As I often write about various aspects of South Indian music, I thought I would do a brief post summarizing how the various topics work together as one: “the” secret behind South Indian music (especially drumming) that reveals its particular genius. So what is the secret? The secret is that underlying South Indian music is a beautiful and often complex mathematical relationship between arithmetic, algebra, and geometry: three aspects of math that come together to produce a fascinating world of speech and rhythm. But before we begin we must know a few basics.
The Essential Principles of Rhythm in Karnatak (or Carnatic) music and South Indian dance are called the Dasa Prana, also known as the 10 vital elements of Karnatak music. These include various ways of indicating rhythm, subdivisions, tempo, classification, and the idea of “time” in general. Through these elements South Indians who study singing, dance, drums, and other instruments learn the essential foundation for expressing themselves.
This means that in musical terms, musicians and dancers learn how to utilize mathematics in three distinct ways: (1) arithmatic, expressed in the art of what is known as solkattu, (2) algebra: expressed in the art of what are known as jati, and (3) geometry: expressed in what are known as yati.
(Note: words like talam, jati, and yati are both singular and plural in the Tamil language, so I will being using them as such: “talam are…”, “a talam is…”.)
First, konakkol and solkattu. The art of studying rhythm through clapping and speaking syllables is known as konakkol: a technique used to teach students how to organize and feel rhythm through speech and hand movement in order to get a better sense of how time flows in South Indian music. Solkattu is one half of konakkol, the art of singing/speaking syllables.
Looking at the Hindu mandala above then, we notice that the flower petals are arranged in a circle. This is an excellent visual metaphor for how beats are visualized in South Indian music, the solkattu syllables being the petals. So having this image in your head will help you eventually see how the mathematics of solkattu, jati, and yati join together. A cycle of solkattu syllables (taka dimi taka jonu) is known as a talam : rhythmic cycles consisting of three beats or more… three syllables = three beats, four syllables = 4 beats, etc., each beat/syllable placed on an imaginary circle like positions on a clock.
TA KI DA
TA KA DI MI
TA DIN GI NA TOM
TA KA DI MI TA KI DA
Each of the talam have names to distinguish each from the other. For example, the most commonly studied south indian rhythm, known as Adi Talam is an eight syllable ergo eight beat cycle, spoken or sung as: “TA KA DI MI TA KA JO NU”. Khanda Eka Talam, for example, is a five-syllable (5 beat) talam spoken or sung as “TA DIN GI NA TOM”. These can be expressed at three speeds (slow, medium and fast), three speeds holding to a strict ratio of [1:2:4], each doubling or halving in speed, essentially creating precise multiplications and divisions of a talam. Imagine a car shifting gears that automatically increase to double the speed. For example, an eight or a seven beat cycle becomes:
TA KA DI MI TA KA JO NU
TAKA DIMI TAKA JONU
takadimi takajonu takadimi takajonu
TA KA DI MI TA KI DA
TAKA DIMI TAKIDA
Having looked at the arithmetic aspect of South Indian drumming (addition and subtraction), we can now look at the second aspect of its genius, its algebraic aspect. Algebra, the study of numbers and mathematical symbols together, plays a particularly fascinating part of Karnatak music many may not be aware of. This is best explained as how groupings of syllables can be assigned to variables like X or Y. These groupings of syllables or beats are known as jati. Therefore a particular jati has a particular number of syllables. For example, “Chatusra” Jati has four syllables, “Tisra” Jati has three, “Misra” Jati has seven, etc. So when we sing an eight beat talam like Adi Talam: TA KA DI MI TAKA JONU, we are singing a talam with two Chatusra jati in it.
To really see the algebra at work, we can look at a fourteen beat cycle called Dhruva Talam. Dhruva is made up of a grouping of 4 beats + 2 beats + 4 beats + 4 more beats, the basis of this talam being the number 4 (Chatusra jati). Expressed as algebra this can be written as X + 2 + X + X. What makes this so interesting is that you can change the jati variable (X) without changing the structure. Thus, you have great creative freedom while also holding to a strict skeletal formula (which is a lot to process in the mind of the drummer). Dhruva Talam, based on Chatusra jati, changes from a fourteen beat cycle… to an eleven beat cycle in Tisra Jati (groupings of three) as the variable: 4 + 2 + 4 + 4 now becomes 3 + 2 + 3 + 3.
TAKADIMI TAKA TAKADIMI TAKAJONU… which becomes:
TAKIDA TAKA TAKIDA TAKIDA
Taking this a step further, Dhruva Talam [X + 2 + X + X] can now become a twenty three beat cycle if we use groupings of seven (Misra jati) as the basis for our talam: 7 + 2 + 7 + 7. If we look at other talam, we see this occurring across the board. For example, the talam called Matthya can be expressed as X+Y+X, so Matthya is a ten beat cycle using Chatusra (4 + 2 + 4), an eight beat cycle using Trisra (3 + 2 + 3), a sixteen beat cycle in Misra (7 + 2 + 7) and so on.
The third and possibly most interesting mathematical aspect of South Indian drumming is expressed in the form of geometry, the art of yati, which are the expansion and contraction of both solkattu and jati, simultaneously, which can be mentally visualized as various shapes: a triangle, upside-down triangle, hour glass, diamond, and so on. Yati are the addition and subtraction of both jati and solkattu syllables simultaneously. Expressed on paper, this tales to form of words creates underlying geometric shapes, for example, Gopuccha yati, upside down triangle shape as the Cow Tail yati
TA DIN GI NA TOM
DIN GI NA TOM
GI NA TOM
The yati known as Damaru, the “Hour Glass Drum” yati, looks like an hourglass, an upside down triangle on to of a regular triangle.
TA KA DI MI
TA KI TA
TA KI DA
TA KA DI MI
So if we return to the Hindu mandala I showed earlier (below), we can now see an excellent visual metaphor for how solkattu, jati, and yati can be conceptually combined into a unified whole, which gives those of us without extensive experience in Karnatak music an effective way of understanding and appreciating this art form.
Thus South Indian musicians create and convey meaning in music through arthimetic, algebraic, and geometric strategies that shape speech, dance, and rhythm in a manner unlike any other in the world. Thus, as essential participants in the art of Karnatak music, these musicians have been preservers of vital intangible cultural properties… an ancient legacy of creative wisdom that has vitalized India, and indeed world heritage, as it has been taught and performed all over the world.