Seamless Korvais –

Elegance in Numbers in

music

Music Academy 20 Dec, 2014

Prelude

Numbers have fascinated man for millennia.

India’s contributions in this area is mammoth ingeneral.

It is therefore unsurprising that Indian rhythm hasled the way in world music when it comes tomusical mathematics.

Even between Indian’s two major classicalsystems, Carnatic culture stands out for not justrhythmic virtuosity but in its sophisticatedapproach towards structured mathematicalpatterns.

Korvai TypesLevel I: Any number taken after appropriate units

after samam to end as required. Ex: (3+3, 5+3, 7)x3after 1. My very first attempt at a korvai at age 5…

Level II: Taken from samam to end at samam with1 or 2 karvais between patterns to fill out theremaining units (in say 32/64/28/40 units in Adi1/2 kalais, Mishra chapu/Khanda chapu etc)

Level III: Same as II but to end a few units after orbefore samam.

Level IV: Same as I or II but with different gatisthrown in.

All these can be termed as man-made korvais

Natural Korvais – seamless eleganceSeamless korvai – DEFINITION: Patterns (of usually two or more parts)from samam to samam/landing point of song that do not have remainderindivisible by 3 in talas or landings indivisible by 3.In other words, these do not have remainder of any number of units notdivisible by 3 (like 2 or 4) which have to be patched up as 1 or 2 karvais it inbetween patterns. These have a grace or sophistication in the numbers thatare obvious only when one is inspired.

Intellectually, they require multi-layered thinking rather than justconventional approaches.

Some of them involve precise and logical patterns but not found inmathematical text books.

I literally stumbled upon most them as some of them are not accessiblethrough intuitive methods.

A couple of them have been in vogue for decades – 6, 8, 10 (or 8+8+8) asfirst part then 3x5, 2x5, (1x5) x3.

Typically, they are in one gati though there are exceptions (but overuse ofmultiple gati will make it a different concept.)

Seamless korvais – amazing options

ADI 2 kalais = 64 units

Challenges: To get 3 khandams (3x5) in Part B, Part A has to be49. Similarly, for 3x6, 3x7 or 3x9 in B, we need A to be 46, 43 or37, none of which is divisible by 3. So, simple approaches willnot work.

1. Simple progressive: These are most obvioustypes. Ex 1: 7+3 (karvais), 6+3….. 1+3 as first part (A)and 5x3 as the second part (B). Ex 2: 7+2…0+2 as Aand B is 5x3 in tishra gati. (A can also be insrotovaha yati)

2.Progressive with addition in multiple parts:

A= (2,3,4)+(2,3,4,5)+(2,3,4,5,6); B=7x3.

Seamless korvais – amazing options3. Inverted progression in 3 parts:

A=3,3,3 + 5X1, B= 3,3+7X2, C= 3+9X3

Another example:

A=2,2,3 + 7x1; B=2,3 + 7x2; C=3 + 7x3(tishra gati)

4. Progressive in second part: A= 6, 6, 6; B = (3x9) +(2x7) + (1x5). Impressive when B is rendered 3 timeswith A alternating between the 9, 7 and 5s.

5. Progressive in each part:

A = (5x3karvais)+(5x2karvais)+5x1;

B= (6x3karvais)+(6x2karvais)+6x2

C = (7x3karvais)+(7x2karvais)+7x3

Seamless korvais – amazing options

5. 3-speed korvais (example for 4 after samam):

(A=7, 2+7, 4+7; B=9x3)x3 karvais;

(A=7, 2+7, 4+7; B=9x3)x2 karvais;

A=7, 2+7, 4+7; B=9x3

Another example, employing second partprogression also (samam to samam):

6+2, 5+2, 4+2, 3+2, (3x5)x3; 6+2, 5+2, 4+2, 3+2, (2x7)x3; 6+2, 5+2, 4+2, 3+2, (1x9)x3

Seamless korvais – dovetailing patterns

The beauty of these are part of A will dovetail into B in aseamless manner.

(a) G,R,S,, R,S,N,, - G,R,SND - GR,S,, RS,N,, - GR,SND -GRS,, RSN,, - GRSND RSNDP SNDPM(b) GR, S, N, S,,, R,,, - GRSND – R,SN, S,,, R,,, - GRSND –SN, S,,, R,,, - GRSND RSNDP SNDPM(c) G,,,,, R,,,,, G,, R,, S,, N,, D,, - G,,, R,,, G, R, S, N, D, -G,R, GRSND RSNDP SNDPM(d) G,,, R,,, S,,, N,,, D – GRSND – R,, S,, N,, D,, P –RSNDP – S,N,D,P,D - GRSND RSNDP SNDPM

It would be obvious that some are 13+5, 13+5 and 13+(3 times5) in various ways. If song starts after +6, various manifestationsof 15+5, 15+5 and 15+(3 times 5) can be created.

Seamless korvais – Boomerang patternsLet’s look at the sequence of numbers: (a) 7, 12, 15, 16….

(b) 6, 10, 12, 12... What are the next numbers?

Typically, these are not part of general math textbooks and donot make sense to most mathematicians. But they are fineexamples of how Carnatic music can transcend science andmath. Remarkably, the series will turn back on itself. I call theseDouble layered progressive sequences which boomerang. Thefirst few numbers are formed using multiplication progressionin (a) are: 7x1, 6x2, 5x3, 4x4. Thus, the next few numbers are15, 12 and 7. Similarly, in (b), they are 10 and 6.

An example of a korvai with this: A= 6x2, 5x3, 4x4; B = 7x3

Another ex: A= 7, 12, 15, 16, 15, 12. B= 3 mishrams C= 3x10(which can be said as ta.. Ti.. Ki ta. Tom (to give an illusion of 7)

The concept of Keyless korvaisAt times, one stumbles upon korvais with no apparent mathematicalrelationship. These cannot be logically deciphered or developed by lockingon to their key (usually the average of their various parts/2nd repeat out of3). Yet, these are elegant beyond words in their simplicity.

1. A 3-part Korvai in 3 speeds: The amazing aesthetics ofthis is mind-boggling – simple when rendered but looks ajungle of numbers when expressed as below!

A = (8+3)x3 + (1x5)x3B = (6+3)x2 + (2x7) x 2C = (4+3)x1 + (3x9) x1

2. A 3-part korvai over 2 cycles (128 units): A stunning set ofpatterns found in nature.

A= [(5+2), (4+2), (3+2)] + (3x5);

B = [(5+2), (4+2), (3+2), (2+2)] + (3x7), C = [(5+2), (4+2),(3+2), (2+2), (1+2)] + (3x9).

Keyless korvais extensions to other talasKeyless methods give scope to execute amazing finishes in

seemingly impossible situations. For instance, a tala likeKhanda Triputa @ 8 units per beat (72 units) or Rupakam,which is already divisible by 3, can hardly offer scope for asamam to + 2 or + 4 finish… Let’s look at a couple ofaesthetic solutions.

1. Khanda triputa – samam to +2 (out of 8) in 2 cycles

A= [(5+2), (4+2), (3+2), (2+2)] + (3x5), B = [(5+2), (4+2), (3+2),(2+2), (1+2)] + (3x8), C = [(5+2), (4+2), (3+2), (2+2), (1+2), (0+2)]+ (3x11).

2. A 3-part Korvai in 3 speeds for same landing as above

A = (11+3)x3 + (1x5)x3 (Can be rendered as G, R, GRSNDPD N,, - GRSND in a raga like Vachaspati)

B = (9+3)x2 + (2x7) x 2C = (7+3)x1 + (3x9) x1

Keyless korvais extensions to other talas3. Khanda triputa – samam to +3 (out of 8)

[A= 7+3 (karvais), 6+3…..1+3, 0+3 B= 7x3] (To be rendered3 times or change B as 5x3, 7x3 and 9x3 each time etc).

4. Mishra Chapu: Samam to -1

[(5x4)+1]x3, [(4x4)+1]x3, [(3x4)+1]x3, [(2x4)+1]x3,[(1x4)+1]x3 (for landings like Suvaasita nava javanti in Shrimatrubhootam)

5. Roopakam: Samam to +2

A= [(5+2), (4+2), (3+2)] + (3x5), B = [(5+2), (4+2), (3+2),(2+2)] + (3x9), C = [(5+2), (4+2), (3+2), (2+2), (1+2)] +(3x13).

(The 3x(5/9/13) can be rendered as just 3x5 all 3 times. Or as3x9, 3x13, 3x17 etc.

Seamless korvais for other talasADI 1 kalai (32 units)

Most korvais in this smaller space require patch work. Someof the most famous ones are even mathematically incorrect.(ta, tom… taka tom.. Takita tom.. + 3x5).

1. Simple progressive: A few years ago, I had introduced

A = 2, 3, 4, 5; B = 6x3.

2.Single part apparently wrong but actually correctkorvai:

GR,-GRS,-GRSN,-GRSNP,-GRSNPG,-GRSNPGR

Typical hearing will make it seem like 1+2 karvais… 5+2karvais and final phrase illogically being 7. In reality, it is2+1, 3+1…6+1 ending in 7.

Seamless korvais for other talas… contdADI 1 kalais = 32 units

3. An elegant solution in 3 cycles for songs starting after 6(34 units/cycle)

A = (3x5) x3; B= (2x6)x3; C = (1x7) x3

4. Several other progressive solutions work beautifullyfor samam to songs starting after 6:

7+7 (karvais), 6+7….2+7 +1 (landing on the song)

The same one can be rendered with 6 karvais for songsstarting on samam.

5. A simple 3-speed solution for 6 after samam:

A = (6x3 + 5x3)x3; B = (6x3 + 5x3)x2

C = (6x3 + 5x3)x1

Seamless korvais for other talas… contdADI 1 kalais = 32 units

6. A progressive 3-speed korvai for 6 after samam:

(7+7+3; 5)x3 karvais; (GR,S,N, DP,D,N, S,, - GRSND)X3

(6+6+3; 5)x2 karvais;

5+5+3; 5,5,5

Roopakam from samam to +3

A= [6, (2+6), (4+6)] B = (5 x 4 karvais + 3x5)

C = [6, (2+6), (4+6)] D = (7 x 4 karvais + 3x7)

E= [6, (2+6), (4+6)] B = (9 x 4 karvais + 3x9)

Note: A, C and E can be any combination divisible by 12

Seamless korvais in other gatisJust as many korvais for Adi can be extended to other talas, they can be extended

to other gatis too. For instance, Adi - Khanda gati (double speed) = 80 units

Eg: GR, SN, DP, DN, S,, - G, R, SND – RS, ND, PM, P D, N,,- R,S |,NDP – SN, DP, MG, MP, D,, - G | ,R,SND – R,S,NDP – S,N,DPM ||

But there are highly interesting possibilities which areoriginal for this like the one I had presented in my soloconcert at the Academy 2-3 years ago: A = (4x5) + (3x7) +(2x9); B= 5+7+9

There is a lovely possibility in 3 gatis:

G,R, SN, S,, - GRSND (tishram)

GR, SN, S,, - G, R, SND (Chaturashram)

GRSN, S,, - G,R,GRSND – R,S,RSNDP – S,N,SNDPM

Seamless korvais with other approachesI had remarked in a mrdanga arangetram about how most of our music is

elementary arithmetic and why percussionists must focus on aesthetics once they have got the patterns right. This got me into thinking about experimenting with

korvais that represent some other math concepts such as a couple below:

1. Fibonachi series: Leonardo of Pisa, known as Fibonacci in1200 AD but attributed to a much earlier Indian mathematicianPingala (450-200 BC). The series is any two initial numbers like 3,4 which are added to get 7. Now, add the last two numbers (4+7)to get 11 and so forth. A korvai in that sequence (in say, Kalyani):

A = G,, - R,,, - G,R,SND – GRSNDPMGRSN – DN,R,, GM,D,, MD,N,, B=G,R,SND – R,S,NDP – D,P,MGR

2. A simple korvai using squares of numbers as first part(3)2+(4)2+(5)2:

A= G,,R,,S,, - G,,, R,,, S,,, N,,, - G,,,, R,,,, S,,,, N,,,, D,,,,

B= 3 mishrams in tishra gati double speed.

Creating Seamless korvaisIt now would be obvious that anyone can create seamless

korvais with the thinking and methods I have shared.

I have used mostly familiar sounding easy patterns to createthese, mainly with melodic aesthetics in mind.

I have shown only a few small samples here, even from theones I have discovered/presented.

Pure rhythmic seamless korvais can deal with typical patternssuited for percussion.

This is a vast exciting new world with tremendous scope toexpand the horizons both melodically and rhythmically.

Each door I’ve opened leads to exhilarating worlds…

Happy exploring!!!