Understanding the srutis

Interview with musicologist Dr K Varadarangan

Dr K Varadarangan, author or two books on microtones, talks about his 'tryst with truth' that could help settle an ancient debate about the octave, and validate the greatness of ancient Indian musicology


Dr K Varadarangan, musician, musicologist and mathematician, has arrived at conclusions that could settle a debate that has engaged the minds of music theorists for centuries.

The excited Bangalorean says he has come "face to face with the truth", and is confident he can once for all explain why Indian musicology believes in the idea that a saptaka (octave in Western music) contains 22 srutis (or microtones). Western musicology divides an octave into 12 tones.

The issues Varadarangan speaks about in this interview call for some acquaintace with music theory, but he also answers questions that lovers of popular music would be curious about -- like whether his discovery could give rise to a new generation of synthesizers with 22 keys in an octave instead of the present 12.

Varadarangan hails from a family of writers. Pu Thi Narasimhachar, his uncle, wrote several operas in Kannada, and was one of the 20th century's big poets in the traditional (as opposed to the modernist) style.

This is what Varadarangan said in an e-mail interview:

You say you have achieve a breakthrough that settles an age-old debate about the number of srutis in an octave. Indian musicology had arrived at 22, and you now say you have mathematical evidence to prove this right. How did you arrive at this number?

The concept of 22 srutis has existed for a long time in Indian musicology. Ancient musicologists, who had deep insight, used special instruments like the dhruva (fixed) veena and chala (movable) veena to arrive at this number. It is difficult, because of their subtlety, to show the existence of these microtones, and this has perhaps led to conflicting observations about the number of srutis within an octave.

The subjectivity vanishes once we formulate the srutis mathematically, when every sruti is quantified precisely. This is exactly what I have done. In my earlier work I had derived the relative frequencies of the 12 swarasthanas by using traditional srutibedha formulas found in our musical literature and known very well to our musicians. I have now prepared a srutibedha chart for these 12 swaras. It is a comprehensive chart that takes each of these 12 swaras as a moorchana swara, and tabulates the new relative frequencies that result from a shift of tonic. For example, suddha madhyama sounds exactly like sadharana gandhara when the tonic is changed to chatusruti rishabha.

When we look at my chart, we find that exactly 11 new frequencies result in addition to the existing 11 srutis (excluding the adhara shadja), thus making it exactly 22 sruthis, no more, no less! For example, anthara gandhara sounds approximately like suddha rishabha when sadharana gandhara is made the tonic. In fact, this approximate sruti is actually a new sruti, which is higher than suddha rishabha by a pramana sruti (also known as a Pythagorian comma).

In this way new srutis result. Note that our musicologists always interpreted sruti as an interval. We should not consider adhara shadja as a sruti by itself. This observation that the additional 11 srutis are embedded in the already existing 11 srutis and reveal themselves when there is a modal shift of tonic, is the essence of my discovery.

As I have shown in my earlier work (readers may kindly refer to my paper Determination of the Relative Frequencies and Fundamental Properties of Swaras by Modal Shift of Tonic published in the Journal of Music Academy, Madras, Vol LXXI, Year 2000), the interval between adjacent swaras in our music scale is not a constant. It is either 90 cents or 114 cents and the difference of 24 cents between these two is known as a 'pramana sruti'. The 11 additional frequencies mentioned above result from the "modulation" of the existing 11 sruthis by the pramana sruti. The 22 srutis computed above tally perfectly with those computed from the cycle of fourths and the cycle of fifths.

They say many previous musicologists like Prof Sambamurthy had "adjusted" a few frequencies to arrive at 22. You now feel this is erroneous and that there is no need to "adjust" anything. Can you explain, for the benefit of non-mathematicians, what the debate is all about?

Prof Sambamurthy's work is remarkable in that he gave the rather vague subject of 22 srutis a clear mathematical foundation. However, there have been two errors in his depiction of the 22 srutis:

The value of pramana sruti is actually 24 cents rather than 22 cents. This error is negligible from a practical viewpoint but theoretically important (see for example, page 47, Vol V of his South Indian Music ). Also, the value of the nyuna sruti should be 66 cents rather than 70 cents. He gives the value of the purna sruti correctly as 90 cents. Since the pramana sruti interval was taken as 22 cents, he had to adjust the frequencies computed by the cycles of fourths and fifths by 2 cents (see page 43 of the above book). The 22 srutis computed by me agree perfectly with the cycle of fourths and fifths and there is no need to adjust anything.

Secondly, a sruti with a relative frequency of 678 cents (which is close to panchama, being lower by a pramana sruti) is omitted from the list of 22 srutis for some mysterious reasons (see page 40 of his book). This frequency comes both in my computation and in the cycle of fourths. He has probably omitted it to fit everything into a 22-sruti system that includes the adhara shadja.

As I have mentioned earlier, the 22 srutis are actually frequency intervals and hence adhara shadja should not be included in their reckoning.

What bearing do you think your discovery will have on the practice of music? Will it prompt musicians to reconsider the srutis they have been singing? And if that happens, will it change the nature of ragas from what we now know them as?

A clear understanding of the 22 srutis will enable us to sing and play the ragas better and with greater feeling. It will also help immensely in teaching our music to students. For example, we can tell the student that the so-called suddha madhyama of raga Begada is actually higher and he should use the next higher variety of madhyama (in the scale of 22 srutis), and so on.

I don't think that we need to reconsider any srutis that we have been singing. On the other hand this tells us what sruti we are using and we become more aware. My discovery may not change the nature of any raga, but may make it easier for us to teach the subtleties in ragas.

What does your study tell you about the history of the 22 srutis?

It is quite likely that many scholars have contributed to the evolution of the 22-sruti system and Bharata was probably the first to describe it systematically. When you look at the subtleties of this sruti system, it appears to me that all of it could not have been conceived by a single person in one shot.

Has any other musical culture arrived at 22? How is our division different from the Western view of the octave?

When we study the evolution of musical scales we do find that more than 12 notes have been conceived by other cultures as well (see, for example, Chapter 5 of the book Science and Music by Sir James Jeans). However, Western music developed on the lines of polyphony and hence an equal temperament scale became an absolute necessity. Once the scale became tempered, there were exactly 12 notes, each placed 100 cents apart. Whatever the key, the frequencies always match, producing harmony in a polyphonic situation. However, because of this equal temperament, the individual notes lost their melody to a noticeable extent. As I have said, the interval between adjacent srutis in our scale of 12 notes is not constant. It is either 90 cents or 114 cents. It is precisely this difference that gives rise to the additional 11 sruthis in our musical scale, totalling 22 srutis. In the equi-tempered scale the difference between adjacent notes is constant, and hence there is absolutely no scope for additional frequencies or srutis to be generated.

Talking of practical applications, do you think your breakthrough could lead to the designing of keyboards with 22 keys in an octave so that they can handle all the nuances of Indian music? The harmonium, the piano and the electronic keyboard, with 12 keys to an octave, cannot capture any of the microtones that Indian music is famous for.

In fact such experimental veenas have been constructed. I don't know whether it is practical to play such an instrument. The point, however, is that it is possible to play the 22 srutis even on an ordinary veena by appropriate pulling of strings. On a keyboard instrument this is not possible and we should have separate keys. But even here, it may not be possible to handle the nuances in ragas because of the instruments' inability to produce the characteristic gamaka or meend. In other words, the 22 srutis are necessary but not sufficient to make the intricate melody of ragas.

What is the next step in your research?

I would like to synthesize the srutis electronically to find out how exactly the 22 srutis sound and interpret the srutis we use in various ragas. I would also like to explore the concept of consonance in greater detail. I feel we have not yet fully understood what really makes a frequency a consonant one.

Do you plan to publish your work in book form? Your last book has caught the attention of serious musicologists all over the world, some of whom contacted you after reading your interview in 'The Music Magazine'.

Yes, definitely. I am already in the process of writing such a book. I must also thank The Music Magazine for giving wide publicity to my earlier work.

Do srutis change between instruments? In other words, would the gandhara of a south Indian veena be different from the same gandhara on the sitar or the nadaswaram or the flute? If yes, what happens during orchestral compositions and jugalbandis? Is there a mismatch of shrutis that our ears cannot discern, or have we learnt to ignore those differences?

I think this is quite possible. We tune our instruments purely by perception and the tuning is within the limit of judgment of the player of the instrument. I have heard many orchestras where I recognize some dissonance because of the perceptional differences of individual players. Also, the same raga (say Mohana/Bhoop) is played differently by Hindustani and Karnatak musicians. Dissonance in orchestral music may be avoided by carefully tuning our instruments to a reference frequency standard. This implies that we understand the 22 srutis correctly so that we can create a frequency standard of our own.

Tell us something about the Indian idea of consonance and dissonance -- the concepts of vadi, samvadi and vivadi -- and how it is different from the Western one.

The concept of consonance is fairly simple. Two notes are said to be consonant if they blend melodiously when sounded together. This concept is the same in Western music too. However, in Indian music this is further elaborated by the introduction of the vadi, samvadi, anuvadi and vivadi swaras. The vadi swara is a note which appears frequently in a raga and is a vital note in that raga. A samvadi swara (relative to a reference note) is essentially a consonant note. A samvadi note of the vadi swara is called an anuvadi swara. The vivadi swara is of course a dissonant note. Indian musicologists have clearly defined consonance and dissonance in terms of the differences in the srutis between them on the 22-sruti scale.

Western music does not have these subtle elements of consonance since they have no concept of the raga. In our music, consonance is intricately linked to the raga. Interested readers may refer to an excellent paper on this subject: Vadi, Samvadi, Vivadi, and Anuvadi Swaras by N Ramanathan, pages 60-82, Vol LIV, The Journal of Music Academy, Madras, 1983.

Anything else you would like to add?

I would like to clear a common misconception that is associated with the relative frequency ratios of our srutis. They say it is very difficult to play or sing a complex ratio such as 729/512 (that of prati madhyama). This is actually a misconception, which is removed once you see it as a number and not a fraction. In fact we cannot produce exactly even the simple ratio of 9/8 (chathussruti rishabha). The point is, while we sing or play we go by perception and not by these numbers. It is however important that we stay near these ideal values. The human voice can really make extremely melodious music because it can hold on to these notes within the accuracy range of the ear. Minor differences are ignored by our ear (actually the brain) and we enjoy our music!

S R Ramakrishna

A scholarly Yahoo! group discusses this interview