2. 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 the
way 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 virtuosity
but in its sophisticated approach towards structured
mathematical patterns.
3. Korvai Types
Level I: Any number taken after appropriate units after
samam to end as required. Ex: (3+3, 5+3, 7)x3 after 1. My
very first attempt at a korvai at age 5…
Level II: Taken from samam to end at samam with 1 or 2
karvais between patterns to fill out the remaining units (in
say 32/64/28/40 units in Adi 1/2 kalais, Mishra
chapu/Khanda chapu etc)
Level III: Same as II but to end a few units after or before
samam.
Level IV: Same as I or II but with different gatis thrown in.
All these can be termed as man-made korvais
4. Natural Korvais – seamless elegance
Seamless korvai – DEFINITION: Patterns (of usually two or more parts) from samam to
samam/landing point of song that do not have remainder indivisible by 3 in talas or landings
indivisible by 3.
In other words, these do not have remainder of any number of units not divisible by 3 (like 2
or 4) which have to be patched up as 1 or 2 karvais it in between patterns. These have a grace
or 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 text
books.
I literally stumbled upon most them as some of them are not accessible through intuitive
methods.
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 will
make it a different concept.)
5. 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 is
divisible 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 second
part (B).
(srgmpdn s,, rgmpdn s,, gmpdn s,, mpdn s,, pdn s,,
dn s,, ns,,) as A and (grsnd rsndp dpmgr) as B
6. 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,) as
A 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,) as
A and (g,r,snd r,s,ndp d,p,mgr) as B
7. 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 B
and (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)
8. 4. Progressive in second part: A= 6, 6, 6; B = (3x9) + (2x7) + (1x5). Impressive when B is
rendered 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
9. 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
10. 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
11. 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,, - 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 times 5) in various
ways. If song starts after +6, various manifestations of 15+5, 15+5 and 15+(3
times 5) can be created.
12. 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 make
sense to most mathematicians. But they are fine examples of how Carnatic
music can transcend science and math. Remarkably, the series will turn back
on itself. I call these Double layered progressive sequences which
boomerang. The first few numbers are formed using multiplication
progression in (a) are: 7x1, 6x2, 5x3, 4x4. Thus, the next few numbers are
15, 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.kitatom
Ta.di.kitatom Ta.di.kitatom ) as B
13. • 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
14. The concept of Keyless korvais
At times, one stumbles upon korvais with no apparent mathematical relationship.
These cannot be logically deciphered or developed by locking on to their key (usually
the average of their various parts/2nd repeat out of 3). Yet, these are elegant beyond
words in their simplicity.
1. A 3-part korvai over 2 cycles (128 units): A stunning set of patterns
found 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,rsndp
d,p,dpmgr) as C
15. 2. A 3-part Korvai in 3 speeds: The amazing aesthetics of
this is mind-boggling – simple when rendered but looks a
jungle of numbers when expressed as below!
A = (8+3)x3 + (1x5)x3
B = (6+3)x2 + (2x7) x 2
C = (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
16. Keyless korvais extensions to other talas
Keyless methods give scope to execute amazing finishes in seemingly impossible
situations. For instance, a tala like Khanda Triputa @ 8 units per beat (72 units) or
Rupakam, 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.] + (Tadikitatom
Tadikitatom 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
17. 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 DPD
N,, - GRSND in a raga like Vachaspati)
B = (9+3)x2 + (2x7) x 2
C = (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 change
B 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 )
18. Keyless korvais extensions to other talas
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 Shri
matrubhootam)
(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 )
19. 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 as
3x9, 3x13, 3x17 etc.
20. Seamless korvais for other talas
ADI 1 kalai (32 units)
Most korvais in this smaller space require patch work. Some of 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.
(Tam. TaTam. TakaTam. TakiTaTam.) – A
(Tadi.kitatom Tadi.kitatom Tadi.kitatom ) - B
21. 2.Single part apparently wrong but actually correct
korvai:
GR,-GRS,-GRSN,-GRSNP,-GRSNPG,-GRSNPGR
Typical hearing will make it seem like 1+2 karvais… 5+2
karvais and final phrase illogically being 7. In reality, it
is 2+1, 3+1…6+1 ending in 7.
22. Seamless korvais for other talas… contd
ADI 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
23. 4. Several other progressive solutions work beautifully for 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 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
24. Seamless korvais for other talas… contd
ADI 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
25. 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
26. Seamless korvais in other gatis
Just 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 original
for 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
28. Seamless korvais with other approaches
I 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 but
attributed to a much earlier Indian mathematician Pingala (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.
29. Creating Seamless korvais
It 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 create these, mainly
with melodic aesthetics in mind.
I have shown only a few small samples here, even from the ones I have
discovered/presented.
Pure rhythmic seamless korvais can deal with typical patterns suited for
percussion.
This is a vast exciting new world with tremendous scope to expand the
horizons both melodically and rhythmically.
Each door I’ve opened leads to exhilarating worlds…
Happy exploring!!!
