19 Aug 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.
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.)