Advanced Mathematics 5: Introduction to Karnatic Rhythms

In today’s class, we’ll move out of the world we’re used to–the Western music world, that is–and move to South India, around the region of Karnāṭaka. We’ll have to learn much of their musical vocabulary, although we’ll try to draw parallels with European classical music as much as possible, since I assume we are all more familiar with it. But, even when parallels are drawn, the two terms are never going to mean exactly the same thing, so it’s important to keep in mind the worlds that separate Western music from Karnatic music.

This video has been the instigator of my interest with Karnatic rhythmic theory, from the amazing Youtube channel MadRasana. Here, V. Shivapriya and B. R. Somashekar Jois perform what is known as a Konnakkol, or Solkattu, a vocal piece that is meant to mimic the sounds from a drum known as mridangam. Their use of varied and complex polyrhythms over an odd time signature, and themes or motives that are never played twice exactly the same way are all hallmarks of the Karnatic music system. Surely, their theory can help us improve, as musicians, and adapting some of their concepts to Western music could make for new and exciting musical experiences.

In my further researches on the topic, I stumbled upon exactly what I was looking for! Rafael Reina published, as his PhD thesis for Brunel University, Karnatic Rhythmical Structures as a Source for New Thinking in Western Music, which you can download for free by clicking this link–or buy his book, Applying Karnatic Rhythmical Techniques to Western Music–, which will serve as the basis for this post that I write. If, after you finish reading this article, you wish to learn more about this system, please go ahead and download the thesis or buy the book, as I don’t intend to cover the whole of Karnatic rhythmic theory exhaustively here; my goal is only to arouse enough curiosity in you that you may learn more by yourselves, if you so desire. I will still try to lay the foundations of Karnatic rhythm, so that it can be useful for any musician, even if you don’t dig deeper into the subject.


Karnatic Metre

As I’ve mentioned before, Karnatic music has a lot of parallels with Western music. First, they use some sort of metre, as we do with measures. Their measure is called a tāla. To avoid drowning into new and foreign vocabulary, I will refer to the terms with their Western music equivalent, but keep in mind that it is always going to behave slightly differently in the context of Karnatic music. For example, the tāla may be vaguely the same as a measure, but it is drastically different in how it is constructed. You can build a measure any way you like, but strict building rules must be applied in its Karnatic music incarnation. Therefore, even though you could hypothetically craft a measure of any number of beats, Karnatic theory forbids some numbers from occurring. For example, you’ll never see a fifteen-beat tāla, because the construction rules of the Karnatic system means that no combination of valid building blocks will be able to give fifteen beats.

Perhaps most importantly, a time signature never changes during a piece (except for a select few musical styles). Compare this to the highly sought after changes in time signatures that are idolized in progressive subgenres of music, and you see just how antipodal the two worlds are. So, how can Karnatic music be of interest to a prog-minded musician? The interesting point lies in how the musicians will play with and against the set metre–the measure and its backbone, the way the beats in it are divided–, using various polyrhythms, accentuations, and phrasings, which I will introduce in this article.

Suladi is one of the most common Karnatic measure, and their construction rules state that they must be made using three building blocks. These building blocks are called angas, and they can be of different sizes. They too have various rules that are well detailed in Reina’s documents. But, for the sake of today’s article, I’ll take the metre Shivapriya and Somashekar Jois use in the video I’ve included at the start of this post.

With a Westerner’s musical eye, this time signature is instantly recognized as \(\frac{7}{8}\), divided in beats of 1, 2, 2, and 2 eighth notes. This is the opposite of most \(\frac{7}{8}\) bars we hear in prog music, which display a 2, 2, 2, and 1 beat subdivision. However, to the Karnatic musician’s eye, this is the Misra Chapu tāla, a fourteen-beat metre using beats of 6, 4, and 4. The Chapu tālas are borrowed from folk music, and they are a special case in Karnatic music, since they don’t use the same building blocks as other tālas and are played at a higher tempo. Both musicians’ points of view are valid, of course, but their words will carry different meanings, and different backgrounds. The Westerner might wonder why the time signature never changes, while the Karnatic musician knows that it never changes within a piece, but both can communicate and play the same thing nonetheless.

So, what is important to remember, as a Western musician, and to learn from Karnatic metre? While not all numbers of beats are possible within Karnatic music theory, there is no reason for the Westerner to abandon the \(\frac{15}{8}\) or \(\frac{19}{16}\) time signatures. If you wish to create a genuine Karnatic piece, you will need to abide by the rules of their system, but for a Western piece with Karnatic music influences, you could just decide to implement other Karnatic concepts, which we’ll discuss later in this article. A measure in Karnatic music never changes throughout a piece. Once again, there is no reason for the Western musician to stick to these strict rules, but it could prove to be an exciting challenge in creative restriction to do so, and it is, after all, the soul of Karnatic music to play around and against a set beat. So, if you wish to include Karnatic music in your compositional or creative process, I highly recommend you stick to one time signature and beat accentuation for an extended period of time. We will see now some concepts of Karnatic music, and you’ll see why this might be important.


Beat Subdivision

Right now, we’ve only talked about the structure of the measure, the unchanging clock, subjacent in most Karnatic music pieces. Even though there wasn’t much to take home for the Westerner, except the creative restriction arising from using one single time signature, the following chapters will be much more interesting. It was necessary, however, to set the stage with solid foundations before going on in more complex talks.

Now, let’s talk about how Karnatic musicians treat a beat. In Western music, every beat is often left on its own (a quarter note), or subdivided using powers of two: the eighth, sixteenth, thirty-second, sixty-fourth, one-hundred-and-twenty-eighth notes, and so on. It’s also possible to use tuplets to subdivide a beat (read our class on polyrhythms and polymetres to catch up on the topic), thus playing three, five, seven, or any other number of notes in a beat. Karnatic music favours uneven beat subdivisions, as one of its goals is to create complex and interesting rhythmic patterns and structures.

These beat subdivisions are called gatis, and each unit of a gati, or note of a tuplet, is called a matra. Moreover, each gati has a different name, but, hopefully, only represents five different beat subdivisions. The names of the different gatis are not important for today’s lesson, as they can be almost perfectly referenced using Western nomenclature. However, if you wish to communicate to Karnatic musicians, please use the right vocabulary, as can be found in Rafel Reina’s thesis and book, and other sources online.

Therefore, the subdivisions of a beat include the triplet (three notes per beat), the quadruplet (four notes per beat), the quintuplet (five), septuplet (seven), and nine-tuplet. Subdivisions of one, two, six, and eight beats are also possible, but are considered as one of these tuplets played in a different speed. Therefore, to have two notes in a beat, you would play quadruplets at half speed; need sextuplets? play triplets at second speed.

As I’ve mentioned earlier, Karnatic music is a vocal practice that mimics the mridangam drum sounds. The Solkattu, or Konnakkol, is the vocal portion of Karnatic music, and it includes a list of syllables to use and how to use them with the different gatis. However, I’ll skip this portion entirely in this article, as I’m more interested in the rhythmic theory that can be applied to Western music instruments than played in a genuine Karnatic manner. Once again, please refer to Reina’s material for a complete and thorough look into all the aspects that I will be overlooking today.

Gati Phrasing

Just as in Western music, every tuplet can be played any number of ways. You can, for example, subdivide a beat in four sixteenth notes, but only play the first, third, and fourth of them, therefore giving the impression that you’ve played one eighth and two sixteenth notes. We might not see it like this intuitively, as Western musicians, but Karnatic music works like this: divide the beat in the required number of notes, and then play all or some of them. The important difference here is that the subdivision is still strongly felt, each note being felt by the musician, even if it is not played. As you can see with the below example, there are seven ways of playing quadruplets. That number increases dramatically to sixty when playing with septuplets.

The different possible ways of playing the Chatusra gati (quadruplets), taken from Rafael Reina’s Karnatic Rhythmical Structures as a Source for New Thinking in Western Music.

One of the inherently different ways musicians are brought up in the two contexts is that Karnatic musicians will learn all these subdivisions and the different ways of playing them first, while a Western musician will start with the most useful ones, or the most common, or the easiest ones before moving on to the next, often times never learning all possible ways of playing all different beat subdivisions in a lifetime, either out of complacency, laziness, or uselessness. The Karnatic musician’s way of learning rhythm gives them strong confidence and a wide phrasing vocabulary that is readily accessible and useful in many different contexts. This is one thing that should be imported to Western musicians, in my opinion, as it could greatly improve the common rhythmic vocabulary of our music.

Once a certain phrasing is achieved, it can never be repeated exactly as it is in the piece. If it is brought back, it must be with something new each time. For example a different development with rests or tied notes, sometimes completely disfiguring the theme in the process, even though it is based on it. Moreover, in Karnatic music, a certain beat subdivision must be used for an entire measure, or tāla. It would be impossible, for example, to play one beat of the bar in triplets, and the next in quintuplets. Tuplet changes must occur only between two different measures. As with some things mentioned above, however, I think this is a rule that is not necessary to import to Western music, unless you plan to write a genuine Karnatic piece. It could be really interesting to use different tuplets in a measure, following most other rules of Karnatic music rhythm.

For this chapter, it’s important to take home the concept of knowing many tuplets in and out. Practice with a metronome playing subdivisions of a beat using triplets, quintuplets, septuplets, and quadruplets, and practice even further by changing from one to another, and playing only some notes, not all, while still keeping in mind the pulse of the tuplet you’re playing. This should give any Western musician a stronger grasp on rhythm and a wider vocabulary that can then be applied to any and all musical context. It’s of utmost importance to play different tuplets one after the other on the same metronome pulse. This way, you will develop a strong feeling of how triplets relate to quintuplets, for example, and how any tuplet relates to any other one in terms of note length. This is an arduous process, but it’s important to be rigorous and thorough.


Accentuations

How you accentuate a phrasing, in Karnatic music, is called jathi. One way of seeing this concept for the Westerner is as polymetres. A simple example is when playing sixteenth notes (quadruplets), but accentuating every three notes, so as to give the impression of a 3:4 polymetre, which will then give a 3:4 polyrhythm feel. This concept is of high importance in Karnatic music, and all students and musicians practice every aforementioned gati using every possible jathi (accentuation): 3, 4, 5, and 7.

Now, we get to see some possibilities for multiple musicians, or voices in a composition. An example of this would be two musicians playing triplets, but one is accentuating every four notes. This is more of a polymetric approach, where the notes of the two musicians are of same length, but they are divided in a different number of notes before accentuation. This is like playing \(\frac{3}{16}\) over \(\frac{4}{16}\).

However, two voices could also play the accentuations at the same time, but use different beat subdivisions. This is the polyrhythmic approach, where the notes of one voice are not of the same length as the notes of the other, but their accentuations happen simultaneously. For a simple example, this is like playing triplets on one hand, and quadruplets on the other, giving off a 3:4 polyrhythm.

Then, there is also the option of keeping the same jathi on different tuplets. Here, the same jathi doesn’t mean that the accentuations will happen at the same time, but that they happen after the same number of notes. The trick here is that the notes of different tuplets will have different durations, therefore the accentuations will be delayed with one another. This will give off a polytempo feel, which is a bit like a mix of polyrhythm and polymetre.

If you’re struggling to keep up with the terms, please go back to Advanced Mathematics 103, where I explain a lot about polyrhythms, polymetres, and polytempo.

Finally, you can use any gati, or tuplet feel, with any jathi, or accentuation type, but they always have to resolve on the 1 beat, the first beat of a measure, also known as tāla sam. The polyrhythm, polymetre, or polytempo device can, however, start anywhere in a measure.

Unlike Western music, Karnatic music doesn’t especially relish the idea of two voices that take a long time to resolve. Being more in favour of rapid and constant change, they will use shorter resolutions, and a wider range of them. I’d like to repeat, however, that it is perfectly fine to use polyrhythms that take longer to resolve in Western music.

It’s important to practice different tuplets with different accentuations precisely and extensively to really engrave these rhythmic devices in your brain. It’s also important to have a firm grasp of a wide variety of them; the more you learn, the more you will be likely to use commonly with ease.


Change through Destruction

Bhedam means “change through destruction”. It is often applied to other musical terms like those mentioned above. It is where common Western notation fails to grasp the entirety of the picture painted by Karnatic music. However, as stated earlier, it is to be expected. I’ve used Western terms to refer to Karnatic concepts during this lesson, so as to ease the approach to this completely foreign musical world only, and didn’t pretend to paint the full picture. To be fully immersed in Karnatic music, it would be necessary to study and play it for months, even years, learning from masters and dedicating oneself fully to this task.

One way to circumvent this irreconcilability, as per Rafael Reina, is to use the beaming of notes relative to the jathi, or accents, instead of the beat subdivision. The beat subdivision would be instead reported by tuplet groupings, above the notation.

A complex quintuplets phrasing with three-note accentuation using regular notation (upper row) and adapted to the jathi (lower row), taken from Rafael Reina’s Karnatic Rhythmical Structures as a Source for New Thinking in Western Music.

The beaming of notes according to the accentuation pattern can be really useful for creating music and making things more easily comprehensible and therefore playable by the musicians supposed to play it. One of the main goal of Karnatic music is to create polyrhythms and polymetres that give the impression of polytempo. This is achieved using many rhythmic concepts, some of which we’ve seen here today, and playing them precisely and consistently is primordial to achieve this goal. Therefore, using a more comprehensible notation, if a bit out of the norm by Western classical standards, can be very helpful.

As with the above example, gati bhedam is the destruction of the gati based on the jathi. The notes are therefore not beamed according to the beat pulse, but rather to the accentuation pattern. Jathi bhedam is the destruction of the accentuation pattern by changing it constantly.

As with most things in Karnatic music, jathi bhedam is subject to many rules. Simply put, once you’ve found a tuplet feel you wish to use, and a number of beats before resolving the pattern, you must choose different numbers that will add up to the same number as the multiplication of the tuplet feel and the number of beats used. Once these numbers are found, you should construct accentuation patterns so that no accent should fall on a beat (until at least the fourth beat), no two consecutive beats are accentuated, the same number doesn’t repeat more than three times consecutively, and that only the resolving tāla sam (first beat) is accentuated. This is quite complex, indeed, but the aim is to fool the listener into thinking a part of the piece is in a different rhythm than it actually is. It is therefore imperative to use complex rhythmic devices to achieve this effect.

Between the accents, you can add as many or as few notes as you wish, as long as the chosen accents are correctly accentuated according to the Karnatic rules. For use in Western music, know that bending or breaking some of these rules would defeat the purpose of this compositional tool, so it’s best to observe them strictly.


Speed Change

I slightly touched on speed changed when I explained the different tuplets of Karnatic music. However, this goes a lot deeper. The concept of speed can be divided into two terms: Anuloma and Pratiloma.

Each of the two terms have different speeds, but they have opposite effects. Let’s take it as a ratio of A:B, A being the number of notes to be played, and B the number of beats in which these notes should be played. A regular triplet will have a ratio of 3:1–three notes in one beat–, while a quadruplet, or four sixteenth notes, will have a 4:1 ratio.

Whenever you apply a speed in Anuloma, you multiply the number A by the speed. For example, a piece in triplets, in Anuloma second speed–multiply 3 by 2–will mean that you will play a 6:1 ratio, or sextuplets; six notes in a beat. In Anuloma third speed, it’s 9:1, and fourth speed, 12:1. This is very different than the dyadic format found in Western music, where a faster speed could be seen as a doubling of notes with powers of 2. That also means that quadruplets in Anuloma third speed will feel like very fast triplets to the Westerner, even though they are inherently different.

Pratiloma is the opposite of Anuloma. When finding a speed in Pratiloma, you will multiply the B number in the A:B ratio by the Pratiloma speed. That means that you will play the same number of notes, but in a different number of beats. If you play triplets in Pratiloma second speed, you will then play a 3:2 ratio: three notes in the span of two beats. Triplets in Pratiloma fourth speed will be three notes in four beats, also known as a triplet of half notes in \(\frac{4}{4}\).

It is also foreseeable to combine both speed concepts and create new ratios, especially in the context of motif development (more on that later). For example, let’s take a 3:1 phrase, in regular triplets. Let’s say we wish to develop it with Pratiloma second speed, giving us a 3:2 pattern: three notes in two beats. For further development, we may want to play that with Anuloma third speed, thus giving way to a 9:2 notes-beats ratio. You could bring this back to the original phrasing by developing it with Pratiloma third speed–giving 9:6–, and then deciding to play only every three notes, giving the illusion of 3:1. Intrinsically, however, the end result will be different than the starting block, since phrasings and syllables will have to be stretched, multiplied, and cut, but based on the notes alone, the rhythm is similar.


Development

I’ll briefly touch on motive development in Karnatic music, but you can find more in-depth information in Reina’s thesis or book, and in many other places online.

Like mentioned before, it’s impossible to bring back a motif that has been played already exactly as it was. Many developmental techniques are available, but most are based on the concept of having a common denominator. This term can refer to many things. For example, it could be keeping the same tuplet feel, but changing the accentuation pattern, or vice-versa. It could also be to keep the same number of played notes between accents.

One way to see it is like branches of a tree. There is the main stem of the tree, the primordial theme or motif of the piece. By following it, you see that it branches off–something needs to change in it–, it needs to be slightly modified before continuing with it. Then, another smaller branch arrives, and it still needs to be developed in order to go further. This process can keep going indefinitely, as long as you have ideas of connecting the theme to another version of itself using any common denominator.


Conclusion

I’ve only brushed the surface of the deep and complex world of Karnatic music. I wanted to wake in the reader a sense of curiosity that might entice them to read and learn on their own via readily accessible documentation or by taking classes with Karnatic music teachers. Links to Reina’s thesis and book will be added after this text. I believe that Karnatic music rhythm theory can help any Western musician improve by challenging their rhythmic mastery and enhancing their vocabulary. It would be very interesting to hear any amount of Karnatic theory inside of Western music, regardless of the amount of rules broken, since it is not a genuine Karnatic piece to begin with, as long as the breaking the rules don’t defeat the purpose of the concept.

Hopefully, you’ve learned a thing or two today, and want to learn more about the beautiful and diverse world of Karnatic music. I’ve only written about the rhythm side of their theory, but Karnatic music is a multifaceted musical dimension with theories dedicated to the use of pitch, modes, and improvisation that all are interesting on their own.


Links

Rafael Reina’s Karnatic Rhythmical Structures as a Source for New Thinking in Western Music
Rafael Reina’s Applying Karnatic Rhythmical Techniques to Western Music


Advanced Mathematics

101: Morse Code
102: Microrhythms
103: Polyamory
104: Notes inégales

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