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 in general.

It is therefore unsurprising that Indian rhythm has led theway in world music when it comes to musical mathematics.

Even between Indian’s two major classical systems,Carnatic culture stands out for not just rhythmic virtuositybut in its sophisticated approach towards structuredmathematical patterns.

Korvai Types

Level I: Any number taken after appropriate units aftersamam to end as required. Ex: (3+3, 5+3, 7)x3 after 1. Myvery first attempt at a korvai at age 5…

Level II: Taken from samam to end at samam with 1 or 2karvais between patterns to fill out the remaining units (insay 32/64/28/40 units in Adi 1/2 kalais, Mishrachapu/Khanda chapu etc)

Level III: Same as II but to end a few units after or beforesamam.

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

All these can be termed as man-made korvais

Natural Korvais – seamless elegance

Seamless korvai – DEFINITION: Patterns (of usually two or more parts) from samam tosamam/landing point of song that do not have remainder indivisible by 3 in talas or landingsindivisible by 3.In other words, these do not have remainder of any number of units not divisible by 3 (like 2or 4) which have to be patched up as 1 or 2 karvais it in between patterns. These have a graceor sophistication in the numbers that are obvious only when one is inspired.

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

Some of them involve precise and logical patterns but not found in mathematical textbooks.

I literally stumbled upon most them as some of them are not accessible through intuitivemethods.

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

Typically, they are in one gati though there are exceptions (but overuse of multiple gati willmake 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 be 49. Similarly,for 3x6, 3x7 or 3x9 in B, we need A to be 46, 43 or 37, none of which isdivisible by 3. So, simple approaches will not work.

1. Simple progressive: These are most obvious types.

Ex 1: 7+3 (karvais), 6+3….. 1+3 as first part (A) and 5x3 as the secondpart (B).

(srgmpdn s,, rgmpdn s,, gmpdn s,, mpdn s,, pdn s,,

dn s,, ns,,) as A and (grsnd rsndp dpmgr) as B

Simple progressive contd Ex 2: 7+2…0+2 as A and B is 5x3 in tishra gati. (A can also

be in srotovaha yati)

(srgmpdn s, rgmpdn s, gmpdn s, mpdn s, pdn s, dn s, ns, s,) asA and (grsnd rsndp dpmgr) as B in tishra gati.

2.Progressive with addition in multiple parts:

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

(s, ns, dns,) + (s, ns, dns, pdns,) + (s, ns, dns, pdns, mpdns,) asA and (g,r,snd r,s,ndp d,p,mgr) as B

Seamless korvais – amazing options

3. Inverted progression in 3 parts:

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

(g,, r,, s,, + grsns) as A and( r,, s,, + g,r,sns r,s,ndn) as Band (s,, + g,r,grsns r,s,rsndp d,p,dpmgr) as C

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 isrendered 3 times with A alternating between the 9, 7 and 5s.

( gr,s,, rs,n,, sn,d,,) + (g,, ,,, r,, ,,, g,, r,, s,, n,, d,, r,, ,,, s,, ,,, r,, s,, n,, d,, p,,s,, ,,, n,, ,,, s,, n,, d,, p,, m,,) first time

then (gm,p,, mp,d,, pd,n,,) + (g, ,, m, ,, p, d, p, m, ,, p, ,, d, n, d, p, ,, d, ,,n, s, n,) as second time

and (gr,s,, rs,n,, sn,d,, ) +( grsnd rsndp dpmgr) as third time

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

(g,, ,,, r,, ,,, s,, ,,, ,,, s,, n,, g,, ,,, r,, ,,, s,, ,,, ,,, d,,n,,s,,n,, g,, ,,, r,, ,,, s,, ,,, ,,,) as

A and (g,, ,,, r,, ,,, g,, r,, s,, n,, d,, r,, ,, s,, ,, r,, s,, n,, d,, p,, s,, ,,, n,, ,,, s,, n,, d,,p,, m,,) as B;

(g, ,, m, ,, p, ,, ,, p, m, g, ,, m, ,, p, ,, ,, n,d,p,m, g, ,, m, ,, p, ,, ,, ) as A and(g,m,g,m,p,d,p, m,p,m,p,d,n,d, p,d,p,d,n,s,n,) as B;

(g,r,s,, sn g,r,s,, dnsn g,r,s,,) as A and (g,r,grsnd r,s,rsndp d,p,dpmgr) as B

Another example, employing second part progression 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 a seamless manner.

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

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

Seamless korvais – Boomerang patterns

Let’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 do not makesense to most mathematicians. But they are fine examples of how Carnaticmusic can transcend science and math. Remarkably, the series will turn backon itself. I call these Double layered progressive sequences whichboomerang. The first few numbers are formed using multiplicationprogression in (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

(Ta….. Ki…..), (Ta,,,, ki,,,, ta,,,,) , (Ta,,, ka,,, di,,, mi,,,) as A and (Ta.di.kitatomTa.di.kitatom Ta.di.kitatom ) as B

• Another ex: A= 7, 12, 15, 16, 15, 12 or 7x1, 6x2, 5x3, 4x4, 5x3, 6x2 and B= 3 mishrams C= 3x10 (which can be said as ta.. Ti.. Ki ta. Tom (to give an illusion of 7)

(g,,,,,, r,,,,,s,,,,, n,,,,d,,,,p,,,, m,,,g,,,r,,,s,,, r,,g,,m,,p,,d,, m,p,d,n,s,r,) as A and (g,r,snd r,s,ndp p,d,nsr) as B and

(g,,r,,sn,d r,,s,,nd,p s,,n,,dp,m ) as C

The concept of Keyless korvaisAt times, one stumbles upon korvais with no apparent mathematical relationship.These cannot be logically deciphered or developed by locking on to their key (usuallythe average of their various parts/2nd repeat out of 3). Yet, these are elegant beyondwords in their simplicity.

1. A 3-part korvai over 2 cycles (128 units): A stunning set of patternsfound in nature.

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

[( gmpdn s,) (mpdn s,) (pdn s,)] + grsnd rsndp sndpm as A

B = [(5+2), (4+2), (3+2), (2+2)] + (3x7)

[( gmpdn s,) (mpdn s,) (pdn s,) (dn s,)] + (g,r,snd r,s,ndp d,p,mgr) as B and

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

[( gmpdn s,) (mpdn s,) (pdn s,) (dn s,) (n s,)] + (g,r,grsnd r,s,rsndpd,p,dpmgr) as C

2. 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

(Ta.. … Di.. … Ta.. Ka.. Di.. Na.. Tam.. … …) + (Ta.. Di.. Ki.. Ta..tom.. ) – A(Di. .. Ta. Ka. Di. Na. tam. .. ) +( Ta. .. Di. .. Ki. Ta. Tom. Ta. .. Di. .. Ki. Ta.Tom.) – B( Takadina Tam..) +( Ta.di.tatikitatom Ta.di.tatikitatom Ta.di.tatikitatom) -C

Keyless korvais extensions to other talas

Keyless methods give scope to execute amazing finishes in seemingly impossiblesituations. For instance, a tala like Khanda Triputa @ 8 units per beat (72 units) orRupakam, which is already divisible by 3, can hardly offer scope for a samam to +2 or + 4 finish… Let’s look at a couple of aesthetic 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).

[Takatakita tam. Takadina tam. Takita tam. Taka tam.] + (TadikitatomTadikitatom Tadikitatom) – A

[Takatakita tam. Takadina tam. Takita tam. Taka tam. TaTam.] + (Tadi . Ki .Ta . tom Tadi . Ki . Ta . tom Tadi . Ki . Ta . tom ) - B

[Takatakita tam. Takadina tam. Takita tam. Taka tam. TaTam. Tam.] +

(Ta di .. Ki.. Ta.. Tom Ta di .. Ki.. Ta.. Tom Ta di .. Ki.. Ta.. Tom ) - C

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, GRSN DPDN,, - GRSND in a raga like Vachaspati)

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

A = (g,, ,,, r,, ,,, g,, r,, s,, n,, d,, p,, d,, n,, ,, ,,) +( g,, r,, s,, n,, d,,)B = ( r, ,, g, r, s, n, d, p, d, n, ,, ,, ) + (g,,, r,,, s, n, d, r,,,s,,,n,d,p,)C= (grsndpd n,,) + (g,r,grsnd r,s,rsndp s,n,sndpm)

3. Khanda triputa – samam to +3 (out of 8)

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

A= Takadimitakita tam.. Takadimitaka tam.. Takatakita tam.. Takadina tam.. Takita tam.. Taka tam.. Ta tam.. Tam..)

B = (Ta.di.kitatom Ta.di.kitatom Ta.di.kitatom )

Keyless korvais extensions to other talas4. 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)

(Ta…. Di…. Ki…. Ta…. Tom Ta…. Di…. Ki…. Ta…. Tom Ta…. Di…. Ki…. Ta…. Tom ),

(Ta... Di... Ki… Ta… Tom Ta... Di... Ki… Ta… Tom Ta... Di... Ki… Ta… Tom ),

(Ta.. Di.. Ki.. Ta.. Tom Ta.. Di.. Ki.. Ta.. Tom Ta.. Di.. Ki.. Ta.. Tom),

(Ta. Di. Ki. Ta. Tom Ta. Di. Ki. Ta. Tom Ta. Di. Ki. Ta. Tom) ,

(Tadikitatom Tadikitatom Tadikitatom )

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. Some of the most famousones 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.

(Tam. TaTam. TakaTam. TakiTaTam.) – A

(Tadi.kitatom Tadi.kitatom Tadi.kitatom ) - B

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

is 2+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

(G,, r,, s,, n,, d,, r,, s,, n,, d,, p,, d,, p,, m,, g,, r,,) – A

(g, m, ,, p, d, p, m, p, ,, d, n, d, p, d, ,, n, s, n,) – B

(gr,,snd rs,,ndp dp,,mgr) - C

4. Several other progressive solutions work beautifully for samam tosongs 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 songs starting 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

(G,, r,, ,,, s,, ,,, n,, ,,, + d,, p,, ,,, d,, ,,, n,, ,,, + s,, ,,, ,,, ; g,, r,, s,, n,, d,,) for the first part

(g, r, ,, s, n, ,, + d, p, ,, d, n, ,, + s, ,, ,, ; g, r, s, n, d, ) for second part

(grsn, + dpdn, + s,,) ; grsnd rsndp dpmgr for third part

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 are originalfor this like the one I had presented in my solo concert at the

Academy 2-3 years ago: A = (4x5) + (3x7) + (2x9); B= 5+7+9

(Ta… Ka… Ta… Ki… Ta… ) + ( Ta.. .. Di.. .. Ki.. Ta.. Tom..) + (Ta. .. Di. ..Ta. Di. Ki. Ta. Tom. ) as A

And (Tadikitatom Ta.di.kitatom Ta.di.Tadikitatom) as B

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 in 1200 AD butattributed to a much earlier Indian mathematician Pingala (450-200 BC). Theseries is any two initial numbers like 3, 4 which are added to get 7. Now, addthe 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 korvais

It now would be obvious that anyone can create seamless korvais with thethinking and methods I have shared.

I have used mostly familiar sounding easy patterns to create these, mainlywith melodic aesthetics in mind.

I have shown only a few small samples here, even from the ones I havediscovered/presented.

Pure rhythmic seamless korvais can deal with typical patterns suited forpercussion.

This is a vast exciting new world with tremendous scope to expand thehorizons both melodically and rhythmically.

Each door I’ve opened leads to exhilarating worlds…

Happy exploring!!!