Archive for the ‘Poetry’ Category

The Musical Journey of Lalgudi G Jayaraman

September 17, 2016

Comments on Biography of Lalgudi

I just happen to finish reading the book – An Incurable Romantic (The Musical Journey of Lalgudi Jayaraman), by Lakshmi Devnath.  The book itself was published in May 2013, just after the sudden demise of Sri Lalgudi on 22nd April 2013. As a tribute to his memory, I am sharing with my readers and many fans of Lalgudi, a few comments I have on this book, which may add up additional features to his phenomenal life of great achievements in the field of Carnatic Music. Page numbers are given for ready reference.

  • (P-23) Sethupathi Vallal Pandithurai Thevar was a great promoter of classical music. He was also a leading figure in preserving Hindu Culture and was the head of a movement against Sharada act, banning child marriages. In those days, it was considered as interference in religious freedom, and people argued that the same reform could be achieved by persuasion than by an act of government. I remember seeing a copy of an old letter written by my grandfather to Pandithurai Thevar, suggesting a meeting on this issue along with Andipatti (Mattapparai) Zamindar. By the way my grandfather L S Raja Ramanatha Iyer was a Veena Vidwan, a contemporary and a friend of Karaikudi brothers. He was also briefly an elected president of Madurai Jilla Board.
  • (P-24) The episode concerning Lady Loka Sundari and Sir C V Raman is quite funny. Smt Lokasundari was trained in Music by Valadi Radhakrishna Iyer, grandfather of Lalgudi. She sang ‘Rama nee Samanam evaru’, apt for the occasion, when the groom C V Raman came for bride introduction function at Madurai, in early 1900s.
  • (P-29) Muthulakshmi Patti is the grandmother of Lalgudi (Wife of Valadi Radhakrishan Iyer. Her poetry is fantastic. Now we know where from Sri Lalgudi inherited his composing genius. Look at this: Vennai unda vaya, Oliyum Maya, kalvanum Niya, Kanniyar Neya, kadir nigar thooya, un manam kaya, anbellam Poyya – in his Charukesi varnam. Compare this with Patti’s: Ennariya Thozharai Anna Thambi enru, Vennai Thayir Palundu, Unna amudam kondu…
  • (P- 44) Lalgudi’s mother Smt Savithri is from Edayathumangalam, near Lalgudi, same village my patti (Grand Mother) Ammalu nee Akhilandeswari, hailed from. I have heard from Smt Savithri Ammal, that it was her father Sundara Sastrigal who was the official priest who married off my patti.
  • (P-64) Lalgudi’s father, Sri V R Gopala Iyer’s Music School in Lalgudi: The group photograph shows my aunt (Mami) Smt. Dharmambal, who had learnt music from Gopala Iyer in her younger days.
  • (P – 116) Speaking of Lalgudi’s tala expertise and control – Once Lalgudi was accompanying the great MD Ramanathan in a concert in Bharatiya Fine Arts in Mumbai. MD elaborated raga Pantuvarali and after the krithi, started niraval and kalpana swaras. After a long passage of swaras, he started his Kuraippu sequence. Lalgudi was accompanying him beautifully and suddenly he stopped his bowing. MD continued his kuraippu for another cycle of talam and looked at Lalgudi puzzled. Lalgudi shook his head ever so slightly, just enough for MD to realize that the particular kuraippu is not going to end at the required beat of the tala. He gestured to audience his acceptance of mis-beat and proceeded further, cutting short his kuraippu and started an entirely new kuraippu sequence smilingly acknowledging Lalgudi’s follow-up. I am referring to this concert again later in a different interesting context.
  • (P 127) Accompanying Madurai Somu – His accompaniment has always embellished vocal concerts in a very imaginative way. In a concert by Madurai Somu, in Shanmukhananda, Mumbai, he was presenting a viruttham on Goddess Meenakshi. In his rich emotional voice he was describing goddess Meenakshi’s alankaram : ‘Vairam…. Vaiduriam…’. Audience were spell bound: ‘mookutthi.. Odyanam..’. Lalgudi beautifully enhanced the imagery by playing chords in a fast tempo. When Somu dropped into an emotional silence, the chords of Lalgudi continued in a low tone bringing out the emotion among the audience also. There was a rapturous applause for the presentation.
  • (P-159) In the CD attached with the book, track-8, Meenakshi Memudham was simply superb. The violin sings. When he plays on two strings we can hear the words. Initially I thought he sings along. He creates this effect repeatedly in his rendering. The CD itself was too good and deserves to enter into all musical archives.
  • (P-161) Lalgudi most certainly re-invented the raga Kalyana Vasantham and the kriti Nadaloludai. Among other Krithis he embellished was Manavyalakim in Nalina Kanthi. He slowed it down and lifted the Krithi from its light music aspect to main Krithi level.
  • (P-164) Regarding Sahithya bhava in Theerada Vilayattu pillai : In another krithi Nee Irangayenil, variety of ways he plays Sei Uyir Vazhumo; first he will play Vazhumo flat, suggesting ridicule and rebuke, then he will play the same thing with harsh bowing showing anger, then the ‘vazhumo…..’ will be prolonged as though crying for help. He says that is how the sangatis are meant to be presented.
  • (P-171) Regarding playing ragamalika, Lalgudi introduced several novelties, including the one mentioned. When he plays ragamalika with his son: he plays a raga, his son picks up the same raga and then seamlessly changes to another raga and ‘passes the baton’ to Lalgudi: Lalgudi takes over ‘on the go’ and after a few phrases changes over to the next raga. Yes, it was a relay-raga-malika. This trend he practiced in many of his concerts.
  • (P-177) Lalgudi’s Tala gnana was tremendous. Once before a concert in Dubai I happened to visit him in the Hotel room he was staying, on the morning of the day of the concert. When I entered the room I saw Lalgudi reciting some jati’s keeping Adi Tala on the hands. His accompanists, Tiruvarur Bhaktavatsalam and Tanjavur Nagarajan were also seated in the room. He demonstrated an elaborate korvai of decreasing counts, in about 8 to 10 avartanas. Instantly he reversed the same korvai to increasing counts for the same avartanas, to the amazement of tala vidwans
  • (P-224) Lalgudi Poornachandar episode: What irked Sri Gopala Iyer, apparently, was his lack of respect – in saying that he had obtained legal clearance for appending Lalgudi to his name. He even dropped the name of P V Rajamannar, former Chief Justice of Madras High Court. As a disciple of Lalgudi and Balamurali, he was shaping up extremely well as a violinist, but unfortunately lost his way in between.
  • (P-239) Duet with GJR Krishnan: It was the first ever duet of LGJ and GJR in Mumbai Shanmukhananda (~1974). There was a rotational power-cut in Mumbai due to power shortage. Concert started at 6.05 PM as usual. Sharp at 7PM, power went off and without any facility of UPS, the sound system also went down. Two emergency lamps appeared from either side of the stage. Auditorium was dim and silent, and the pure and melodious sounds of violins continued crystal clear for the next half hour (music-unplugged) and then, the power returned. At the end of the piece, audience applauded the artists for the way they handled the situation and using it to enhance the audience experience.
  • (P-257) It was in 1984. We were having a concert tour of Lalgudi in the Gulf. We were having two concerts in UAE one in Dubai and another in Abudhabi. I was driving Sri Lalgudi to Abudhabi and my wife was also with me. The whole two hours of driving we had a feast of his vocal music along with his anecdotes. He sang the whole Charukesi varnam enjoying his own lyrics and music immensely. We reached the auditorium in Abudhabi just in time. But the accompanists, who were travelling in a different car lost their way a little bit and arrived almost 10 minutes late. But Lalgudi took the stage on time and started warming up. Just then accompanists also arrived at the stage. Lalgudi’s comment cheered up the audience: “Sri Ramabhadran and Sri Vinayakram were always accompanying me faithfully but this time they failed to do so. Surely, they will make up for this on the stage.”
  • (P-265) When his disciple, Sow Akhila was in Dubai, Lalgudi used to stay with her whenever he visits Gulf either for concert or on transit. During one such visit, we were blessed to watch the video of Jaya Jaya Devi ballet along with him and his comments.
  • (P-335) On 9th March 2008, The Music Academy, Madras awarded Sri Lalgudi, the Special Life Time Achievement Award, a one-off award for the first time ever in the ninety years history of the Academy. During the occasion the president of the Academy Sri N Murali said that non-award of Sangita Kalanidhi title to Lalgudi, can be compared to Mahatma Gandhi not getting a Nobel Peace Prize. Sri Murali was proud that they did better than Nobel foundation, by seeking ‘to erase the mistake and the aberration’ and ‘in conferring the Special Life Time Achievement Award’ for Sri Lalgudi. But all said and done, I am still feeling bad to see that Lalgudi’s portrait is not seen anywhere in the lobbies of the Academy, not even among the portraits of Sangita Kalanidhis. Will the Academy take steps to erase this aberration too?
  • The book is a very interesting read for a biography. It has been a very well researched material with all interesting references. Some of the intrigues, conflicts and challenges in the world of Carnatic music have been brought out along with Lalgudi’s mature responses for the same. I felt the book could have included a few more comments from the rasikas including a few Lalgudi fans. Being an ardent fan of Lalgudi, for my own satisfaction, I am adding below a few of my experience and interactions with Lalgudi.
  • V R Gopala Iyer: Lalgudi’s father was really a genius. Here is my own experience with him. In my younger days (1977), I was once visiting Lalgudi Sir along with my wife and 2-year old daughter. As Lalgudi sir had gone out, we were greeted by Sri Gopala Iyer. When we were talking to him, my daughter was playing with the grilled door of their house. She was rattling the bolt of the door. When I asked her to keep quiet, Gopala Iyer remarked: Kuzhandai kaila dhaivatham, Panchamam Vilayadikirathe! (Daivath and Pancham are the playthings in her hands) – When I looked somewhat puzzled, he explained: ‘The child is playing with Da and Pa, Thapa,(i.e.) the bolt ! When he came to know the child’s name as Sriranjani, he again made a remark: Pancham Varjyam ! (In raga Sriranjani, the swara panchamam is absent). Pancham can also mean in Tamil, scarcity or poverty.
  • During a lecdem Lalgudi was asked whether Saint Thyagaraja’s nadopasana composition, ‘Nabhi Hruth Kanda Rasana’ applies to instrumentalist also, he confirmed that even for instrumentalist the Nada emanates from Nabhi, then Hruth and Kanda and then it echoes on the instruments instead of Vocal cords as in the case of vocalist. “Set of vocal cords is their instrument and violin is my vocal cords; Otherwise source of Nada is the same” as per Lalgudi.
  • Once Lalgudi was accompanying the great MD Ramanathan in a concert in Bharatiya Fine Arts in Mumbai. (Referred earlier). It was immediately after MD received the title Padmashri from GOI. Before the concert there was a felicitation for MD where many people spoke of him and his music. MD was all the time nudging Lalgudi to speak, which he was refusing. When it was MD’s turn to reply, he openly acknowledged about his unkind references to awards, when Lalgudi received his Padmashri ahead of MD; it was almost an apology. The concert that followed was really memorable without any slightest sign of discard.
  • When Lalgudi came to Mumbai for a concert in Shanmukhananda he had arrived two days earlier. As he was comparatively free, I invited him to our apartment for coffee. When he was there I took the opportunity of taking his blessings for my wife who was a budding vocalist. He asked her to sing raga Todi. After elaborating the raga, she was about to sing one of the trinity krithis. He asked for a different one and then again a different one. Finally he settled for Thygaraja Krithi, ‘Dachu kovalena’. When she was singing a particular sangati, she was asked to repeat the same sangati several times, as apparently he liked it very much. In the next day’s concert he played this krithi and played the same sangati, looking smilingly at us in the audience.  [My wife says pranams to her (Late) Guru Sangita Kala Acharya, Vidwan S Ramachandra Bhagavatar of Shanmukhananda – on whom Sri Lalgudi also had a high regard].

Today, 17th Sept 2016, is the 86th birth anniversary of Sangita Vidwan, Violin Maestro, Padma Bhushan, Lalgudi Sri G Jayaraman (1930 – 2013). My thanks are due to the author Smt Lakshmi Devnath for making me live through Lalgudi experience all over again. There are a lot more musical incidences to remember him on this day. Luckily his music still lives in the form of his many recordings, many compositions and in many of his disciples including his children, Sri GJR Krishnan and Sow Viji.

Ref: An Incurable Romantic (The Musical Journey of Lalgudi Jayaraman), by Lakshmi Devnath, Harper Collins Publishers India (2013)

Dan Brown’s Inferno, the hell

August 19, 2016

I just finished (Aug 2016) reading the novel Inferno by Dan Brown. When I finished reading The Amber Room by Steve Berry, I wrote a blog about similar mysteries that abound in ancient and medieval India. I invoked English language writers of Indian origin to write such mysteries a-la Dan-Brown-Style, but with Indian artifacts and mysteries. (Please refer to my blog.

https://lvnaga.wordpress.com/2014/01/28/ancient-mystery-thriller/). My thirst for the same was quenched somewhat by a novel ‘Deluge – Agasthya’s Secrets’, by Dr. Ramesh Babu of Chennai. (https://lvnaga.wordpress.com/2016/03/22/dr-ramesh-babu-indian-version-of-dan-brown/).

This new novel ‘Inferno’ by Dan Brown is based on a biological ‘terrorism’ of scientific age placed in the surroundings of medieval mysteries of Ottoman Empire covering present day Florence of Italy and Istanbul of Turkey.

This novel is heavily based on Italian poet Dante Alighieri’s master piece ‘The Devine Comedy’ consisting of three cantos – Inferno, Purgatory and Paradise, describing the path of the dead/soul towards hell, punishment and finally perhaps the heaven. There are many articles available in the net comparing this piece of literature with similar ideas represented in ancient Hindu scriptures by Saint Veda Vyasa in his Garuda Purana. Some of them even suggest that Dante was inspired by this description of Hell (and the travel of soul with its pseudo-body through the hell to the Paradise). I wonder if any other Indian  literature has given annotations of this work in Sanskrit or any other modern languages. I am vaguely familiar with a story in Mahabharat where King Yutishtra, with all his integrity and moral equipoise, is made to undergo a horrible view of the Hell, as a punishment for his abetment of a lie in killing Aswatthama in the war. Any extract of description such a view, is it available, I wonder.

(Those who have not read this novel ‘Inferno’ and are planning to read the same, may please avoid reading further, to retain the mystery and suspense of the Novel.)

In this novel there are some unexplained ambiguities as below

  • How can a single type of vector virus would do equal harm to the fertility of both men and women? Evidently, their reproductive systems are quite different.
  • Neither it is necessary to affect random one-third of both male and female population equally, to reduce the birth rate by a third.
  • Hence it is better to say that the vector-virus modifies the DNA of whole population but switches on at random only in one-third of male population. This will reduce the birth rate by one-third and gradually reduce the population by a third, as this DNA-Virus from even unaffected males gets inherited by the subsequent population. It will be again switched on at random in one third of males in every generation.
  • Though a lot of anxiety is expressed by all the characters in the novel about this biological ‘terror’, it appears to be a very humane way of controlling the population. It is same as vasectomy and tubectomy, which are of course, voluntary. This type of population control is normally adopted in animals and pests.
  • The characters in the novel, opposed to this type of ‘terrorism’ initially, come around and accept the same and think of making it reversible.

However the novel is quite interesting and highly readable. I understand it is also coming as a movie soon with Indian actor Irfan Khan in the role of Provost, the off-shore expediter and the secondary antagonist in the novel.

PS: I remember playing a board game in my village (India) on the nights of Gokul Ashtami and Shivratri known as Parama Pada Shobanam. It is a kind of a snake and ladder game where we start from hell and pass through several evil images and then on to happy images and finally to the heavenly images of Gods.(i.e. Inferno, Purgatory and Paradise).  On the way we encounter many snakes (named after villains of Hindu epics) and ladders of good conduct and behaviors. I wish somebody adds an image of this board to this blog. (LVN)

 

 

Rhythmatics

February 27, 2016

Rhythmatics

Fibonacci – Hemachandra Sequence

Some of my readers will m remember, one Krishnagiri Kittappa, the official percussionist of Oho Productions in the great Tamil romantic comedy of 1960s, Kathalikaa Neramillai (No time for romance). He was initially a self-taught mridangam (Drum) player. He wanted to learn to play Tabla also. He went to a Tabla player to learn the same. He was started on his first lesson, of course, in Teen Tal (or Triputa Tal in Carnatic music) of 8 beats.

Na Din Dinnah – Na Din Dinnah

Na Din Dinnah – Na Din Dinnah

Na Din Dinnah – Na Din Dinnah  . . . . . . .

This went on for quite some time. Our man got bored of playing the same rhythm. It is the same 1,1,2 – 1,1,2 all the time for the 8-beat cycle. Why not 1,2,1, – 1,1,2, he thought.

Din Dinnah Din – Na Dhin Dhinna

Din Dinnah Din – Na Dhin Dhinna

Then, why not 2,1,1-1,2,1

Dinnah Din Din – Din Dinnah Din

Good. Now he further thought about how many such combinations of 1 and 2 (Din and Dinnah), he can make in a cycle of 8 beats. Ancient Indians have already thought about this and so, I gave him the answer as 34 different combinations. He was surprised. So many? How did they calculate?

Ancient Indians always depended on recursive technique in solving such problems. They started from 1-beat, then to 2-beats, 3-beats etc.

No. of Beats (n) Syllables – 1 & 2 Combinations Total Combinations Kn
1 Din 1 1
2 Din, Din

Dinnah

 

2

 

2

3 Din Din Din

Dinnah Din

Din Dinnah

 

 

3

 

 

3

4 Dinnah (+ 2-beats)

Din  (+ 3-beats)

2

3

 

5

Now they generalized:
(n+1) Beats Dinnah +  (n-1) beats

Din + (n) beats

K(n-1)

K(n)

K(n+1) = K(n-1) + K(n)

Therefore,

K4 = K2 + K3 = 2 + 3 = 5

K5 = K3 + K4 = 3+ 5 = 8

K6 = K4 + K5 = 5 + 8 = 13

K7 = K5 + K6 = 8 + 13 = 21

K8 = K6 + K7 = 13 + 21 = 34
Hence we have 34 combinations of 1-2 in an 8-beat cycle.

Now look at the series K1, K2, ….. Kn:

1, 2, 3, 5, 8, 13, 21, 34, 55 ……

This is the famous Fibonacci series ‘invented’ by Fibonacci (alias Leonardo Pisano Bogollo) in 13th Century AD. Ancient Indians knew about this, at least, a thousand years before him. Fibonacci himself acknowledges this fact. Fibonacci also helped spread Hindu- Arabic Numerals (like our present numbers 0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). The olden day knowledge route was from India to Alexandria (Egypt) to Europe.

Susantha Goonatilake (Ref-2) writes that the development of the Fibonacci sequence ” is attributed in part to Pingala (200 BC), later being associated with Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c. 1150). Parmanand Singh cites Pingala’s cryptic formula misrau cha (“the two are made together”) and cites scholars who interpret it in the context as saying that the cases for ‘n’ beats (Kn+1) is obtained by adding [Short or 1] to Kn cases and [Long or 2] to the Kn−1 cases. He dates Pingala before 450 BC ”.

“However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135). The sequence is also discussed by Gopala (before 1135 AD) and by the Jain scholar Hemachandra (c. 1150) “. Fibonacci was born only in 1170 AD.

Prof. Manjul Bhargava (R Brandon Fradd Professor of Mathematics, Princeton University, USA) gave a Lec-Dem on Music & Mathematics at the Music Academy, Madras during their annual conference 2015, on 31st December 2015. Being a Tabla player himself, he dealt with the above aspect of rhythm variations in detail. His talk was the inspiration for me to write this blog.

References:

  1. https://www.mathsisfun.com/numbers/fibonacci-sequence.html
  2. https://en.wikipedia.org/wiki/Fibonacci_number
  3. Manjul Bhargava’s Lec-Dem at the Music Acedemy, Madras (2015)

L V Nagarajan

 

Meru Prastarah (or Pascal’s Triangle ?!?)

October 21, 2014

Meru Prastarah (or Pascal’s Triangle ?!?)

Let me start with an ancient (1000 CE) Sanskrit text as below:

Anena ekadvyaadilaghukriyasiddhyartham, yaavadabhimatam prathama prastaravat meruprastaram darsayati, uparistadekam chaturasrakoshtam likhitva, tasya adhastat ubhayatordhaniskrantam koshtadwayam likhet, tasyapiadhastatrayam tasyapiadhastatccaturtyamevam yaavadabhimatam sthanamiti meruprastarah tasya prathame koshte ekasamkhyam vyasthapyalakshanamidam pravarttayet, tatra dvikoshtaayaampanktau ubhayo koshtayorekaikamankam dadyaat, tatastritiyaayaam panktau, paryantakoshtayorekaikamankam dadyaat, madhyamakoshtethuparikoshtadvayaankamekikrtya purnam nivesayediti purnasabdarthah, chaturtyampanktavapi, paryantakoshtayorekaikamankam sthapayet, madhyamakoshtayothuparakoshtadvayaankamekikrtya purnam trisamkhya rupam sthapayeth,  uttaraataraapyevameva nyaasah, tatra dwikoshtaayaam pankatauekaakshrasya prastaarah,……. tritiyayaam pankatau, dviakshrasya prastaarah, chaturtyaam pankatau, triakshrasya prastaarah, ….

The above is not in praise of any god of Jain, Budha or Hindu religion. It is not a religious text at all. It is a text describing a method for constructing a mathematical table. Ancient Indian Mathematician Pingala (200 BC) in his Chandahsutra had given the rules for formation of different chandahs (≈ musical meters) for Sanskrit prosody. Another ancient Indian mathematician Halayudha (1000 CE) has given the explanation and commentary on this work by Pingala. Given above is a selected portion of his commentary. For some reasons unknown to me, ancient Sanskrit texts always use composite words very frequently. These words need to be broken into individual words properly to obtain the intended meaning of these words. Here is an attempt to translate the above text into English with proper separation of words.

Anena ekadvyaadi laghu kriya siddhyartham, yaavadabhi matam

(To get every combination of one, two, etc. syllables as required)

Prathama prastaravat meru prastaram darsayati.

(from first row onwards , the meru tabulation will be shown below)

Uparihi tad ekam Chaturasrakoshtam likhitva,

(At the top itself one square cell is drawn)

Tasya adhah tat ubhayato ardhani skrantam

(Below this row let us have a pair, half over lapping)

Koshtadwayam likhet.

(Two cells are drawn)

Tasyapi adhah tat trayam

(Again the row below will have three)

Tasyapi adhah tat chaturtyam,

(Again its next line will have four)

Evam yaavadabhi matam sthanam

(same way, up to the  required stage, cells are constructed)

iti meru prastarah.

(This is called Meru Prastara or Meru-Tabulation)

Tasya prathame koshte eka samkhyam

(Its first stage-cell will hold the number 1)

Vyvasthapya lakshanamidam pravarttayet

(From here on, the following is the way it grows)

Tatra dvikoshtaayaam panktau

(in its twin-cell row)

ubhayo Koshtayoh eka ekam ankam dadyaat

(the pair of cells holds numbers 1,1)

Tatah tritiyaayaam panktau, paryanta Koshtayoh Eka ekam ankam dadyaat

(then in the 3rd row, the extreme cells will hold numbers 1,1)

Madyama koshteth, upari koshtadvayah ankam eki krtya purnam nivesayeth

(middle cell takes the added value of the two cells above)

Iti purnasabdarthah

 (Thus completes the table for 2nd power)

Chaturtyam panktau api, paryanta Koshtayoh eka ekam ankam sthapayet

(then in the 4th row also, the extreme cells will hold numbers 1,1)

Madyama koshtayoth, upara koshtadvayah ankam eki krtya purnam

(middle cells take the added values of the two cells above each)

Trisamkhya rupam sthapayeth

(this completes the 3rd power)

Uttara utaaro api evameva nyaasah,

(next and next stages also follow the same rule)

tatra dwikoshtaayaam pankatau, eka akshrasya prastaarah

(Here the twin-cell row gives one syllable table)

tritiyaam pankatau, dvi akshrasya prastaarah

(the 3rd row gives two syllables table)

chaturtyaam pankatau, tri akshrasya prastaarah

(Thus 4th row gives three syllables table)

And so on.

Meru

If we follow the above step by step construction given so clearly by Halayudha (1000 CE), we get the above pyramid or Meru in Sanskrit, (stands for a mountain with a peak). What do we have here? It is the same as Pascal’s Triangle, “discovered” by Blaise Pascal (1623-1662 CE).

This table gives in every nth line the coefficients (a+b)**(n-1). i.e. the second line gives coefficients of (a+b) as 1,1; the second line gives 1,2,1, as coefficients of (a+b)2.; the third line gives 1,3,3,1 as coefficients of (a+b)3 and so on.

However Halayudha gives credit for this table to Pingala (200 BC). He claims to have derived this table from Pingala’s cryptic clue which he translates to a set of rules, as below (with a and b as the two syllables to be combined, in any n-syllable chandah):

  1. First write down all (‘n’ number of)  b’s as the first combination
  2. In the next line, replace the first ‘b’ with an ‘a
  3. At the same line, replace all letters to the left of this new ‘a’ with ‘b
  4. For the next and the subsequent lines repeat the steps 2 & 3.
  5. Continue as above till we arrive at a line with all a’s,

This can be clearly seen as a binomial expansion (a+b)n staring with bn and ending in an. Halayudha later puts these results on a table known as Meru Prasthara. He later gives a step-by-step method as above, for constructing this table without specifically going through the above rules. This Meru Prastarah traveled to China and the Chinese mathematician Yang Hui reported it in the thirteenth century, although his work was unknown in Europe until relatively recent times. The Meru Prastarah traveled to Europe a little later through Arabia, Egypt and Greece and gets “discovered” by Pascal in 17th century CE, 600 years after Halayudha. We are blaming all the time ‘the lack of scientific temper’ among Indians.

Ref:

  1. Binomial Theorem in Ancient India – By Amulya Kumar Bag, History of Science, Ancient Period Unit II, No.1, Park Street, Calcutta-16 (1966)
  2. Journey Through Genius – The Great Theorems Of Mathematics – by William Dunham – Wiley Science Editions, John Wiley & Sons Inc.(1990)
  3. Probability in Ancient India, by C K Raju., ckraju.net, 2011.

Connected Topics:

Meru Prastarah

Baudhayana’s Circles

Square Root of Two

Sine of an Angle

LVN/ Oct, 2014

The River – நதி

June 23, 2013

                         நதி

தாயின்று எழுந்து நீராவியாய், பின்

வான்நின்று பொழியும் நல்மேகமாய்,

பூநின்று செல்லும் நீரோட்டமாய், நதி

தாயொன்றி மகிழும் கடல் கூடியே. –  1

தான்நின்று பல்லோர்க்கும் அமுதாகி, நதி

தாள்சென்று அடையாது நஞ்சுற்றே

உயிர்குன்றி ஒசிந்து உணர்வற்று, தன்

உடல்குன்றி வீழ்ந்து ஓய்ந்ததுவே. –  2

ஆஒற்றிக் கரந்த பால் எனினும்

அதன்கன்றிர்க்கும் ஓரளவு ஈவது போல்

உயர்குன்றில் விழுந்த நதி நீரும், சிறிது

தாய்சென்று அடைவதே தருமம் அன்றோ. –  3

வேரின்றி வளராது விருட்சம், தன்

காலின்றி வாழாது கால்நடைகள்

நீரின்று அமையாது உலகு, எனின்

வானின்று அமையாது ஒழுக்கு. –  4

English Translation

Rising from its source as vapours

Falling from the benevolent clouds

Flowing through earth as streams – rivers

Folding joyously into the laps of seas. – 1

Holding its flow to feed thousands, but

Stalling on its way with filth and toxins

Losing its life, form and feeling – river

Falling a victim to greed and neglect. –  2

Tending the cow and drawing the milk, but

Leaving a bit for its calf to drink – like wise

Allowing the waters to reach its source – river

Ending its flow in a holy communion. –  3

No growth for trees without their roots

No life for animals without their feet

No human race without the waters -never

Any peace sans water resources –  4

(The last two lines are adopted from Tirukkural-Tamil)

I am re-publishing this poem after seeing the man-made disaster in Uttarakhand.

L V Nagarajan – 23 June 2013

Thirukkural – 178

October 19, 2012

Thirukkural – 178

அஃகாமை செல்வத்திற்கு யாதெனின் வெஃகாமை

வேண்டும் பிறன்கைப் பொருள்

Ahkkamai selvatthiRku yadhenin vehkkamai

VeNdum piran kai PoruL

 

Ahkkamai – conservation, retention

Selvam – Ones prosperity, wealth

Vehkkamai – not to crave for, not to grab or not to seize

Piran Kai PoruL – belongings in other’s hands.

 

What is the best way to retain one’s own prosperity? One should not yearn for, or grab, or seize the belongings in other’s hands.

You are considered rich only when there is no need for  you to earn more. Desire to earn by unfair means renders you among the poor.

 

Wealth in your hand should never decay – then

Grabbing from others should never be the way

– by L V Nagarajan

 

Kamba Ramayanam – 1

September 27, 2010

 Kamba Ramayanam

 Ref: Tamil Virtual Academy –  http://www.tamilvu.org/

Kamba Ramayanam, as popularly known, is composed by Kambar, who is dated between 855 and 1185 AD. The name given to the work by the poet is RAmAvathAram. He has followed the original work of Valmiki Ramayanam closely. It is revered as one of the major pieces of Tamil literature along with Thirukkural and SilappadhikAram. For more information the website above may be referred to.

Many thousands of Tamils would have enjoyed the emotions of the story of Rama as told by Kambar. Many hundreds of Tamils would have enjoyed, in addition, the beauty of Tamil language as handled by Kambar. Many people would have also enjoyed the poetic beauty and imagination of Kambar. It is as though, the nectar (of Ramayana) is stored in a gold cup (in the form of  Tamil language) which itself is decorated with diamonds (of beautiful Tamil Poetry). Myself, being just an insect, happened to squat on the edge of this dazzling cup and to lick a few traces of this nectar. In the following series of articles, I propose to share my delight with others.

In this first post, four verses are presented. They are describing the way Sita was decorated in preparation for her marriage with Lord Rama,. The verses and the summary in Tamil were taken from the above quoted site. The summary in English is mine.

Thanks to http://www.suratha.com/reader.htm for enabling conversion of the Tamil Texts to Unicode.

பால காண்டம்

கோலம்காண் படலம் 

1223 – தோழிமார் சீதையைச் சூழ்தல் 

பரந்த மேகலையும் கோத்த

                                         பாத சாலகமும் நாகச்

சிரம் செய் நூபுரமும் வண்டும்

                                          சிலம்பொடு சிலம்பி ஆர்ப்பப் 

புரந்தரன் கோல் கீழ் ஆனோர்

                                          அரம்பையர் புடைசூழ்ந்து என்ன 

வரம்பறு சும்மைத் தீம் சொல்

                                           மடந்தையர் தொடர்ந்து சூழ்ந்தார்

பரவின மேகலையாகிய இடையணியும், கோக்கப்பட்ட பாத சாலகம் எனும் காலணியும், பாம்பின் தலைவடிவில் முகப்புடைய நூபுரம் என்னும் காலணியும்,  கைவளையல்களும் காற்சிலம்பும் ஒலித்து ஆரவாரமுண்டாக்க; இந்திரன் ஆட்சிக்குட்பட்ட அரம்பை  முதலிய தேவப் பெண்கள் பக்கங்களில் சூழ்ந்துவந்தாற் போல; அளவு கடந்தவர்களாய், இனிய ஒலிக் குதலைச் சொல்லினராய். தோழிப் பெண்கள் பலர், பிராட்டியின் மருங்கே இடையீடின்றிச் சூழ்ந்து சென்றனர். 

Resembling the dancers of the heaven in King Indra’s court, the friends of Sita were surrounding her all the time. They were wearing broad waist bands, bejeweled footwear, cobra shaped armlets and beaded anklets, all jingling together along with their sweet giggles. 

1204 – சீதையைத் தாதியர் அணிசெய்தல் 

அமிழ் இமைத்துணைகள் கண்ணுக்கு

                                       அணியென அமைக்குமா போல்

உமிழ் சுடர்க் கலன்கள் நங்கை

                                       உருவினை மறைப்பது ஓரார் 

அமிழ்தினைச் சுவை செய்து என்ன

                                      அழகினுக்கு அழகு செய்தார் 

இமிழ் திரைப் பரவை ஞாலம்

                                      ஏழைமை உடைத்து, மாதோ! 

(கண்கள்) மறைதற்குக் காரணமான இரண்டு  இமைகளும் அக்கண்களுக்கு அழகென்று (இறைவனால்) அமைக்கப்பட்டிருப்பது போல;  (சீதைக்கு  அணிசெய்யும்  மகளிர்) ஒளிவீசும் ஆபரணங்கள் சீதையின் (அழகிய) வடிவத்தை மறைக்கும் என்னும் உண்மையை உணராதவர்களாய்;  (இயல்பாகவே, அத்துணைச் சுவைகளையும்    பெற்றிருக்கிற) அமிழ்துக்கு. (கற்கண்டு சர்க்கரை முதலியவை  கொண்டு) மேலும் சுவை கூட்டும் முயற்சியைப் போல; (இயற்கையிலேயே பேர்  அழகுவாய்ந்த சீதையெனும்) அழகிற்கு (மேலும்  புதிதாக) அழகு  செய்யத் தொடங்கினார்கள்;  அலை ஒலிக்கும் கடல்சூழ்ந்த இந்நிலவுலகத்து  மக்கள் (நன்மை புரிவதாக  நினைத்துப் பிழைபுரியும்) அறியாமையுடையவர்களே.     . 

Though the eye-lids cover the eyes, they are considered ornamental to the eyes. Same way the sparkling jewels hide her youthful body. Totally unaware of this, just like the futile attempt of adding taste to the already sweet divine nectar, the maids are trying to beautify, the already beautiful (Sita). It was appalling to see the ignorance of the people living in this world covered by the seas. 

1208 – கழுத்தணி அணிதல் 

கோன் அணி சங்கம் வந்து

                                    குடி இருந்த அனைய கண்டத்து

ஈனம் இல் கலன்கள் தம்மில்

                                   இயைவன அணிதல் செய்தார்

மான் அணி நோக்கினார்தம்

                                  மங்கலக் கழுத்துக்கு எல்லாம் 

தான் அணி ஆன போது,

                                 தனக்கு அணி யாது மாதோ?

தலைவனான  திருமால் தாங்கும் சங்கு (இங்கு) வந்து  குடியிருந்தாற் போன்ற (சீதையின்) கழுத்தில் குற்றமற்ற ஆபரணங்களில் பொருந்துவனவற்றைப் பூட்டினார்கள்; மானின் விழிகளையுடைய (உலகத்து) மங்கையர் மாங்கலிய நாணோடு கூடிய கழுத்துகளுக்கு எல்லாம்;  (திரு என்னும் அடையோடுகூடித் திருமாங்கல்யம் என்று) தானே ஓர் ஒப்பற்ற அலங்காரமாய் இருக்கும்போது;  தன் கழுத்துக்கு ஏற்ற அணியாவது எது? (ஒன்றும் இல்லை.)

Her neck looks like it (already) wears the white conch normally worn by Vishnu. (Still) more of such choicest of the spotless jewels were worn, by maids, on her neck. The women of this world, with beautiful deer-like glances, wear around their necks. Sita’s images as holy ornaments. Then what else can be an ornament for her (neck)?

1219 – பிராட்டிக்குக் காப்பிடுதல் 

நெய் வளர் விளக்கம் ஆட்டி, 

                                நீரொடு பூவும் தூவித், 

தெய்வமும் பராவி, வேத

                               பாரகர்க்கு ஈந்து செம்பொன்

ஐயவி நுதலில் சேர்த்தி,

                               ஆய் நிற அயினி சுற்றிக், 

கைவளர் மயில் அனாளை

                               வலம் செய்து காப்பும் இட்டார். 

(அணிசெய்து முடித்த தோழியர் பிராட்டிக்குக் கண்ணேறு படும் என்று) நெய் நிறைந்த விளக்கொளியைச் சுற்றிக்காட்டி, புனித நீரினோடு கூடிய பூக்களையும் முன்னால் இறைத்து; தெய்வத்தினையும் வழிபட்டு, வேதங்களைக் கரைகண்டறிந்த  அந்தணாளர்க்குச் தானம் அளித்து;  செம்பொன் போன்ற சிறுவெண் கடுகின் பொடிகளை நெற்றியில் தீற்றி, ஆராய்ந்தெடுத்த செந்நிற ஆலத்தி நீரைச் சுழற்றி; கைகளினால் எடுத்து வளர்க்கப்பட்ட மயிலைப் போன்ற பிராட்டியை வலப்புறஞ் சுற்றிக் காப்பு நாணையும் இட்டனர். 

The maids showed ghee-lamp to her. They showered on her, flowers and scented water. They prayed to God. They gave charity to brahmins steeped in vedic knowledge They sprayed golden dust on her fore-head. They surrounded her with saffron colored arati. Went around the peacock-like Sita and cast the evil eyes away by putting saffron on her forehead.

I am not sure whether my English summary did justice to the beauty of Kambar’s poetry. Here is one more attempt to bring out its beauty. Smt Gowri Nagarajan set these verses to music as a part of a dance opera enacted by students of Sri S. Natarajan of Melattur Bhagavata Mela. (I need to write about Sri S. Natarajan and Melattur Bhagavata Mela in a separate post). The verses have been tuned into popular ragas, Vasantha, Kalavathi and Maduvanthi. You may listen to the musical version of these poems by clicking below.

Azhakinirku

Thirukkural – 302

August 31, 2010

செல்லா இடத்துச் சினந்தீது செல்லிடத்தும்
இல்அதனின் தீய பிற.

CellA idatthu china(m)theedhu cellidatthum

il adhanin theeya pira.

CellA Idatthu – in places beyond your reach or control

China(m) theedhu – Anger is harmful

Cellidatthum – even in places under your reach/control

adanin theeya pira – more harmful than it (i.e. anger)

il – one cannot find

One should not get angry in places not under one’s reach or control, since the same will not be tolerated. It may recoil on you. In addition, you will lose your purpose and also your goodwill for the future. This is easy to see. But, one should also understand that, even in places under one’s reach and command, there is nothing more harmful than anger. It has the danger of being highly disproportionate and will create ill will in your own camp.

In high places, wrath recoils, you know; 

In your own den, it’s a wild fire, you blow.

Ma Ganga

April 5, 2009

Please see below  a write-up on the condition of our holy Ganga.

 

Ganga River

Extracted From: http://www.gits4u.com/water/ganga.htm


          Today, over 29 cities, 70 towns, and thousands of villages extend along the Ganges’ banks. Nearly all of their sewage – over 1.3 billion liters per day – goes directly into the river, along with thousands of animal carcasses, mainly cattle. Another 260 million liters of industrial waste are added to this by hundreds of factories along the river’s banks.  Municipal sewage constitutes 80 per cent by volume of the total waste dumped into the Ganges, and industries contribute about 15 percent. The majority of the Ganges pollution is organic waste, sewage, trash, food, and human and animal remains. Over the past century, city populations along the Ganges have grown at a tremendous rate, while waste-control infrastructure has remained relatively unchanged. Recent water samples collected in Varanasi revealed fecal-coliform counts of about 50,000 bacteria per 100 milliliters of water, 10,000% higher than the government standard for safe river bathing. The result of this pollution is an array of water-borne diseases including cholera, hepatitis, typhoid and amoebic dysentery. An estimated 80% of all health problems and one-third of deaths in India are attributable to water-borne diseases.


            The sacred practice of depositing human remains in the Ganges also poses health threats because of the unsustainable rate at which partially cremated cadavers are dumped. In Varanasi, some 40,000 cremations are performed each year, most on wood pyres that do not completely consume the body. Along with the remains of these traditional funerals, there are thousands more who cannot afford cremation and whose bodies are simply thrown into the Ganges. In addition, the carcasses of thousands of dead cattle, which are sacred to Hindus, go into the river each year. An inadequate cremation procedure contributes to a large number of partially burnt or unburnt corpses floating down the Ganga.

 
            The industrial pollutants also a major source of contamination in the Ganges. A total of 146 industries are reported to be located along the river Ganga between Rishikesh and Prayagraj. 144 of these are in Uttar Pradesh (U.P.) and 2 in Uttrakhand. The major polluting industries on the Ganga are the leather industries, especially near Kanpur, which use large amounts of Chromium and other toxic chemical waste, and much of it finds its way into the meagre flow of the Ganga.  From the plains to the sea, pharmaceutical companies, electronics plants, textile and paper industries, tanneries, fertilizer manufacturers and oil refineries discharge effluents into the river. This hazardous waste includes hydrochloric acid, mercury and other heavy metals, bleaches and dyes, pesticides, and polychlorinated biphenyls highly toxic compounds that accumulate in animal and human tissue.

    
            However, industry is not the only source of pollution. Sheer volume of waste – estimated at nearly 1 billion litres per day – of mostly untreated raw sewage – is a significant factor.  Runoff from farms in the Ganges basin adds chemical fertilizers and pesticides such as DDT, which is banned in the United States because of its toxic and carcinogenic effects on humans and wildlife. Damming the river or diverting its water, mainly for irrigation purposes, also adds to the pollution crisis.
 

 

I was very disturbed to read the above report. If this is the condition with Ganga, the holiest of our rivers, what about other rivers?

Please read my poem on rivers by clicking on the link below

rivers

Please read my blog https://lvnaga.wordpress.com/2008/08/08/once-there-were-rivers/

L V Nagarajan/ 5th April 2009

Iyarkkaiyin Navarasam – The 9-Moods of the Nature

September 11, 2008

Iyarkkaiyin Navarasam – The 9-Moods of the Nature

A Poem published in Thendral, a tamil magazine published from Bay Area, SFO.