and PRE- PRIN upper-Sa. T On the string divided into three parts, Pa will be at the first one-third point (from the left). 10-12 Gāndhāra -2017 (Ga) is located at the midpoint between Sa and Pa. By dividing the distance between Sa and Pa into three parts, Ri is placed at the first point from the left. Dhaivata (Da) is taken as the midpoint between Pa and the upper-Sa. By dividing the distance between Pa and the upper- Sa into three parts, Ni is placed after leaving two parts from Pa. He further gives the 22-śruti positions of unequal pitch intervals along with their traditional names, details of which can be read in the original20. The taut string of the Vīṇā is held at the two ends and plucked near the right support. The above statement of Ahobala translates into placing the frets for the middle register at (1, 8/9, 5/6, 3/4, 2/3, 7/12, 5/9, 1/2) of the full length of the string measured from the right end support to produce the seven svaras in their ascending order and then the dviguṇa. Since the fundamental natural frequency of a string is inversely proportional to its length the above values indicate the śuddha svara scale of Ahobala to be (1, 9/8, 6/5, 4/3, 3/2, 12/7, 9/5, 2). Naturally there are other ways of dividing the length to produce more number of differing notes which may or may not lead to a pleasant rāga scale. It is also evident that several ways of partitioning the string, holding the dviguṇa property as invariant was known to Bharata and his successors. But for some special reason they did not like to precisely discretize the octave into finite number of unconnected points, although the seven or the twenty one strings of the ancient Vīṇā were separated physically. All the texts abound in discussions on how to combine, compare, discriminate, organize and accept or reject the svara pattern to generate innumerable melodies. What is glaringly evident in all the texts on music, starting from the Vedic period, is the symbiotic relation nurtured between the Vīṇā and the human voice. This experimental approach seems to have peaked during 1200-1700 CE. A philosophical question that has occupied the Hindu mind since ancient times has been whether the svara gamut is contrived artificially or is it something already existing in the cosmic nāda space only to be manifested through human voice? The genesis of this question is traceable to the metaphysical distinction made between dhwani and nāda. The former is the primordial all pervasive sound that is the cause of all creation21. A form of this dhwani known as nāda leads to music but it is again of five types. The ati- sūkṣma-nāda (most subtle sound) is in the Heart; the sūkṣma-nāda (subtle sound) is in the cave (of the Heart). The avyakta-nāda (non-manifest sound) is at the level of the jaw, while it Saṅgīta Pārijāta Printed and Published by R.S.Gondhalekar, Jagaddhitecchu Press. Pune, 1897 20 21 kuÉÌlÉrÉÉåïÌlÉÈ mÉUÉ ¥ÉårÉÉ kuÉÌlÉÈ xÉuÉïxrÉ MüÉUhÉqÉç| AÉ¢üÉliÉÇ kuÉÌlÉlÉÉ xÉuÉïÇ eÉaÉiÉç xjÉÉuÉUeÉ…¡ûqÉqÉç || (Bṛhaddeśī v. 11) 7  22 PRE-. PRIN is vyakta (manifest) at the throat centre and it is kṛtrima or artificial at the mouth T The Heart referred here is not the biological organ on the left side of the chest, but the Heart-2017 10-12 in the subtle body that is extolled in the Vedas23. The 22-śrutis of music are also conventionally stated to originate at this location due to 22-nodes present in the Heart-chakra that is perceptible in yogic meditation24. The question of artificiality of musical notes might have arisen due to academic interest also, since not all persons are capable of producing the musical svara sequence satisfactorily, but inarticulate sounds of some birds appear to follow a natural pattern if heard intently. Nāradīyaśikṣā traditionally revered as the Vedic ancillary text representing the interface between Vedic and laukika music, states that the peacock calls in ṣaḍja (Sa), the krauñca (heron) calls in madhyama (Ma) and the koel calls in pañcama (Pa)25. We need not dwell on this concept other than noting that Bharata, Mataṅga, and Śārṅgadeva cite this model whereas later authors particularly Rāmāmātya26, Somanātha, Ahobala do not quote this, even though they were equally interested with the question of natural existence or otherwise of the musical notes. These three musicologists excelled in their experimentation, construction and classification of a variety of Vīṇā instruments. Rāmāmātya followed by Somanātha reports that in the four stringed instruments tuned as Sa-Pa-Sa-Ma in two registers some svaras are spontaneously excited even when the assigned frets are not pressed on the neighbouring string. The presence of the second and third harmonic corresponding to the dviguṇa-Sa and the Pa could be recognized aurally by plucking the open Sa-string and theorized to be svayambhū or self-generated. By a series of such arguments, Somanātha concludes all the notes in the octave to be natural and not of human creation or imagination. Besides recognizing the presence of higher harmonics in a svara sound, he also reports sympathetic vibration on neighbouring strings tuned to nearly the same śruti. The concept of saṁvādi svaras (consonant notes) known from Bharata’s time, get a new interpretation nearer to the modern theory of vibration of strings by the experiments of Somanātha. Śruti and Svara 22 xÉÔ¤qÉÉå lÉÉSÉå aÉÑWûÉuÉÉxÉÏ WØûSrÉå cÉÉÌiÉxÉÔ¤qÉMüÈ| MühPûqÉkrÉå ÎxjÉiÉÉå urÉ£üÈ AurÉ£üxiÉÉsÉÑSåzÉMåü|| M×ü̧ÉqÉÉå qÉÑZÉSåzÉå iÉÑ ¥ÉårÉÈ mÉgcÉÌuÉkÉÉå oÉÑkÉæÈ | CÌiÉ iÉÉuÉlqÉrÉÉ mÉëÉå£üÉ lÉÉSÉåimĘ́ÉqÉïlÉÉåWûUÉ|| (Bṛhaddeśī v.24-25) Nārāyaṇa Sūkta in the Taittirīya Āraṇyaka (10.13) of the Kṛṣṇa Yajurveda. 23 iÉxrÉ ²ÉÌuÉÇzÉÌiÉpÉåïSÉÈ ´ÉuÉhÉÉcNíÓûiÉrÉÉå qÉiÉÉÈ| WØû±ÔkuÉïlÉÉQûÏxÉÇsÉalÉÉ lÉÉŽÉå ²ÉÌuÉÇzÉÌiÉqÉïiÉÉÈ || (Saṅgīta Ratnākara I.3.8) 24 25 Nāradīyaśikṣā, Published by Srī Pītāmbarāpīṭha Samskṛta Pariṣad, Datiya,1964. 26 Rāmāmātya grandson of Kallinātha is famous for his treatise on music Svara-mela-kalānidhi ( c 1550 CE). 8  PRE-PRIN A point that has been vigorously discussed, in all the texts, is the difference between śruti and T svara. Is śruti the cause of svara or the other way round? Svara are seven, but10-12 -2017 is there only one śruti or are there precisely 22-śrutis? Simha-bhūpāla says that there are 22 different types of nāda denoted as śruti, but also points out the confusion existing about the meaning of the word used27. Even though śruti and svara were held to be different, as Mataṅga says svaras get depicted always by śruti28, theorists looked for physical demonstrations to discriminate the above two words and to clarify their meanings. Abhinavagupta (10th cent CE) the great philosopher from Kashmir in his illuminating commentary Abhinavabhāratī on BNS explains the above technical terms clearly referring to the stringed Vīṇā. According to him svara is that pleasant audible sound with anuraṇana emanating from excitation at the corresponding śruti position. Śruti is just the audible peculiarity arising immediately with sound and it is not any divisible part of sound29. Śārṅgadeva also refers to anuraṇana and defines śruti and svara (SR I.3.24) in the same way as Abhinavagupta. Somanātha through his experiments with strings points out that svaras are not only self- emanating but have the property of anuraṇana. That is, with any svara there exists ‘follower sound’ which refers to presence of overtones and higher harmonics. On the other hand śruti is just any sound in the interval of the octave without anuraṇana. In the absence of a tuning fork to demonstrate śruti as the flat sound value due to a pure Sine wave, the above are to be considered as fairly clear explanations. The subtlety in this argument is due to the fact that the sound values of both are relative with respect to madhya-Sa taken as unity. Whereas svaras as dependent variables have nearly specified locations, śruti as the independent variable is just any point in the interval [½, 4] spanning the three registers and beyond. Ahobala points out that as per experts śrutis are abundant, placed at hair-tip interval, in the octave. It is recognized, he says, that there are 22 śrutis in the human voice and in the Vīṇā based on the Sa-Pa interval relation30. As we go through the texts, we find that essentially the numbers 7 and 22 are taken to be conventional to illustrate how svaras can be produced to a first approximation by dividing the octave, but the nāda of svara is not same as śruti-nāda. The Nāradīyaśikṣā figuratively explains that variation of svara over śruti cannot be known, iÉxrÉ lÉÉSxrÉ ²ÉÌuÉÇzÉÌiÉxÉÇZrÉMüÉ pÉåSÉ pÉuÉÎliÉ | iÉå cÉ ´ÉÑÌiÉxÉÇ¥ÉrÉÉ EcrÉliÉå |......iÉ§É ´ÉÑiÉåÈ LMüiuÉÉlÉåMüiuÉ ÌuÉwÉrÉå qÉWûiÉÏ ÌuÉmÉëÌiÉmĘ́ÉÈ|| 27 28 wÉQèeÉÉSrÉÈ xuÉUÉÈ xÉmiÉurÉerÉliÉå ´ÉÑÌiÉÍpÉÈ xÉSÉ | AlkÉMüÉUÎxjÉiÉÉ rɲiÉç mÉëSÏmÉålÉ bÉOûÉSrÉÈ || (Bṛhaddeśī v. 36) 29 uÉrÉÇ iÉÑ ´ÉÑÌiÉxjÉÉlÉÉÍpÉbÉÉiÉmÉëpÉuÉzÉoSmÉëpÉÉÌuÉiÉÉåÅlÉÑUhÉlÉÉiqÉÉ ÎxlÉakÉqÉkÉÑUÈ zÉoS LuÉ xuÉU CÌiÉ uɤrÉÉqÉÈ |... ... ´ÉÑÌiÉ¶É lÉÉqÉ ´ÉÉå§ÉaÉqrÉÇ uÉæsɤÉhrÉÇ rÉÉuÉiÉÉ zÉoSålÉÉåimɱiÉå | ... ... zÉoSÉuÉrÉuÉÉå lÉ ´ÉÑÌiÉÈ CirÉÑ£üqÉåuÉ || (Abhinavabhāratī Commentary on BNS) MåüzÉÉaÉëurÉuÉkÉÉlÉålÉ oÉyurÉÉåÅÌmÉ ´ÉÑiÉrÉÈ xqÉ×iÉÉÈ| uÉÏhÉÉrÉÉÇcÉ iÉjÉÉ aÉɧÉå xÉÇaÉÏiÉ¥ÉÉÌlÉlÉÉÇ qÉiÉå|| 30 qÉkrÉå mÉÔuÉÉåï¨ÉUÉoÉ®uÉÏhÉÉrÉÉÇ aÉÉ§É LuÉ uÉÉ | wÉQèeÉmÉÇcÉqÉpÉÉuÉålÉ ´ÉÑÌiɲÉïÌuÉÇzÉÌiÉÇ eÉaÉÑÈ || (Saṅgīta-pārijāta v. 42, 43) 9  smoothPRIN but the text prescribes that the transit from svara to svara should be PRE- T like the movement of shadow and sunlight31. This is as good as a mathematical condition 10-12 -2017 that svara must vary as a continuous function of śruti. The principle of similarity between the human body and the stringed Vīṇā instrument goes deep back into the Vedic period as pointed out in the beginning of this article. Musicologists have explored this doctrine further since this is all about the tonal quality of the voiced svara that is the basic ingredient of rāga. What was suspected to exist in voiced svara was confirmed by them on the string so that through a feedback the two modes of music could get mutually enriched. Modern science helps us to visualize the presence of overtones and higher harmonics through frequency domain analysis. As an aid to follow the previous discussion, the frequency spectrum of three svaras as sung by a musician is shown in Fig. 132. Pa Sa Ma Fig1. Spectrum of voiced svara Sa, Ma and Pa. The svara are very nearly centred on Sa = 180 Hz; Ma=240 Hz; Pa=270 Hz. Observe that in the first graph of Sa higher harmonics namely, dviguṇa-Sa and dviguṇa-Pa are present. The peaks are not spiked and pointed like arrow heads. Sound due to a narrow band of frequencies around the corresponding primary śruti and also due to dviguṇa harmonics would be present in any good quality svara. The term śruti can be taken as non-dimensional proxy for the modern frequency. Rāga Characterization Indian music from the time of Bharata or earlier has had syllabic symbols for the seven notes as Sa, Ri, Ga, Ma, Pa, Da, Ni that can be vocalized with or without the pronunciation of the vowels and also produced on the string. A select number of svaras out of the seven pure and twelve vikṛta svaras are arranged and combined in myriad ways leading to thousands of tāna, mūrchana, alaṅkāra and still other patterns for voice training and practice on the Vīṇā. Exposition of this part is invariably based on sophisticated mathematical methods of 31 xuÉUÉixuÉUxÉÇ¢üqÉxiÉÑ xuÉUxÉÎlkÉqÉlÉÑsoÉhÉqÉç| AÌuÉÎcNû³ÉÇ xÉqÉÇ MÑürÉÉïiÉç xÉÔ¤qÉÇ NûÉrÉÉÅÅiÉmÉÉåmÉqÉqÉç|| Nāradīyaśikṣā (I. 6.18) 32 This is taken from the M.Sc Dissertation, Spectral Analysis of Gamaka Svaras by Karuna Nagarajan, SVYASA University, 2006. Figures personally communicated by Dr.Karuna Nagarajan. 10 permutation and combination for forming chains of svaras of varied length and PRE- PRIN hence canTbe 10-12-2017 precisely described. But, any definition or description of rāga will be as incomplete as one can possibly describe in words the flowing river, the blowing wind or the thunder storm. A very general statement due to Mataṅga is that rāga is a special imaginative sound sequence that is aesthetically pleasing and melodious to captivate the minds of the listeners33. However, musicologists with keen hearing and long observation have found several special normative features as essential for depicting any rāga out of a selection of svaras. Bharata lists ten general characters for what he denotes as Jāti (genus) the forerunner of the later rāga of Mataṅga and Śārṅgadeva. The ten characters considered fundamental to describe a particular rāga and also to discriminate one from another (i.e. with a given name) are graha, amśa, nyāsa, apanyāsa, mandra, tāra, ṣāḍavita, auḍuvita, and alpatva, bahutva. All the ten points with further divisions and subdivisions are described in SR and the two commentaries in great detail. The first four characters are about the starting, major, intermediate and ending notes. The next four refer to the lower and upper registers and six- and five-svara possibilities. The last two are the most intriguing importance of which is missed in literal translations. Alpatva-Bahutva Rāga depiction can happen either with a song already composed or extempore without any words. The former is the nibaddha (kalpita) music while the latter is the ālāpana. For any Rāga with a defined āroha-avaroha (scale), its individuality is infused through the concept of alpatva and bahutva of the svaras. Except for the alpatva-bahutva characteristic, the other eight listed above can be classified as in the case of tāna, mūrcchana, alṅkāra that are deterministic svara groups34. The first jāti-rāga named Ṣāḍjī exemplified through a song with svara-prastāra appears in the seventh section of the first chapter of SR. The notes of the song of twelve lines to be sung row by row are shown here in Table 1. Kallinātha, in his commentary explains that this example is useful in understanding the property alpatva- bahutva. A§É xuÉUxÉÇZrÉÉ AsmÉiuÉoÉWÒûiuÉ mÉËU¥ÉÉlÉÉrÉ ÍsÉZrÉiÉå | wÉQèeÉÉÈ wÉOèû̧ÉÇzÉiÉç; GwÉpÉÉ ²ÉSzÉ; aÉÉlkÉÉUÉ ÌuÉÇzÉÌiÉÈ; qÉkrÉqÉÉ A¹Éæ; mÉÇcÉqÉÉ A¹Éæ; kÉæuÉiÉÉÈ wÉÉåQûzÉ; ÌlÉwÉÉSÉ ²ÉSzÉåÌiÉ ÍqÉÍsÉiÉÉ ²ÉSzÉÉå¨ÉUÇ zÉiÉqÉç|| rÉÉåÅxÉÉæ kuÉÌlÉÌuÉzÉåwÉxiÉÑ xuÉUuÉhÉï ÌuÉpÉÔÌwÉiÉÈ| UgeÉMüÉå eÉlÉÍcɨÉÉlÉÉÇ xÉ UÉaÉÈ MüÍjÉiÉÉå oÉÑkÉæÈ || 33 34 A table is available for the 18-Jāti group with name, aṁśa, nyāsa, apanyāsa, mūrcchana, ṣāḍava, auduva in the book Sangeeta Ratnakaram: A Study by R.R.Ayyangar, Wilco Publishing House, 1978, Bombay. 11  PRE-PRINT He shows the relative frequency of the svaras by counting their occurrence in the table as 10-12-2017 Sa=36; Ri=12; Ga=20; Ma=8; Pa=8; Da=16 and Ni=12; the sum total is 112. He does not differentiate between long and short notes and the registers. He also points out that the aṁśa (important) svara Sa is the most frequent. This art of mixing svara in different proportions to derive melodies goes back to Bharata. Śārṅgadeva in the first chapter of SR presents illustrative examples, like the above, for all the 18 jāti-rāgas of BNS demonstrating the alpatva-bahutva property as explained by Kallinātha. It is easy to see from the above example the ratio of the counts to the sample size 112 is, in modern data analysis, an estimate for the probability of occurrence of the seven svaras. Here, the svara is seen to be treated as a discrete random variable with seven possible outcomes with different probabilities. Starting from such elementary discrete structures, more complex compositions are described and transition to continuous variation of alpatva-bahutva property is achieved in rāgālāpana. Table 1. Swara Prastāra of song in the Rāga Ṣāḍjī [Lower dot: Mandra; Upper dot: Tāra; others: Madhya] Sā Sā Sā Sā Pā NiDa Pā DaNi Rī GaMa Gā Gā Sā RiGa DaSa Dā RiGa Sā Rī Gā Sā Sā Sā Sā Dā Dā Nī NiSa. NiDa Pā Sā. Sā. Nī Dā Pā DaNi Rī Gā Sā Gā Sā Dā. Da.Ni. Pā. Sā Sā Sā Sā Sā Sā Gā Sā Mā Pā Mā Mā Sā Gā Mā DaNi NiDa Pā Gā RiGa Gā Gā Gā Gā Sā Sā Sā Sā Dā Sā Rī GaRi Sā Mā Mā Mā Dā Nī Pā DaNi Rī Gā Rī Sā RiGa Sā Rī Gā Sā Sā Sā Sā In the second chapter of SR more than two hundred Rāgas are described some of them with elaborate examples. A major contribution of Śārṅgadeva is the illustration of the structure of several rāgas through songs and direct ālāpana, as a succession of sounds arising out of different groupings of svaras with the property of alpatva-bahutva in a pleasant manner. These examples are actually long sample data showing the svaras to be used in a particular rāga and their proportion and mix. SR does not report any quantified proportions, but that 12 can be easily estimated from the samples as explained above by Kallinātha. PRE- PRIN There are T further divisions to the ten principal characters of rāga and many more delicate10-12 -2017 features are described in SR to introduce continuous variability in a nuanced fashion in rāga presentation. Only a few such artifices can be noted here. Alpatva is of two kinds known as langhana and anabhyāsa35. The first is to leap over a svara whereas the latter is infrequent use. Popularly langhana is interpreted as skipping a svara. But even in the pentatonic and hexatonic scales, transitions in the rāga are continuous. Hence, Śārṅgadeva is careful to define laṅghana as īṣat-sparśa that is infinitesimally vanishing touch most probably observed with deleted svara36. Kallinātha explains this further as the absence of (svara) form produced by effort37. The second type of alpatva is anabhyāsa that is infrequent use. Hence alpatva is better translated as low probability. The opposite of the above is bahutva with alanghana and abhyāsa that refer to emphasis and frequent use of particular svaras. Thus, this pair of property is both qualitative and quantitative and also to be effected relative to each other. For every rāga, svaras having low probability are specified in SR. This automatically implies the remaining svaras to be relatively more frequent. It is not that always the probabilities of the svaras have to be unequal. For example in the Rāga Mālavaśrī the seven svaras have equal weight38 and this is confirmed by Kallinātha also in his commentary39. This example illustrates that among all the theoretical characters it is the probabilistic property of alpatva- bahutva that is central to Rāga individuality. The other features such as sañcāra and antaramārga, ṛju and vakra and variation in speed introduce more variability into the sound structure so much so the pitch position of a particular svara in any rāga will not be single valued. Already we have seen in Fig.1, the basic svara is not a strict Sine wave, but it is only nearly so, such that its energy is spread in a narrow band carrying overtones contributed by closely spaced śrutis or frequencies. Additionally ornamentation due to gamaka which is defined as a delicate modulation around a svara is a must to please the listeners40. This modulation is further classified into fifteen types of refined movements such as quivering, throbbing, wavering, shaking, oscillating, swinging, sliding, and rotating with further combinations thereof. There are too many possible technical variations that cannot be discussed here except to remark that all of these add variability to the rāga on different time AsmÉiuÉÇ cÉ Ì²kÉÉ mÉëÉå£üqÉlÉprÉÉxÉÉŠ sɆ¡ûlÉÉiÉç | AlÉprÉÉxÉxiuÉlÉÇzÉåwÉÑ mÉëÉrÉÉå sÉÉåmrÉåwuÉmÉÏwrÉiÉå|| (SR I.7.50) 35 DwÉixmÉzÉÉåï sɆ¡ûlÉÇ xrÉÉimÉëÉrÉÈ iÉssÉÉåmrÉaÉÉåcÉUqÉç | EzÉÎliÉ iÉSlÉÇzÉåÅÌmÉ YuÉÍcɪÏiÉÌuÉzÉÉUSÉÈ|| (SR I.7.51) 36 sɆ¡ûlÉqÉÏwÉiÉç xmÉzÉïÈ xuÉUxrÉ xjÉÉlÉ-mÉërɦÉ-M×üiÉ-xuÉÃmÉ-lrÉÔlÉiÉÉ || (Kallinātha on SR above) 37 ÌSlÉxrÉ MåüzÉuÉmÉëÏirÉæ qÉÉsÉuÉ´ÉÏxiÉSÒ°uÉÉ | xÉqÉxuÉUÉ iÉÉUqÉlSìwÉQèeÉÉÇzÉlrÉÉxÉwÉQèeÉpÉÉMçü || (SR II. 2.53) 38 qÉÉsÉuÉ´ÉÏsɤÉhÉå - xÉqÉÉ xuÉUÉ rÉxrÉÉÈ xÉÉ xÉqÉxuÉUÉ | xuÉUÉhÉÉÇ xÉqÉiuÉÇ A§É AsmÉiuÉoÉWÒûiuÉM×üiÉ uÉæwÉqrÉUÌWûiÉiuÉÇ ÌuÉuÉͤÉiÉqÉç|| 39 40 xuÉUxrÉ MüqmÉÉå aÉqÉMüÈ ´ÉÉåiÉ× ÍcɨÉxÉÑZÉÉuÉWûÈ | (SR III.87a) 13  of ālāpana scales. Finally it is the imagination of the singer that decides the detailingPRE- PRINinTa session, so much so musicologists neither visualize nor explain rāga as a10-12 -2017 deterministic process in time. If musicologists are expected to analyze the best of the music heard by them to delineate the theory behind Rāga depiction, we can see that the texts starting from BNS, with the available vocabulary at their disposal, foreshadow concepts that are basic to the Theory of Probability. Rāga as a random process may sound unusual and even jarring for those who connect with the aesthetics of rāga music only through emotion. But Śārṅgadeva following his predecessors not only analyzes classical music in terms of a hierarchically complex set of sound structures but also describes through similes the tantalizingly unpredictable nature of Rāga while describing the procedure of ālāpana in the third chapter of SR. His approach is analytical as one can realize from hundreds of technical words used to delineate not only the grammar of Rāga but also the sound quality of voice and classification of good and bad singers. To this already heavy vocabulary Somanātha adds a further set of technical terms and a unique system of notations in his text for rendering rāga music on the Vīṇa. Ālāpana or ālapti is defined in SR as exhibition (prakaṭīkaraṇam) of the Rāga. Some broad guide lines are stated for this aural depiction that can be vocal or presentation on the string. Four modes of transit are mentioned as dwelling states. From the sthāyisvara up to but excluding the fourth svara (dvyardha) and return is the first transit. The second is the one including the fourth svara. The third transit is dwelling in between the fourth and the eighth svara to return to the sthāyi. The fourth would include the eighth note (dviguṇa) and occasionally above to return back to the end note that is like a stationary point. In this process conditions previously stated such as alpatva-bahutva, langhana, nigraha, praveśa and varieties of ornamental gamaka as found pleasing, have to be maintained. The exposition is done in calm, composed and leisurely pace, little by little, bringing out the temporal pattern of Rāga as a whole that is sometimes hidden, sometimes prominent but always modulated within the region of attraction prescribed by the ascent and descent of the svaras. Śārṅgadeva cryptically but effectively states xiÉÉåMüxiÉÉåMæüxiÉiÉÈ xjÉÉrÉæÈ mÉëxɳÉæoÉïWÒûpÉÌ…¡ûÍpÉÈ| eÉÏuÉxuÉUurÉÉÎmiÉqÉÑZrÉæ UÉaÉxrÉ xjÉÉmÉlÉÉ pÉuÉåiÉç || (SR 3.196) 14  stoka PRIN The word stoka-stokaiḥ has the meaning of little-by-little, and the wordPRE- T in Indian mathematics stands for a small unit of time equal to about 5.35 seconds4110-12 . The-2017 phrase bahubhaṅgi is equally meaningful. This refers to another important point that with all general characters of the rāga remaining same a refreshingly different realization is possible in each session. Each musician can also render the same rāga in individual style with several little- by-little embellishments. Thus the theoretical characterization of rāga music as developed in the Sanskrit texts is nothing but stochastic. Discussion In modern times, Ramachandran42 was perhaps the first person to point out somewhat hesitantly “Hitherto single figures have been given by various writers as the value of the śrutis in each raga. But it may be clearly seen that in each raga a note assumes different shapes……It is more or less a general tendency for the śruti of a particular note to appear sharp when that note is relinquished for a higher note and to appear flat when a descent is made for that note…..And gamakas afford a wide choice of śrutis in the treatment of a note.” He found with the help of a sound analyzer that in the Rāga Kanakāṅgi the svara Ri appears to take at least two values 10/9 and 256/243. He experimented with singers and found in some cases three different śruti values were used for the same svara. He seems to take svara to be a Sine function which has two or three distinct frequency values that are rational fractions. But as per the ancient texts the variation over the śruti is generally continuous and hence careful analysis should show any svara to occupy a narrow band of śruti values in songs set to rāga by learned composers and also in appealing ālāpana performances. This fact has been experimentally verified by Komaragiri43 by analyzing time series of several rāga performances by three famous contemporary musicians. In Fig. 1 three isolated svara were depicted using modern recording and analysis techniques to find that their peak śruti values, although nearly equal to (1, 1.33, 1.5) still show considerable spread. In the presentation of any rāga by an inspired maestro, in either the southern or the northern style, the svara flow will be continuously weaving patterns in time. In such a case to find the śruti value of a svara would be, as aptly said by Nārada, like finding the path of fish in water and flight of birds in air44. In recent years there is increased interest in computational methods to The Gaṇita-sāra-sangraha of Mahāvīrācārya, (Ed.) M. Rangācārya. Madras Govt.Press, Madras 1912. 41 42 N.S.Ramachandran (ibid) 43 M.M.Komaragiri. Departures from the Acoustical Parameters in the Intonation of South Indian Musical Intervals. Canadian Acoustics Vol.40 No.4, 2012, pp.3-11. 44 rÉjÉÉmxÉÑcÉUiÉÉÇ qÉÉaÉÉåï qÉÏlÉÉlÉÉÇ lÉÉåmÉsÉprÉiÉå| AÉMüÉzÉå uÉÉ ÌuÉWû…¡ûÉlÉÉÇ iɲiÉç xuÉUaÉiÉÉ´ÉÑÌiÉÈ || Nāradīyaśikṣā (I. 6.16) 15 identify rāgas of classical music based on actual recordings of performances 45 As an T PRE-. PRIN off shoot of such investigations one can construct pitch histogram of a rāga which -2017 shows 10-12 the number of times finely divided śruti bins in an octave get occupied in a sample rāga performance over a short time period. In Fig.2, two such plots are shown for the number of occurrences of different svaras over the śruti interval in the performance of two popular rāgas. The X-axis is in Cents defined as Cent = 1200 log2(śruti) Thus, the middle register covering the śruti interval [1, 2] corresponds to the interval [0, 1200] Cents. The figure shows that the histograms do not peak at every one of the theoretically prescribed svaras even though their contribution is present in the distribution. Also interesting to note is the very small contribution from svara that are popularly said not to belong to that particular Rāga. This seems to happen since in gamaka phrases, neighbouring śrutis and even slight movements of another rāga (svara-kāku, rāga-kāku) are permitted but with very low probability. Description of such niceties is available in the 3rd Chapter (prakīrṇādhyāya) of SR and the commentaries. The relative heights and spreads in the figures are visual representations of the alpatva-bahutva character Fig.2. Pitch Histograms showing the property of alpatva-bahutva or relative probability of occurrence of the seven prescribed svaras for two popular Rāgas of Karnatik Music in current practice. [Figure after Prof. Hema Murthy; IITM, Chennai Accessed on 21-11-2017; https://www.google.co.in/Hema-Murthy-slides-3rd-CompMusic-workshop.pdf ] 45 G.K.Koduri, J.Serra and X.Serra. Characterization of Intonation in Carnatic Music by Parametrizing Pitch Histograms. Proc. 13th Intnl. Soc. For Music Information Retrieval Conference (ISMIR 2012) pp.199-204. 16  PRE-PRINT of the particular rāga within the time interval of its exposition and also as discriminated 10-12-2017from another rāga. More importantly, from a historical view point, this is a modern generalization of the svara counting method first demonstrated by Kallinātha six centuries ago. The variation of the pitch in Fig. 2 is continuous and hence for proper comparison between two ragas the figures have to be normalized such that the area under the curve is unity. This will be in fact an estimate of the first order probability density function of the continuous random variable Svara of that particular Rāga. Each Rāga will be characterized by one such multimodal probability density function with several peaks. It must be noted that this is only a first order mathematical description or model with no claims for completeness in explaining rāga music in all its delicate intricacies. Like with any stochastic process joint density functions of higher order and time wise variation of statistical properties greatly matter. Among the structural characters of a rāga, apart from the ten properties stated by Bharata, antaramārga (internal path) is considered important by all authors. This is a very general phrase and only some vague information of this is available in SR and the commentaries. Roughly speaking this allows use of svaras displaced from their assigned positions to be used in between other svaras as though they have alpatva (low probability) character. Seen from the perspective of a stochastic process antaramārga may be referring to internal relation among svaras in time. The simplest measure of this second order property is the autocorrelation function. How is this antaramārga concept brought into the education of music students? This is where Śārṅgadeva highlights the importance of vāggeyakāras who are accomplished scholars having the ability to compose songs, set them to rāga music and also sing, for the sustenance of the classical tradition. Thus the nibaddha-saṅgīta or composed music for which the svara-prastāra is made available by tradition is indispensable not only for training purposes but also for the theory of rāga music. We have seen the example of Rāga Ṣāḍjī already. SR provides several such examples with samples of ālāpana also as a sequence of svaras. One can analyse these to some extent by assigning śruti values to the svaras. Out of curiosity in Fig. 3(a) the time series of the gīti in the Rāga Ṣāḍjī given in SR as per Table 1 is plotted by taking the short syllable as a reference (mātrā) time unit. The autocorrelation of the rāga as it evolves in time is shown in Fig. 3(b). 17  PRE-PRINT 10-12-2017 Fig. 3 (a) Time series of the song in Rāga Ṣāḍjī Fig. 3(b) Autocorrelation It is observed that the prastāra is strongly correlated with the autocorrelation decaying very slowly. It would be interesting to compare the above with simple songs currently in use. Saint Purandaradāsa (c 1550 CE) considered the grandsire of Karnatic music systematized teaching at the elementary level to begin with sarale-varase, janti-varase, and alamkāra series. He also composed hundreds of songs in different rāgas, many of which are popular to this day. His gīti compositions śrī-gaṇanātha and kereya nīranu in the Rāga Malahari are the very first songs that students learn even now. These are shown in Fig 4 (a) and Fig 5 (a) as time series plotted with the prescribed svaras. The points are joined by lines to get a visual impression of the svara pattern. Fig 4(b) and Fig 5(b) show the autocorrelation function which for up to two steps appears to be decreasing geometrically but later shows slow decay, but faster than the song of Śarṅgadeva above. Fig. 4(a) Śrī Gaṇanātha…time series plot Fig. 4 (b) Autocorrelation Fig. 5(a) Kereya Nīranu…time series plot Fig 5(b) Auocorrelation 18 These types of songs are simple, without gamaka embellishment and speed PRE- PRIN variations. T These can be faithfully presented even on keyboard instruments using the nominal svara 10-12 -2017 positions. These are introduced in music classes after training in the strictly deterministic and periodic alaṅkāra strings. The point to be noted is that the autocorrelation decreases in two different ways caused by two different internal arrangements among the same svara groups indicating the two songs to be sample presentations of the same rāga among many other possibilities. Next, in the graded order of complexity students are trained in swarajate and varṇa compositions. In Figure 5 (a,b) the time series and autocorrelation for a varṇa composition in the Rāga Hamsadhwani are shown. These results are by their very nature approximate since the time series is artificially constructed with discrete svara positions as given in printed text books and class notes. Nevertheless, one can get an idea of possible second order property of a rāga. Here also after two or three steps, the autocorrelation decreases too slowly, indicating long time memory in the composition as far as the song structure is concerned. HDVRN ACF 1 0.8 Sample Autocorrelation 0.6 0.4 0.2 0 -0.2 0 5 10 15 20 25 30 35 40 45 50 Lag Fig. 5(a) Hamsadhwani Varṇa Time series. Fig. 5(b) Autocorrelation Note: Mandra-Sa =0.5, Madhya-Sa=1, Ri=1.125, Ga=1.25, Pa=1.5, Ni=1.875; Tāra-Sa=2 Composed songs mark time with the help of external cyclic tāla measure to produce a feeling of rhythm in the background of the rāga. This may introduce hidden trends leading to slow decay of the autocorrelation as seen above. In ālāpana without tāla the external time keeping is removed so that the time series of rāga can evolve in different time scales, but staying within the boundary defined by the āroha-avaroha and the ever present gamaka. In this perspective with or without a vocalized song the theoretical description in the texts of any Rāga corresponds to a sample time series of the continuous random variable Svara evolving over its sample space namely, the śruti interval [½, 4] as per a pre-defined and culturally imbibed or intuitively innovated probability measure. Summary and Conclusion 19  PRE-PRIN In this paper we have briefly reviewed Sanskrit texts by Bharata, Dattila, Mataṅga, and a few T later musicologists on classical music, to bring out that the theory of melody or10-12 -2017is rāga music based on the principle of uncertainty of svara positions in the octave. The dviguṇa limit or interval of the octave is precise, but the positions of the seven primary and twelve modified svaras are actually treated as variables except for their sequential order within the octave. This is not to say that the Vīṇā string positions as stated by Ahobala are invalid. As Ahobala and before him Somanātha points out, the visual fret positions are of great help for learners to get a feeling to the tonal variability of a svara, around its central or dominant śruti value. It is generally accepted that rāgālāpana or melody-depiction, for want of a better word, is the real test of a musician. The textual theory can be seen as a pedagogical support to train the future musician to carry out ālāpana, which will be an extempore performance without accompaniments except for the drone or ādhāra-śruti. It is with reference to this basic pitch level that the ālāpana will be gauged and perceived as an aesthetically appealing production of a particular Rāga. To educate and train a learner for this purpose a heavy hierarchy of increasingly complex sound structures, ranging from the discrete to the continuous, periodic to the stochastic, have been developed by the classical tradition layer by layer. The preliminary teaching is with deterministic discrete svara groups. These patterns are without exception pre-determined and hence repeat periodically after a prescribed interval. Such structures are musical and may even exhibit some elements of rāga but they are not yet rāga music. Rāga enters for the first time through the probabilistic alpatva-bahutva property in the increasingly complex musical patterns of songs sung to external time keeping with the help of tāla. Rāgālāpana is a further generalization without external time support where the artist has the freedom to bring out the features of the rāga as a sample time series with its mathematically structured aural pattern that is emotionally enjoyable and satisfying to the performer and the listener. Apart from a variety of other factors, Bharata and his successors highlight unexpectedness and particular types of svara mix as the key to bring out rasa-bhāva or emotional appeal in melodies. Further investigations are necessary to understand higher order organization of rāga music, beyond the ancient alpatva-bahutva and the modern first order pitch histogram. This is a rich area for further exploration that will help analytical understanding of how emotions (rasa) of listeners get evoked in particular rāgas by their special aural patterns. This knowledge tradition combines experimentation on the Vīṇā with vocal music in a mutually beneficial feedback relationship. While rigorous training, aptitude and intuition play 20  PRE-PRIN obviously decisive roles in shaping a musician, knowledge of the grammar expounded T in the 10-12-2017 texts adds sophistication and refinement to the art without obstructing the originality of the performer. The historically well evidenced culture of classical rāga music as a dialogue between the artist and the audience, independent of any human language, is well preserved in the texts by expounding the theory repeatedly with minor additions and alterations to keep up with contemporary practices. An amazing edifice of Sanskrit texts has been built by our ancient musicologists to explain the theory behind classical music. A formidable intellectual foundation that is partly speculative and narrative but systematic and analytical using mathematical and experimental methods is available for present day students, scholars and practitioners of music. This has been possible only because the art and the science of rāga music were encouraged to develop together as lakṣya and lakṣaṇa with mutual dependence in the traditional academic ecosystem. While the scholarly growth of musicology is an achievement of the Indic knowledge system with its roots going as far back as the Vedas and even though classical music has large number of votaries to this day all over the country, there is visible stagnation in the theoretical tradition with no important Sanskrit text on music appearing after mid-19th century. It is hoped that the relevance of the inquisitive mind set and scientific ideas left behind by the textual corpus will be recognized by our educational institutions including Sanskrit Universities, to bridge the above gap, so that Indian classical music as a fine art can rejuvenate itself side by side with its traditional partners namely, mathematics and experiments with musical instruments. ***** 21