A Modern Introduction to Music – 13

By Anjum Altaf

I hope by now readers have fully internalized the most essential characteristic of music. It is not the frequency of a swara that is important; rather, it is the interval between swaras or the ‘musical distance’ between them that is critical. One can start from any frequency; as long as the subsequent swaras are at the right distance, one would be in the realm of music.

We had started this series with the claim that while all music is sound, not all sound is music. In doing so we had made the distinction between music and noise. We are now in a position to elaborate on this distinction.

Think of construction in which the building block is a brick. If we dump a load of bricks on a plot of land we would have an untidy sight to behold. However, if we arrange the bricks according to a plan or pattern we could have a very elegant structure. The building blocks of music are the swaras. If we toss out swaras without attention to the distance between them, we would have noise. On the other hand, if we choose the swaras well, place them correctly, and determine a judicious path from one to the other, we will have music.

Pursuing this analogy, we can think of music as being an attractive building constructed from swaras. Different arrangements of these swaras would give us different types of buildings. In one conceptualization, we can think of a raaga as a musical building; different raagas represent different types of buildings all of which are constructed from the same set of swaras. It will be this analogy that I will use to motivate an understanding of raagas in a subsequent installment.

At this point, I would like to step back and tie up a few loose ends about musical distance. Recall from a previous installment that we had simplified the notion of musical distance to the extreme by positing just two possible variants – the full step and the half step. Thus, starting from an arbitrarily chosen reference (which we called Sa and denoted as S) we had the following shudh swaras in a saptak:

Sa Re Ga Ma Pa Dha Ni [S]

where [Sa] is the Sa of the subsequent saptak with a frequency double that of the reference Sa.

These are denoted as follows:

S R G M P D N [S]

The musical distance between these swaras required for consonance was determined to be as follows:


Keep in mind that this was an approximation. The steps are not exactly equal in size. For example, the first step from S to R represents the interval 9/8 (1.125) while the second step from R to G represents the interval 5/4 (1.111). When musicians tune their instruments, they tune them in accordance with the exact intervals. The approximations make the exposition simple and the pattern easy to remember.

Now recall that we showed with the help of the keyboard that if we picked a different reference swara, we would have to introduce additional keys in order to maintain the required pattern of steps and half-steps mentioned above. We need five additional keys which in fact create a saptak made up of 12 half-steps. This is done by inserting a new half-step key between every two full steps.

Now we have the 12 swara saptak:

S r R g G M m P d D n N [S]

The distance between each swara is now one half-step and there are 12 half-steps (but not of the same size) in the saptak.

There are two interesting digressions here for the reader interested in theory. They pertain to the peculiarities of the Western and Indian traditions of music.

First, it is important to keep in mind that the Western tradition is instrument-oriented in which a keyboard instrument like the piano plays a central role. By contrast, the Indian tradition revolves around vocal music and its accompanying instruments are primarily string instruments (leaving the percussion instruments out of the discussion for the moment).

Now think of the implications for retuning instruments when the reference swara or key is changed. It is relatively easy to retune string instruments but very cumbersome to retune keyboard instruments like the piano. In fact it is impractical to retune a piano repeatedly. A solution for this problem was found by sacrificing a little bit of the integrity of sound for a whole lot of convenience.

The solution was to make each of the 12 half-steps on the keyboard of exactly the same size. With this change it would not matter which key was picked as the starting one because the musical distances would be the same independent of the choice. It is easy to find the interval represented by the size of this modified half-step. Since all half-steps are now of exactly the same size if we multiply the starting frequency 12 times by the interval of the half-step the resulting frequency should be twice the starting frequency:

1 * (x) ** 12 = 2 where ** signifies that x is raised to the power of 12 or multiplied by itself 12 times.

It follows that x = 12th root of 2 and this can be found on a calculator to be 1.0595.

It also follows that now the size of every full-step is the same with a value (1.0595 * 1.0595) = 1.1225. Thus the interval between S and R is 1.2225 instead of the true value of 1.125 and the interval between R and G is also 1.2225 instead of the true value of 1.111, and so on. These variations are too small to affect the consonance of the music although there has to be a subtle loss of integrity. The resulting scale of 12 equal half-steps is called the equally-tempered scale while the scale with true intervals is called the just-tempered scale. The equally-tempered scale is the compromise that allows pianos and keyboards to be played from any key without the need to be retuned for every change.

In contrast to the keyboard, the human voice is infinitely flexible. It does not jump from one swara to another but glides continuously between them and can rest anywhere in the interval. Thus it is quite possible for a vocalist to decide that a particular composition sounds better with a slightly flatter swara than that provided by the standard variant. For example, instead of a flat or Komal Re, the performer could prefer an even flatter variant called Ati-Komal Re.

This is the logic underlying the discussion of microtones or shrutis. It is claimed that there are 22 and not just 12 pleasant intervals in a saptak that can be distinguished by the human ear. Thus instead of there being one swara between S and R there are 3 shrutis in that interval. Many musicologists have specified exact ratios for these shrutis (for example, the interval 256/243 is given for Sa to Ati-Komal Re) although it is not the interval that determines the resting place but the appeal of the sound to the ear of the performer.

The important thing to keep in mind is that while a keyboard has fixed resting places, for the human voice these are a matter of choice. (It is possible to design keyboard instruments with more than 12 keys in an octave but they become very difficult to play.) Also to be kept in mind is the fact that in a composition, a vocalist (or string instrumentalist) is unlikely to choose more than one shruti of a swara, i.e., he/she would either choose a Komal Re or an Ati-Komal Re but not both. This discussion is enough to give a sense of the place of microtones or shrutis in the Indian tradition.

With this we conclude the discussion of the physics of music. From here on we will be talking about the language of music and the musical constructions that we have equated to raagas.

More soon.

Those who wish to see and hear a demonstration of shrutis to get a feel for what is involved can do so here on YouTube.


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8 Responses to “A Modern Introduction to Music – 13”

  1. Arun Pillai Says:


    Thank you for a unique and fabulous introduction to the physics of music. This material should be elaborated and combined with the material to follow into a book.

    This reminds me of a question that was discussed earlier on your blog: is Indian musical space discrete or continuous? What would be your response now?

    • Anjum Altaf Says:

      Arun: Thanks for the encouragement. The goal you have suggested can be kept in mind although it would take quite a bit of work to attain it.

      The exploration of the physics of music certainly puts us in a better position to discuss whether the Indian musical space is discrete or continuous. From a strictly technical perspective, it is discrete because whatever the number of shrutis, it is not infinite. The number of resting points is finite; ergo, the space is discrete.

      From an aural perspective, the musical space comes across as continuous because the melody does not jump from one resting place to another but glides through them.

      An illustration might give us something concrete to talk about:

      There was an old Indian film song that stuck in my mind because it sounded different from most other film songs: http://www.youtube.com/watch?v=XitG1Oj0jkE

      It was perhaps a quarter century later when I was listening to Mozart’s Symphony No. 40 that I made the connection. The song had that jumpy staccato feel because it was mounted on a Western foundation: http://www.youtube.com/watch?v=aZD9nt_wsY0

      Listen to any other Indian film song and note how different it comes across from the first song. The sense of continuity is much stronger: http://www.youtube.com/watch?v=2w9SBGATr3w&feature=related

  2. Kabir Says:

    Is the difference between just temperament and equal temperament perceptible to the ear?

    Also, since the harmonium is a western instrument, then it is equally-tempered, right? So, then a lot Indian music also uses the equally-tempered scale, not allowing for the use of the shrutis.

    • Anjum Altaf Says:

      Kabir: Yes, the difference should be perceptible to the trained ear especially if the training has been exclusively on one temperament. It might be difficult for Indian musicians who have learned music with a great deal of reliance on the harmonium.

      The harmonium is a western instrument but has been greatly adapted for Indian music. While most harmoniums are equally-tempered, there is nothing that prevents tuning them to just intonation if a performer wants a harmonium for his or her exclusive use. Also, 22 shruti harmoniums have been available for some time.

      Here is a useful piece by the inimitable Parrikar that you would find of interest. The bottom line is that the harmonium is fine as an accompanying instrument because it gives a rich sustained sound and its function is not to reproduce a swara by swara copy of the vocalist. The most important advice is to avoid the harmonium while learning music in order to preserve the ear’s sensitivity to the intonations that are possible for the human voice.


  3. sakuntala Says:

    Good question. Actually when one plays Indian music on the harmonium, one uses the equal temperament scale but develops techniques whereby one quickly alternates between adjacent keys to give the impression of swaras in between (which the keys cannot play) Also, now there is a huge controversy about Carnatic music on the piano (of two leading pianists of the south, one uses benders to create gamakas, the other says he does not , but will play only those ragas which do not call for subtle 22 srutis — Malkauns (Hindolam) Bhoop (Mohanam) etc — and we had a day long seminar on this at Mumbai recently, and the consensus was that as long as one does not try ragas like Sahana (which depend on 22 srutis for getting the raga’s full flavor) Indian music on piano can also be enjoyed…..

  4. Arun Pillai Says:


    I think you have pointed to a very satisfying resolution to the dilemma of discrete vs. continuous. “Ontologically” (i.e. in terms of what “is” or what “exists”), the space is discrete but “phenomenologically” (i.e. as “experienced”), it is continuous. This is similar to a motion picture: it is made up of stills and so is ontologically static but is experienced as moving (owing to the persistence of vision) and so is phenomenologically dynamic. In music, the resting points are finite because the number of consonant ratios is finite, but sliding/gliding notes allow intermediate pitches to be voiced as well. So there is also some difference between music and film. Of course, there are also limits on how much our ears can discriminate.

  5. Vinod Says:


    Just a minor clarification point needed. The jumpiness in the old Hindi song in that link was due to the music, right? Not the singing itself; at least not as jumpy as the music makes it be. Am I right in aurally observing that the music in the Mozart piece was matched not by the instrumental from the Indian piece but by the vocalization?

    The link to the Mozart piece is done incorrectly. It still points to the old hindi song.

    I’m reminded once again of that post in this blog where you described western music as architecture and Indian music as a painting. Is that because of what you mention here about discreteness and continuity in the notes change?

    One more question – Indian classical music lovers like to boast about the superiority of Indian classical over western classical. Is that just pride given what you’ve just described about the 22 swaras in Indian classical and only 12 in western?

    • Anjum Altaf Says:

      Vinod: Most of your questions would be addressed by the video clip I have linked in response to your other comment. I do wish to talk about your last question. I recall we have gone over this issue of superiority and concluded it is not the right perspective. These are different traditions with great histories and we have to engage with them on their own terms. There is nothing that prevents a keyboard being developed with 22 keys in the octave but it would be awkward to play. Western music doesn’t need that because it is instrument-based and layered differently with the development of harmony – many voices singing different notes at one time. Indian music is voice-based and limited to one note at a time. It needs the melodic richness that comes from the 22 shrutis to make it attractive. The use of ornamentation and embellishment in the two traditions is different. Comparing them would be equivalent to saying that Indian marriages are superior to Western marriages, or vice versa. It would not make much sense.

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