Katapayadi Sankhya

KaTaPaYaDi Sankhya For Melakartha Ragas:
Definition and Some interesting facts about this system:
For laymen,
For those who are from an IT background hopefully you know what a hash means!! Here I present the first known hashing algorithm in the world, which of course is again from ancient India !! Well, we Indians are the world pioneers in Mathematics, as Einstein said ‘We owe a lot to the Indians who taught us how to count’

OK, So let us refresh what a hash is

A hash is a one way representation of a text in the form of a unique number. The text might be a single word, or a sentence or an entire document! The hashing algorithm converts it into a unique number

To use the sankhya, take the first two syllables of the name of the ragam, and locate the corresponding columns on the table. Then take the two numbers and reverse them to get the mela number.

Following verse found in Sankaravarman's Sadratnamala explains the mechanism of the system
नज्ञावचश्च शून्यानि संख्या: कटपयादय:मिश्रे
तूपान्त्यहल् संख्या न च चिन्त्यो हलस्वर:

Transiliteration
nanyāvacaśca śūnyāni saṃkhyāḥ kaṭapayādayaḥmiśre
tūpāntyahal saṃkhyā na ca cintyo halasvaraḥ

Translation: na (न), nya (ञ) and a (अ)-s i.e. vowels represent zero. The (nine) integers are represented by consonant group beginning with ka, ṭa, pa, ya. In a conjunct consonant, the last of the consonants alone will count. A consonant without vowel is to be ignored.

Katapayaadi Varga - Consonants
1 2 3 4 5 6 7 8 9 0
ka क kha ख ga ग gha घ nga ङ ca च cha छ ja ज jha झ nya ञ
ṭa ट ṭha ठ ḍa ड ḍha ढ ṇa ण ta त tha थ da द dha ध na न
pa प pha फ ba ब bha भ ma म - - - - -
ya य ra र la ल va व śha श sha ष sa स ha ह - -

A quick introduction to Swaras before venturing into Ragas:
Raaga - The progression of swaras in some specific order. Each Raaga has Arohanam and Avarohanam.
Swaram - The fundamental constituent of a Raaga. There are 7 Swarams. Sa Ri Ga Ma Pa Dha Ni. Sa and Pa are called Prakruthi Swaras, for there is only one type of Sa and Pa. (its equivalent of C and G in Western Music).
1) Note that there are 3 variations each for Ri, Ga, Dha and Ni (R1, R2, R3, G1, G2, G3, D1, D2, D3, N1, N2 and N3)
Here please note that Ri2 == Ga1, Ri3 == Ga2, Dha2 == Ni1, Dha3 == Ni2, where "==" is the Equivalence relation.
2) M has 2 varieties - M1 and M2 (Shuddha Madhyamam and Prati Madhyamam)
3) How do we have 72 melakarthas?
Sa - 1,
Ri Ga - (R1 G1, R1 G2, R1 G3, R2 G2, R2 G3, R3 G3)= 6,
Ma, (Ma1, Ma2) = 2,
Pa - 1,
Dha Ni - (D1 N1, D1 N2, D1 N3, D2 N2, D2 N3, D3 N3) = 6

Multiply the possibilities to get the overall possibilities = 1 * 6 * 2 * 1 * 6 = 72. So we have the Melakartha scheme which is 72.

Simply put a combination of 7 swaras with its varieties once each, renders to us the Melakartha raga - each Melakartha Raga is a sampoorna Raga (contains all 7 swaras).

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 katapayadi sankhya. The swaras 'Sa' and 'Pa' are fixed, and here is how to get the other swaras from the melakarta number.
1) Melakartas 1 through 36 have Ma1 and those from 37 through 72 have Ma2.
2) 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.
3) '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
4) '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
Some examples:
1) Raga DheeraShankarabharanam: The katapayadi scheme associates dha9 and ra2, hence the raga's melakarta number is 29 (92 reversed). Now 29 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.
2) Raga MechaKalyani
From the coding scheme Ma 5, Cha 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.
3) Exception for Simhendramadhyamam
You can see that, as per the above calculation we should get Sa 7, Ha 8 giving the number 87 instead of 57 for Simhendramadhyamam. This should be ideally Sa 7, Ma 5 giving the number 57. So it is believed that the name should be written as Sihmendramadhymam (As in the case of Brahmana in Sanskrit).

Slightly offtrack,
The following phrase found in "sadratnamAla" a treatise on astronomy,
bhadram budhi siddha janma gaNita Sraddha@h mayadbhUpagi@h
when decoded yields
4 2 3 9 7 8 5 3 5 6 2 9 5 1 4 1 3
which when reversed gives
3 1 4 1 5 9 2 6 5 3 5 8 7 9 3 2 4
which is readily recognised as the digits in "pi" (except that the 17thdigit is wrong - it should be 3) :-)!
(source: The article "The Katapayadi Formula and Modern Hashing Technique"by Anand V Raman, appearing in "Computing Science in Ancient India", editedby T.R.N.Rao and Subhash Kak, published by the Center for Advanced ComputerStudies, University of South Western Louisiana, Louisiana LA 70504)

The Complete Melakartha Raga List

CHAKRA 1 - INDU
1 Kanakangi S R1 G1 M1 P D1 N1 S S N1 D1 P M1 G1 R1 S
2 Ratnangi S R1 G1 M1 P D1 N2 S S N2 D1 P M1 G1 R1 S
3 Ganamurti S R1 G1 M1 P D1 N3 S S N3 D1 P M1 G1 R1 S
4 Vanaspati S R1 G1 M1 P D2 N2 S S N2 D2 P M1 G1 R1 S
5 Manavati S R1 G1 M1 P D2 N3 S S N3 D2 P M1 G1 R1 S
6 Tanarupi S R1 G1 M1 P D3 N3 S S N3 D3 P M1 G1 R1 S
CHAKRA 2 - NETRA
7 Senavati S R1 G2 M1 P D1 N1 S S N1 D1 P M1 G2 R1 S
8 Hanumatodi S R1 G2 M1 P D1 N2 S S N2 D1 P M1 G2 R1 S
9 Dhenuka S R1 G2 M1 P D1 N3 S S N3 D1 P M1 G2 R1 S
10 Natakapriya S R1 G2 M1 P D2 N2 S S N2 D2 P M1 G2 R1 S
11 Kokilapriya S R1 G2 M1 P D2 N3 S S N3 D2 P M1 G2 R1 S
12 Rupavati S R1 G2 M1 P D3 N3 S S N3 D3 P M1 G2 R1 S
CHAKRA 3 - AGNI
13 Gayakapriya S R1 G3 M1 P D1 N1 S S N1 D1 P M1 G3 R1 S
14 Vakulabharanam S R1 G3 M1 P D1 N2 S S N2 D1 P M1 G3 R1 S
15 Mayamalavagaula S R1 G3 M1 P D1 N3 S S N3 D1 P M1 G3 R1 S
16 Chakravakam S R1 G3 M1 P D2 N2 S S N2 D2 P M1 G3 R1 S
17 Suryakantam S R1 G3 M1 P D2 N3 S S N2 D2 P M1 G3 R1 S
18 Hatakambari S R1 G3 M1 P D3 N3 S S N3 D3 P M1 G3 R1 S
CHAKRA 4 - VEDA
19 Jhankaradhwani S R2 G2 M1 P D1 N1 S S N1 D1 P M1 G2 R2 S
20 Natabhairavi S R2 G2 M1 P D1 N2 S S N2 D1 P M1 G2 R2 S
21 Keeravani S R2 G2 M1 P D1 N3 S S N3 D1 P M1 G2 R2 S
22 Kharaharapriya S R2 G2 M1 P D2 N2 S S N2 D2 P M1 G2 R2 S
23 Gowrimanohari S R2 G2 M1 P D2 N3 S S N3 D2 P M1 G2 R2 S
24 Varunapriya S R2 G2 M1 P D3 N3 S S N3 D3 P M1 G2 R2 S
CHAKRA 5 - BANA
25 Mararanjani S R2 G3 M1 P D1 N1 S S N1 D1 P M1 G3 R2 S
26 Charukesi S R2 G3 M1 P D1 N2 S S N2 D1 P M1 G3 R2 S
27 Sarasangi S R2 G3 M1 P D1 N3 S S N3 D1 P M1 G3 R2 S
28 Harikambhoji S R2 G3 M1 P D2 N2 S S N2 D2 P M1 G3 R2 S
29 Dheerashankarabharanam S R2 G3 M1 P D2 N3 S S N3 D2 P M1 G3 R2 S
30 Naganandini S R2 G3 M1 P D3 N3 S S N3 D3 P M1 G3 R2 S
CHAKRA 6 - RUTU
31 Yagapriya S R3 G3 M1 P D1 N1 S S N1 D1 P M1 G3 R3 S
32 Ragavardhani S R3 G3 M1 P D1 N2 S S N2 D1 P M1 G3 R3 S
33 Gangeyabhushani S R3 G3 M1 P D1 N3 S S N3 D1 P M1 G3 R3 S
34 Vagadheeswari S R3 G3 M1 P D2 N2 S S N2 D2 P M1 G3 R3 S
35 Soolini S R3 G3 M1 P D2 N3 S S N3 D2 P M1 G3 R3 S
36 Chalanata S R3 G3 M1 P D3 N3 S S N3 D3 P M1 G3 R3 S
CHAKRA 7 - RISHI
37 Salagam S R1 G1 M2 P D1 N1 S S N1 D1 P M2 G1 R1 S
38 Jalarnavam S R1 G1 M2 P D1 N2 S S N2 D1 P M2 G1 R1 S
39 Jhalavarali S R1 G1 M2 P D1 N3 S S N3 D1 P M2 G1 R1 S
40 Navaneetam S R1 G1 M2 P D2 N2 S S N2 D2 P M2 G1 R1 S
41 Pavani S R1 G1 M2 P D2 N3 S S N3 D2 P M2 G1 R1 S
42 Raghupriya S R1 G1 M2 P D3 N3 S S N3 D3 P M2 G1 R1 S
CHAKRA 8 - VASU
43 Gavambodhi S R1 G2 M2 P D1 N1 S S N1 D1 P M2 G2 R1 S
44 Bhavapriya S R1 G2 M2 P D1 N2 S S N2 D1 P M2 G2 R1 S
45 Subhapantuvarali S R1 G2 M2 P D1 N3 S S N3 D1 P M2 G2 R1 S
46 Shadvidhamargini S R1 G2 M2 P D2 N2 S S N2 D2 P M2 G2 R1 S
47 Suvarnangi S R1 G2 M2 P D2 N3 S S N3 D2 P M2 G2 R1 S
48 Divyamani S R1 G2 M2 P D3 N3 S S N3 D3 P M2 G2 R1 S
CHAKRA 9 - BRAHMA
49 Dhavalambari S R1 G3 M2 P D1 N1 S S N1 D1 P M2 G3 R1 S
50 Namanarayani S R1 G3 M2 P D1 N2 S S N2 D1 P M2 G3 R1 S
51 Kamavardhini S R1 G3 M2 P D1 N3 S S N3 D1 P M2 G3 R1 S
52 Ramapriya S R1 G3 M2 P D2 N2 S S N2 D2 P M2 G3 R1 S
53 Gamanasrama S R1 G3 M2 P D2 N3 S S N3 D2 P M2 G3 R1 S
54 Viswambari S R1 G3 M2 P D3 N3 S S N3 D3 P M2 G3 R1 S
CHAKRA 10 - DISHI
55 Syamalangi S R2 G2 M2 P D1 N1 S S N1 D1 P M2 G2 R2 S
56 Shanmukhapriya S R2 G2 M2 P D1 N2 S S N2 D1 P M2 G2 R2 S
57 Simhendramadhyamam S R2 G2 M2 P D1 N3 S S N3 D1 P M2 G2 R2 S
58 Hemavati S R2 G2 M2 P D2 N2 S S N2 D2 P M2 G2 R2 S
59 Dharmavati S R2 G2 M2 P D2 N3 S S N3 D2 P M2 G2 R2 S
60 Neetimati S R2 G2 M2 P D3 N3 S S N3 D3 P M2 G2 R2 S
CHAKRA 11 - RUDRA
61 Kantamani S R2 G3 M2 P D1 N1 S S N1 D1 P M2 G3 R2 S
62 Rishabhapriya S R2 G3 M2 P D1 N2 S S N2 D1 P M2 G3 R2 S
63 Latangi S R2 G3 M2 P D1 N3 S S N3 D1 P M2 G3 R2 S
64 Vachaspati S R2 G3 M2 P D2 N2 S S N2 D2 P M2 G3 R2 S
65 Mechakalyani S R2 G3 M2 P D2 N3 S S N3 D2 P M2 G3 R2 S
66 Chitrambari S R2 G3 M2 P D3 N3 S S N3 D3 P M2 G3 R2 S
CHAKRA 12 - ADITYA
67 Sucharitra S R3 G3 M2 P D1 N1 S S N1 D1 P M2 G3 R3 S
68 Jyotiswaroopini S R3 G3 M2 P D1 N2 S S N2 D1 P M2 G3 R3 S
69 Dhatuvardhini S R3 G3 M2 P D1 N3 S S N3 D1 P M2 G3 R3 S
70 Nasikabhooshani S R3 G3 M2 P D2 N2 S S N2 D2 P M2 G3 R3 S
71 Kosalam S R3 G3 M2 P D2 N3 S S N3 D2 P M2 G3 R3 S
72 Rasikapriya S R3 G3 M2 P D3 N3 S S N3 D3 P M2 G3 R3 S