Katapayadhi Sankhya
Ka-ṭa-pa-yā-di (Devanagari: कटपयादि) system (also known as Paralppēru, Malayalam: പരല്പ്പേര്) of numerical notation is an ancient Indian system to depict letters to numerals for easy remembrance of numbers as words or verses. Assigning more than one letter to one numeral and nullifying certain other letters as valueless, this system provides the flexibility in forming meaningful words out of numbers which can be easily remembered.
Haridatta's Grahacāraṇibandhana
Here is an actual verse of spiritual content, as well as secular mathematical significance:
“gopi bhagya madhuvrata
srngiso dadhi sandhiga
khala jivita khatava
gala hala rasandara”
The translation is as follows: “O Lord anointed with the yoghurt of the milkmaids’ worship (Krishna), O savior of the fallen, O master of Shiva, please protect me.”
Now the interesting fact is that when you start numbering the consonants with the respective numbers from go = 3, pi = 1, bha =4 , ya = 1 , ma = 5 , duv = 9 and so on. you will end with the number 31415926535897932384626433832792.
Laghubhāskariyavivarana written by Sankaranārāyana in 869 CE
Sarikamavarman Sudratna
nanyāvacaśca śūnyāni saṃkhyāḥ kaṭapayādayaḥ
miśre tūpāntyahal saṃkhyā na ca cintyo halasvaraḥ
Raagas
The melakarta ragas of the Carnatic music is named so that the first two syllables of the name will give its number. This system is sometimes called the Ka-ta-pa-ya-di sankhya. The Swaras 'Sa' and 'Pa' are fixed, and here is how to get the other swaras from the melakarta number.
Melakartas 1 through 36 have Ma1 and those from 37 through 72 have Ma2
The other notes are derived by noting the (integral part of the) quotient and remainder when one less than the melakarta number is divided by 6
'Ri' and 'Ga' positions: the raga will have:
Ri1 and Ga1 if the quotient is 0
Ri1 and Ga2 if the quotient is 1
Ri1 and Ga3 if the quotient is 2
Ri2 and Ga2 if the quotient is 3
Ri2 and Ga3 if the quotient is 4
Ri3 and Ga3 if the quotient is 5
'Da' and 'Ni' positions: the raga will have:
Da1 and Ni1 if remainder is 0
Da1 and Ni2 if remainder is 1
Da1 and Ni3 if remainder is 2
Da2 and Ni2 if remainder is 3
Da2 and Ni3 if remainder is 4
Da3 and Ni3 if remainder is 5
Simhendramadhyamam
You can see that, as per the above calculation we should get Sa \leftrightarrow 7, Ha \leftrightarrow 8 giving the number 87 instead of 57 for Simhendramadhyamam. This should be ideally Sa \leftrightarrow 7, Ma \leftrightarrow 5 giving the number 57. So it is believed that the name should be written as Sihmendramadhyamam (As in the case of Brahmana in Sanskrit).
Mechakalyani
From the coding scheme Ma \leftrightarrow 5, Cha \leftrightarrow 6. Hence the raga's melakarta number is 65 (56 reversed). 65 is greater than 36. So MechaKalyani has Ma2. Since the raga's number is greater than 36 subtract 36 from it. 65-36=29. 28 (1 less than 29) divided by 6 : quotient=4, remainder=4. Ri2 Ga3 occurs. Da2 Ni3 occurs. So MechaKalyani has the notes Sa Ri2 Ga3 Ma2 Pa Da2 Ni3 SA
Dheerashankarabraranam
The katapayadi scheme associates dha\leftrightarrow9 and ra\leftrightarrow2, hence the raga's melakarta number is 29 (92 reversed). Now 29 \le 36, hence Dheerasankarabharanam has Ma1. Divide 28 (1 less than 29)by 6, the quotient is 4 and the remainder 4. Therefore, this raga has Ri2, Ga3 (quotient is 4) and Da2, Ni3 (remainder is 4). Therefore, this raga's scale is Sa Ri2 Ga3 Ma1 Pa Da2 Ni3 SA.
Ka-ṭa-pa-yā-di (Devanagari: कटपयादि) system (also known as Paralppēru, Malayalam: പരല്പ്പേര്) of numerical notation is an ancient Indian system to depict letters to numerals for easy remembrance of numbers as words or verses. Assigning more than one letter to one numeral and nullifying certain other letters as valueless, this system provides the flexibility in forming meaningful words out of numbers which can be easily remembered.
Haridatta's Grahacāraṇibandhana
Here is an actual verse of spiritual content, as well as secular mathematical significance:
“gopi bhagya madhuvrata
srngiso dadhi sandhiga
khala jivita khatava
gala hala rasandara”
The translation is as follows: “O Lord anointed with the yoghurt of the milkmaids’ worship (Krishna), O savior of the fallen, O master of Shiva, please protect me.”
Now the interesting fact is that when you start numbering the consonants with the respective numbers from go = 3, pi = 1, bha =4 , ya = 1 , ma = 5 , duv = 9 and so on. you will end with the number 31415926535897932384626433832792.
Laghubhāskariyavivarana written by Sankaranārāyana in 869 CE
Sarikamavarman Sudratna
nanyāvacaśca śūnyāni saṃkhyāḥ kaṭapayādayaḥ
miśre tūpāntyahal saṃkhyā na ca cintyo halasvaraḥ
Raagas
The melakarta ragas of the Carnatic music is named so that the first two syllables of the name will give its number. This system is sometimes called the Ka-ta-pa-ya-di sankhya. The Swaras 'Sa' and 'Pa' are fixed, and here is how to get the other swaras from the melakarta number.
Melakartas 1 through 36 have Ma1 and those from 37 through 72 have Ma2
The other notes are derived by noting the (integral part of the) quotient and remainder when one less than the melakarta number is divided by 6
'Ri' and 'Ga' positions: the raga will have:
Ri1 and Ga1 if the quotient is 0
Ri1 and Ga2 if the quotient is 1
Ri1 and Ga3 if the quotient is 2
Ri2 and Ga2 if the quotient is 3
Ri2 and Ga3 if the quotient is 4
Ri3 and Ga3 if the quotient is 5
'Da' and 'Ni' positions: the raga will have:
Da1 and Ni1 if remainder is 0
Da1 and Ni2 if remainder is 1
Da1 and Ni3 if remainder is 2
Da2 and Ni2 if remainder is 3
Da2 and Ni3 if remainder is 4
Da3 and Ni3 if remainder is 5
Simhendramadhyamam
You can see that, as per the above calculation we should get Sa \leftrightarrow 7, Ha \leftrightarrow 8 giving the number 87 instead of 57 for Simhendramadhyamam. This should be ideally Sa \leftrightarrow 7, Ma \leftrightarrow 5 giving the number 57. So it is believed that the name should be written as Sihmendramadhyamam (As in the case of Brahmana in Sanskrit).
Mechakalyani
From the coding scheme Ma \leftrightarrow 5, Cha \leftrightarrow 6. Hence the raga's melakarta number is 65 (56 reversed). 65 is greater than 36. So MechaKalyani has Ma2. Since the raga's number is greater than 36 subtract 36 from it. 65-36=29. 28 (1 less than 29) divided by 6 : quotient=4, remainder=4. Ri2 Ga3 occurs. Da2 Ni3 occurs. So MechaKalyani has the notes Sa Ri2 Ga3 Ma2 Pa Da2 Ni3 SA
Dheerashankarabraranam
The katapayadi scheme associates dha\leftrightarrow9 and ra\leftrightarrow2, hence the raga's melakarta number is 29 (92 reversed). Now 29 \le 36, hence Dheerasankarabharanam has Ma1. Divide 28 (1 less than 29)by 6, the quotient is 4 and the remainder 4. Therefore, this raga has Ri2, Ga3 (quotient is 4) and Da2, Ni3 (remainder is 4). Therefore, this raga's scale is Sa Ri2 Ga3 Ma1 Pa Da2 Ni3 SA.
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