Earlier today, I had a little chat with composer Stace Constantinou on Twitter. He was exploring various permutations of a series of notes; a technique used in serial composition. It got me thinking about how tabla players use permutations quite a lot in our improvisations. He suggested I write a blog post on the subject, and so, here we are! (I'll have to write a blog post on how we actually structure our kaida-palta improvisations, which is a theme-and-variation form; I'll save that for another time.)
In playing tabla, it often happens that a pattern emerges in the bols. For example: Dha Te Te Dha Te Te Dha Dha. When this happens, a eureka moment occurs - the possibility of permutations being immediately apparent. The permutations in this instance would be:
Dha Te Te Dha Te Te Dha Dha (the original)
Dha Te Te Dha Dha Dha Te Te
Dha Dha Dha Te Te Dha Te Te
This is a simple example, and without the use of any complex formulas, we can see that there are three possibilities. If DhaTeTe = A, and DhaDha = B, then the three possibilities are:
AAB (our original version)
ABA
BAA
(Of course, the DhaDha motive could be split into two separate Dha sounds, and the TeTe could be split as well, but that would not be an idiomatic approach to permutations, as far as the tabla is concerned.)
The next example which pops up often is AAAAB, for example: Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa TaKa, or written another way: (Dha TiRa KiTa) x 4 (Dha TiRa KiTa TaKa). That's four groups of three, and one group of four. Here are the possible permutations:
AAAAB (the original)
AAABA
AABAA
ABAAA
BAAAA
Translating into tabla bols:
Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa TaKa (the original)
Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa TaKa Dha TiRa KiTa
Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa TaKa Dha TiRa KiTa Dha TiRa KiTa
Dha TiRa KiTa Dha TiRa KiTa TaKa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa
Dha TiRa KiTa TaKa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa
For most, it's probably obvious that there would be five possible permutations of this pattern, however if you weren't sure, you could use this mathematical formula, which I learned from my senior guru-bhai, Bob Becker, of the renowned percussion ensemble, Nexus. (If you are curious about rhythm and mathematics on a high level, have a look at his book, Rudimental Arithmetic.)
There are five items in all, so we must calculate 5! (5 factorial), which means 1 x 2 x 3 x 4 x 5, which gives us a large number of possibilities (120). However, there are four items which are identical (Dha TiRa KiTa), and so we need to eliminate those permutations, as they would sound the same as others, so we divide by 4! (1 x 2 x 3 x 4), leaving us with 5. Amazing, eh? Here's the math:
5! / 4! = (1 x 2 x 3 x 4 x 5) / (1 x 2 x 3 x 4) = 5
Another example that pops up often is AAAABB, for example: Ta Ki Ta Ta Ki Ta Ta Ki Ta Ta Ki Ta Tin Na Tin Na, or (Ta Ki Ta) x 4 and (Tin Na) x 2. How many possible permutations now? It's impossible to guesstimate; we need to use the formula.
There are 6 items.
4 are identical (Ta Ki Ta)
2 are identical (Tin Na)
Therefore:
6! / (4! x 2!) = (1 x 2 x 3 x 4 x 5 x 6) / (1 x 2 x 3 x 4 x 1 x 2) = 15
15 possibilities, wow! I'm not going to write them all out for you. Let's leave this as homework for you to figure out on your own. The five "simple" permutations will be the same as above, when we had only five possible permutations. Those would be:
AAAABB (the original)
AAABBA
AABBAA
ABBAAA
BBAAAA
There are ten more. Have fun finding them!
Now, in performance, we would never play through ALL permutations, as it would soon become predictable and boring. However, as a practice tool, it's invaluable. First, from the technical standpoint, to have practiced all possible permutations, means you will be technically prepared to play any of them. Secondly, playing through all the possible permutations might help you discover certain rhythms and combinations that you may not have otherwise discovered. Thirdly, after having practiced permutations enough, they become intuitive, meaning that we can leave the math behind and let our musical intuition and creativity guide the way.
These permutation techniques can also be valuable to the practice and performance of other instruments, whether applied to scales, melodies, rudiments, or any musical phrases. I hope that this article helps lead to some interesting musical explorations for you!
In playing tabla, it often happens that a pattern emerges in the bols. For example: Dha Te Te Dha Te Te Dha Dha. When this happens, a eureka moment occurs - the possibility of permutations being immediately apparent. The permutations in this instance would be:
Dha Te Te Dha Te Te Dha Dha (the original)
Dha Te Te Dha Dha Dha Te Te
Dha Dha Dha Te Te Dha Te Te
This is a simple example, and without the use of any complex formulas, we can see that there are three possibilities. If DhaTeTe = A, and DhaDha = B, then the three possibilities are:
AAB (our original version)
ABA
BAA
(Of course, the DhaDha motive could be split into two separate Dha sounds, and the TeTe could be split as well, but that would not be an idiomatic approach to permutations, as far as the tabla is concerned.)
The next example which pops up often is AAAAB, for example: Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa TaKa, or written another way: (Dha TiRa KiTa) x 4 (Dha TiRa KiTa TaKa). That's four groups of three, and one group of four. Here are the possible permutations:
AAAAB (the original)
AAABA
AABAA
ABAAA
BAAAA
Translating into tabla bols:
Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa TaKa (the original)
Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa TaKa Dha TiRa KiTa
Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa TaKa Dha TiRa KiTa Dha TiRa KiTa
Dha TiRa KiTa Dha TiRa KiTa TaKa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa
Dha TiRa KiTa TaKa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa Dha TiRa KiTa
For most, it's probably obvious that there would be five possible permutations of this pattern, however if you weren't sure, you could use this mathematical formula, which I learned from my senior guru-bhai, Bob Becker, of the renowned percussion ensemble, Nexus. (If you are curious about rhythm and mathematics on a high level, have a look at his book, Rudimental Arithmetic.)
There are five items in all, so we must calculate 5! (5 factorial), which means 1 x 2 x 3 x 4 x 5, which gives us a large number of possibilities (120). However, there are four items which are identical (Dha TiRa KiTa), and so we need to eliminate those permutations, as they would sound the same as others, so we divide by 4! (1 x 2 x 3 x 4), leaving us with 5. Amazing, eh? Here's the math:
5! / 4! = (1 x 2 x 3 x 4 x 5) / (1 x 2 x 3 x 4) = 5
Another example that pops up often is AAAABB, for example: Ta Ki Ta Ta Ki Ta Ta Ki Ta Ta Ki Ta Tin Na Tin Na, or (Ta Ki Ta) x 4 and (Tin Na) x 2. How many possible permutations now? It's impossible to guesstimate; we need to use the formula.
There are 6 items.
4 are identical (Ta Ki Ta)
2 are identical (Tin Na)
Therefore:
6! / (4! x 2!) = (1 x 2 x 3 x 4 x 5 x 6) / (1 x 2 x 3 x 4 x 1 x 2) = 15
15 possibilities, wow! I'm not going to write them all out for you. Let's leave this as homework for you to figure out on your own. The five "simple" permutations will be the same as above, when we had only five possible permutations. Those would be:
AAAABB (the original)
AAABBA
AABBAA
ABBAAA
BBAAAA
There are ten more. Have fun finding them!
Now, in performance, we would never play through ALL permutations, as it would soon become predictable and boring. However, as a practice tool, it's invaluable. First, from the technical standpoint, to have practiced all possible permutations, means you will be technically prepared to play any of them. Secondly, playing through all the possible permutations might help you discover certain rhythms and combinations that you may not have otherwise discovered. Thirdly, after having practiced permutations enough, they become intuitive, meaning that we can leave the math behind and let our musical intuition and creativity guide the way.
These permutation techniques can also be valuable to the practice and performance of other instruments, whether applied to scales, melodies, rudiments, or any musical phrases. I hope that this article helps lead to some interesting musical explorations for you!