A Modern Introduction to Music – 10

By Anjum Altaf

At last we are in a position to answer two fundamental questions: First, why are there so few elements in the musical alphabet? And second, why have widely dispersed civilizations separately discovered the same musical alphabet?

Recall that the range of frequencies that are audible to the human ear extends from about 20 Hz to about 17,000 Hz. This is a huge continuous range that can accommodate an infinite number of stopping points. But as was mentioned earlier, the ear cannot distinguish very small differences in frequencies and of those that it can distinguish, not all combinations are musical or pleasing to hear.

As we mentioned in the last installment, consonant frequencies (those that sound pleasant together) are related to each other by the ratios of small integers. A lot of experimentation must have gone into the discovery of the sequence of frequencies (which are inversely related to string lengths as mentioned before) that are the most pleasing to the ear. Relative to any arbitrary starting frequency (which is normalized to 1 in order to keep the exposition general), the sequence of frequencies is as follows:

1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8 and 2

For example, we can take the starting frequency to be 200 Hz as in the last installment. Then the adjacent frequency would be 200*9/8 = 225 Hz, and so on.

Note a very important feature here: The final frequency in the normalized sequence is 2 (400 Hz in our numerical example). This is an exact multiple of the starting frequency (its second harmonic) after which the same sequence of ratios would repeat. Thus, in our numerical example, the next frequency after 400 Hz would be 400*9/8 = 450 Hz. We can also see from this that the repeating sequence can begin with any frequency: if our starting frequency had been 225 Hz instead of 200 Hz the sequence of consonant frequencies would have ended at 2*225 = 450 Hz instead of at 400 Hz.

Now strings vibrate in the same way everywhere and sound travels in the same way everywhere (on earth, at least) and the construction of the human ear is the same everywhere. Therefore, it is no surprise that widely separated civilizations discovered this same sequence of relative frequencies that sound pleasing to the human ear.

All that remains now is to give identification tags or names to these relative frequencies (just as we give names to our children and do not call them one, two, three, etc.). The names used in the Indian system were the following, starting with the reference frequency: Shadja, Rishab, Gandhar, Madhyam, Pancham, Dhaivat, and Nishad. These are the celebrated seven swaras (collectively called the sargam) and they constitute the saptak, the set that comprises the seven swaras.

These seven swaras also have nicknames (because we refer to them lovingly as a part of vocal music) as follows: Sa, Re, Ga, Ma, Pa, Dha, and Ni. And they are further abbreviated when used for notation as follows: S, R, G, M, P, D, and N. (Go back to the relative frequency sequence and confirm that the frequency of Pa is exactly one-and-a-half times the frequency of Sa.)

What happens after Ni? As mentioned above, we arrive at a frequency that is 2*Sa and the sequence begins to repeat. This first frequency of the next set is also called Sa but it is the Sa of the next higher saptak, called tar saptak in Indian music. In notation, this Sa is written with a dot on top to distinguish it from the reference Sa. Similarly, one can proceed in the reverse direction from the reference Sa and reach the Sa of a lower saptak, called mandra saptak in Indian music. In notation, this Sa is written with a dot under it to distinguish it from the reference Sa.

In our numerical example where the reference Sa was 200 Hz, the Sa of the mandra saptak would be 100 Hz and the Sa of the tar saptak would be 400 Hz.

Thus the swara sequence in Indian music is the following (I regret I am unable to add the distinguishing dots using this word processor):

[S R G M P D N] [S R G M P D N] [S R G M P D N] ….. Each set [S….N] constitutes one saptak.

The typical human can vocalize the range of swaras from the M or P of the mandra saptak to the M or P of the tar saptak when the reference swar is the S of the middle or madhya saptak. It is considered a great achievement for a vocalist to have a range spanning the entire three saptaks – Ustad Bade Ghulam Ali Khan sahib was reputed to have this ability.

Readers who have some familiarity with music would know that a piano has more than three saptaks – in fact it has eight, while a regular keyboard has four. This is because instruments can generate frequencies both higher and lower than the human throat can.

We can conclude here that the musical alphabet is exceedingly simple with only seven elements in it. Thus it should not be very hard to learn. In a forthcoming installment we will talk about music as a language.

There remains one detail to specify and one mystery to unravel. First, why is the reference frequency, Sa, arbitrary and not fixed? Second, readers who have seen keyboards will no doubt know that there are 12 and not seven keys in the repeating pattern of keys. (Note that there is no keyboard instrument in Indian music – the ubiquitous harmonium is a 19th century Austrian import that was long banned on All India Radio because of its alien origin.) The seven white keys are augmented by five black keys (see the piano keys  below and note the octave marked 1 to 8 on the white keys; also note the five auxiliary black keys in the octave).

We need give a rationale for the non-specificity of Sa and provide an explanation for the five extra frequencies that have invaded the saptak before we are ready to talk about music as a language.

More soon.

I don’t intend to talk about Indian music via the language of Western music but it would be of interest to readers to know the names given to these frequencies in the Western tradition: Do, Re, Mi, Fa, So, La, Ti, Do (these are abbreviated for notation as C, D, E, F, G, A, B, C). In Western music, the higher Do is included in the sequence and the set is known as an octave (a collection of eight notes). Listen to Julie Andrews in The Sound of Music teaching children the alphabet of music.

Now listen to Ustad Bade Ghulam Ali Khan. Note the range of his voice and how he uses the sargam in his singing both using the swar names and articulating them as pure frequencies (using the vocalization ‘aaa’ for each – this is called aakar.)


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10 Responses to “A Modern Introduction to Music – 10”

  1. Sohail Kizilbash Says:

    Interesting. You are touching this subject so please explain why it is called a set of 7 swaras when there are actually 12. Secondly what is that website which analyses sound made by you and tells you what swara it is?

    • Anjum Altaf Says:

      Sohail: I will get to this in the next installment. I am glad that the questions are coming from the readers. There was a comment earlier on the blog that an audience retains only 20% of the information communicated by an expert in a lecture format. The retention rises to 80% when the audience actually engages with the content. This is really the best way to internalize the concepts and that is why one should not rush through the material. To be very honest, this type of question fascinates me more than the music itself. This is probably the reason I didn’t learn much of performance music. I would ask the ustaads these kinds of questions and lose interest when they did not have convincing explanations.

      The best software of the type you have inquired about is TuneIt. I have bought it a number of times and most of my learning comes from using it. It is primarily intended to tune instruments but it can be used to train the voice in the way you have mentioned. Its beauty is that it renders music visual which makes an incredible difference to the conceptualization. If I had the ability to use it in this series, it would be a totally different experience.

  2. Kabir Says:

    So you have finally explained how Sa and Re are related to each other. The frequency of Re is related to the frequency of Sa in a ratio of 9/8. From one octave to the next, the frequency doubles. That’s why we can recognize it to be the same note, but one octave higher.

    It is interesting to get a scientific explanation for what musicians learn intuitively and through years of practice.

    • Anjum Altaf Says:

      Kabir: Yes, its the ratios that are important, not the absolute frequencies. We will explore in the next installment why this is important in the Indian tradition that is dominated by vocal music.

      You are also right about the notes that are one octave apart (the second is twice the frequency of the first). They generate the same sound sensation because the lower note generates a strong second harmonic which has the same frequency as the second note. Therefore in terms of tonal quality they remain the same, only the frequencies are different – it is the same Sa (or Re or Ga, etc. ) but at twice the frequency.

      Many people are puzzled by this but an analogy with numbers might help. We can count from 1 to 9 and then get to ten to which we need to assign a symbol. We can assign it the symbol X. The brilliance of the Arabic/Indian number system was to introduce the concept of zero. Now we count from 0 to 9 and denote ten by 10 – the same 1 now has a value ten times its original value. Similarly we have the same 1 in 100 but this has a value ten times the previous one. Without this clever invention we would have something as cumbersome and clunky as the Roman system that is now only good for decoration.

      Similarly, if we didn’t recognize that the tonal quality remains the same when frequency doubles, we would have new names for the notes in the second and higher octaves and end up with a sargam as complex as the Roman numeral system. In counting, the value increases by a multiple of ten every time a zero moves one position to the left. In music, a note doubles in frequency every time it moves up one octave but in terms of tonal quality it remains the same note.

  3. Umair Says:

    You lost me in the following statement,

    “The typical human can vocalize the range of swaras from the M or P of the mandra spatak (typo?) to the M or P of the tar spatak when the reference swar is the S of the middle saptak.”

    Why is the range from M or P of the mandra saptak to the M or P of the Tar saptak. What about the rest of the swaras; S, R, G and D, N?

    Also is there a name for the original reference Saptak?

    • Anjum Altaf Says:

      Umair: The saptak that the human voice can handle with the most ease is called the madhya saptak. Naturally it falls in the middle. Descending lower into the mandra saptak or ascending higher into the tar saptak are both difficult for those without trained voices. The range typically required for singing is the one I mentioned. All the notes in between have to be articulated. You would get this easily if you play these notes on a keyboard or harmonium and try and match your voice to them.

  4. Abhay Says:

    My compliments on the series!

    You talk about strings vibrating the same way everywhere and go on to say, “…it is no surprise that widely separated civilizations discovered this same sequence of relative frequencies that sound pleasing to the human ear.”

    To my mind, at least, it is still surprising that people across civilisations should have, independently, found the same ratios “pleasing” to their ears!

  5. rahul Says:

    Thank you very much for this information. I was really in a mess with sargam frequency range, like if they are separated uniformly on the frequency domain etc. I really love indian classical but never learnt it. Now i have plenty of plenty knowledge about this. But can you tell me, why aren’t the notes separated uniformly. It will be very helpful. Thank you once again for this wonderful writing.

    • Anjum Altaf Says:

      Rahul: I am pleased that you have found the explanation helpful. The difference between the just-tempered scale and the equally-tempered scale is explained in the 13th post in the series. If you continue reading where you left off, you will figure out why there is the difference.

      Spacing the notes uniformly is convenient but it comes at the cost of purity of sound. Most Indian accompanying instruments are stringed and can be re-tuned relatively easily – you must have seen the tuning interval when the tanpura or sarangi is tuned before the beginning of a performance. Therefore Indian artists prefer the pure, just-tempered scale. Many Western instruments use the keyborad (like the piano) which are very cumbersome to re-tune. They work better with the equally-tempered scale.

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