A Modern Introduction to Music – 11

By Anjum Altaf

In the last installment we established the relative frequencies of the seven swaras in the saptak and thereby their relationship to each other. We concluded with two questions: Why was the frequency of the reference note (Sa) not fixed at some absolute value? And, where did the five black keys on the piano octave come from, i.e., Are there 7 or 12 swaras in the saptak?

As often happens, the answer to one question provides a clue to the answer of another. The answer to the first question is really quite intuitive. Vocal music is the dominant genre in the Indian tradition and, as everyone by now will know, every human being has a distinctive voice. Not only that, the voice of the same individual changes over time. Thus, it should not be a surprise that every individual’s reference frequency (the frequency that he/she finds most comfortable to anchor his/her singing around) would differ to some extent. Very simply, this is the reason for not assigning Sa a pre-determined absolute frequency. Only if all human beings were clones could this have been possible.

Let’s explore this a bit more. In the last installment we mentioned that in order to sing most compositions, vocalists (starting from the reference S of the middle saptak) needed to have a range going down to the P of the lower saptak and up to the P of the upper saptak, i.e., a vocal range covering 15 swaras (from the lower to the upper P, the frequency would go up four-fold).

An individual with a deep voice would have no problem with the lower notes but find it difficult to reach the higher ones while one with a high-pitched voice would face the reverse situation. Such individuals adapt by picking a convenient reference frequency that enables them to cover the entire range. Thus the reference note of the first individual would have a higher frequency than the reference note of the second. Typically, men choose a higher reference frequency than women. One can now appreciate the logic of having relative frequencies – both singers would be vocalizing the lower P but the two P’s would not have the same frequency. The human ear recognizes a swara not as an absolute frequency but only on the basis of its relative distance from the reference swara.

Once the performer has made the choice of his/her reference note, the accompanying instruments (taanpura, sarangi or violin, and tabla in the case of vocal music) are tuned in accordance with the frequency of this reference note Sa. The frequencies of rest of the swaras are now fixed in keeping with the ratios mentioned earlier and they are tuned accordingly.

Now imagine what happens when the instruments are tuned for one individual and he/she is followed on the stage by another. Since the reference frequency would change, every string of the instruments would need to be re-tuned. This provides the point of entry for the second question with which we started. Because we are greatly helped by a visual perspective, I will rely on the keyboard to explore the answer.

We need to go back first to the relative frequencies of the swaras in the saptak:

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

These are the relative frequencies of succeeding swaras in relation to the frequency of the reference note which we called Sa. From this sequence it is very easy to calculate the ratios between adjacent swaras (these are called the intervals between adjacent swaras). For example, the interval between R and G and R is (5/4)/(9/8) = 10/9 (i.e., the frequency of R has to be increased by a factor of 10/9 to arrive at the frequency of G). Proceeding in this manner we can see that the sequence of intervals between adjacent swaras is as follows:

9/8, 10/9, 16/15, 9/8, 10/9, 9/8, 16/15

The point to note is that the intervals between swaras are not equal and that we only have to work with three intervals – 9/8, 10/9, and 16/15. From here some simplification follows that helps us in our exposition.

Note that the interval between S and R is 9/8 = 1.125; the interval between R and G is 10/9 = 1.111; and the interval between G and M is 16/15 = 1.067. From the auditory perspective, 1.125 and 1.111 are reasonably close to be treated similarly – they are both termed as one full step (or just a step) between swaras. Thus we can write that from S to R is one step and from R to G is also one step.

However, from the auditory perspective 1.067 is too small to be treated as a full step. But note that 1.067*1.067 = 1.137 which is just over one full step. In other words, two intervals of 16/15 raise the frequency as much as one full step. It is therefore natural to term the interval 16/15 as a half-step.

Now we can write the sequence of intervals of consonant swaras in simplified form as follows:

STEP, STEP, HALF-STEP, STEP, STEP, STEP, HALF-STEP

This is how we get from the reference S to the S of the next saptak:

S to R      One Full Step
R to G     One Full Step
G to M    One Half Step
M to P    One Full Step
P to D     One Full Step
D to N    One Full Step
N to S    One Half Step

Now look at the schematic of the keyboard that was given in the last installment and which is reproduced here (keep in mind that the Western C is the equivalent of the Indian S):

Imagine that the black keys don’t exist for the moment (cover them with a piece of paper). If one begins with the key marked 1 (C or S) one can proceed to the higher C using just the white keys (marked 1 to 8). In order for the intervals to be in accordance with the requirements of consonance these white keys would follow the tuning pattern Step-Step-Half Step-Step-Step-Step-Half Step. Thus the intervals between keys 3 and 4 and 7 and 8 are half steps; all the other intervals are full steps.

Now suppose a vocalist comes along and decides that for his her accompaniment key 5 (G or P) would be his/her reference note, i.e, what was P for the previous individual will be the S for the new one (remember what is critical to music are the intervals between swaras not the absolute frequencies).

Let us take key 5 as our starting point and see what sequence of steps and half steps we encounter as we move up the keyboard:

Key 5 to 6      One Full Step
Key 6 to 7     One Full Step
Key 7 to 8     One Half Step
Key 1 to 2     One Full Step (Note Key 8 is the same as Key 1 of the next octave – they are both C or S)

Key 2 to 3     One Full Step (the white keys of the next octave are not numbered but you can do so yourself)

Key 3 to 4    One Half Step
Key 4 to 5    One Full Step

Note that changing the reference frequency makes us move away from the interval sequence we require for the swaras in a saptak to be consonant. The required interval sequence is:

STEP, STEP, HALF-STEP, STEP, STEP, STEP, HALF-STEP

The sequence that results when we start from key 5 (turning the old P into S) is:

STEP, STEP, HALF STEP, STEP, STEP, HALF STEP, STEP

We are fine till the fifth interval – the two sequences are identical. Then instead of {STEP, HALF STEP} we get {HALF STEP, STEP}. How can this problem be resolved? The black keys represent a solution to this problem.

Look at the schematic again to visualize the problem and the solution. The interval between the white keys 3 and 4 is a half step. But now, with the changed reference note, we need a full step here. The solution is to insert a new key (marked in black) at a half step above the white key 4. Now if we go from key 3 to this new black key we would move the frequency up by a full step. And from this new black key to the white key 5 is now a half step. So by inserting this black key and using it, we convert the {HALF STEP, STEP} sequence to a {STEP, HALF STEP} one and thereby fulfill the interval requirement for consonance.

Proceeding in a similar way by choosing other keys as starting points, we will discover where additional keys have to be inserted in order to have a consonant sequence no matter what the starting point. What is happening in effect, if you look at the schematic, is that a new half step is being added between every full step. These additional half step keys are the new black keys. Note that there are no black keys between the white keys 3 and 4 and 7 and 8 because they were half steps in the first place.

This can also provide an explanation for the distinctive 2-3 pattern of the black keys. It simply follows from the pattern of the white keys:

STEP, STEP, HALF-STEP, STEP, STEP, STEP, HALF-STEP

The first set of two full steps get two black half step keys; the second set of three full steps gets three black half step keys. Thus the saptak now has 7 primary swaras and 5 auxiliary swaras yielding a total of 12 swaras. In effect we can now go from one saptak to the next in 12 half steps. And this clever invention can accommodate all the changes of reference notes possible without losing the consonance of musical sounds. Instead of relying on just the white keys we would be using one or more black keys as needed.

This has been a particularly involved installment but it does answer a difficult question that need not detain every reader. We will defer till the next installment the description of the names given to these new swaras which will complete our alphabet and allow us to discuss the language of music.

More soon.

Note: I am not making these things up nor did I know them at the outset. I am using material that is too technical to be useful to most readers at this point. At the end of the series I will provide a full list of the sources that I have consulted.

 

Tags: Appreciation, Indian, Music