Mathematics in Music is fascinating. The world of Korvais are intriguing and interesting. A short but extremely insightful introduction into this world has been shared in this presentation.
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). Ex 2: 7+2…0+2 as A
and B is 5x3 in tishra gati. (A can also be in
srotovaha yati)
2.Progressive with addition in multiple parts:
A= (2,3,4)+(2,3,4,5)+(2,3,4,5,6); B=7x3.
6. Seamless korvais – amazing options
3. 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 times
with 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
7. 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 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
8. 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.
9. 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
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)
10. 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 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
2. 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);
B = [(5+2), (4+2), (3+2), (2+2)] + (3x7), C = [(5+2), (4+2),
(3+2), (2+2), (1+2)] + (3x9).
11. 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).
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
12. Keyless korvais extensions to other talas
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).
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)
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.
13. 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.
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.
14. 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
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
15. 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
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
16. 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
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
17. 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.
18. 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!!!
