Saptak, Tanpura and Harmonics

Prelude

I always wondered about how the ‘saptak’ evolved. I also wondered about ‘tanpura’. Why the other musical cultures do not need tanpura. My quest led me to some fascinating findings. that are presented.

I have deliberately retained the Sanskrit terms since we are used to them and conve precisely what the are supposed to convey.

Saptak is a foundation of all music. In order to understand music we must know the structure of Saptak. As the term implies a Saptak consists of seven shuddha swars. Shadja or sa is called the ‘aadhar’ swar. All the remaining swars bear a fixed relation to sa. In this sence all the other swars originate from sa. In other words sa is the ‘janak’ of other swars.

In the ‘natural harmonic scale’ re, ga, ma, pa, dha, and ni are related to sa as given below.

If sa is assumed to have a frequency say 240 Hz then re is 270 Hz, ga is 300 Hz, ma is 320 Hz, pa is 360 Hz, dha is 400 Hz and ni is 450 Hz. Sa of upper saptak is then 480 Hz. Expressed in terms of ratios of re to Sa with aadhar sa (tonic) the saptak can be tabulated as,

An important feature of this saptak is, pa is one and a half times sa and Sa is two times sa. This feature is common to many other musical scales. But the key question is, ‘where do the fractions come from?’ As this saptak is called ‘natural harmonic scale’, there is something natural about it. What is that? We get the answer if we understand what are harmonics.

Harmonics

If you pluck a stretched wire, a taar on a tanpura, it vibrates with a certain frequency ‘f’ that depends on the tension on the wire, it’s length, material and thickness. This is true if it was only a stretched wire. In case of string instruments the situation is not so simple. The wire is sitting on a ‘ghodi’ (bridge) that is not flat. The ghodi is resting on a peculiarly shaped hollow body of the tanpura. As a result the sound produced by the body of tanpura is a mixture of frequencies consisting of f and its integral multiples viz. 2f, 3f, 4f, 5f etc. These multiples are called the harmonics. 2f is the second harmonic, 3f is the third harmonic and so on. Same is true for wind instruments like bansuri or percussion instruments like tabla. The percentage of higher harmonics in this mixture of frequencies decreases with higher harmonics, but this harmonic rich or ‘javaridar’ sound is extremely pleasing to hear. (try adjusting the cotton thread between the ghodi and wire on a tanpura. The sound suddenly comes alive when the thread is properly positioned.) .

Coming back to harmonics, if f = 100 Hz. (for simplicity), the harmonics have frequencies,

2nd harmonic = 200 Hz
3rd harmonic = 300 Hz
4th harmonic = 400 Hz.
5th harmonic = 500 Hz

Harmonics higher than 5th have very low intensity but they do affect the quality or ‘naad’ of sound.

Recall that a saptak repeats when frequency doubles. Therefore to get a swar in the next saptak multiply the frequency by 2. To go to the previous saptak divide the frequency by 2. Now our tanpura wire tuned to 100 Hz produces a sound that ‘contains’ sounds of frequencies 200Hz, 300 Hz, 400 Hz, and 500 Hz. If 100 Hz is the sa of mandra saptak then 200 Hz is the sa of madhya saptak and 400 Hz is the sa of taar saptak. Thus 2nd harmonic, 200 Hz is the sa of madhya saptak. 3rd harmonic, 300 Hz, that lies in madhya saptak, is one and a half of 200 Hz. This is pa of madhya saptak. The corresponding pa in mandra saptak will be half of 300 Hz, or 150 Hz. The 3rd harmonic thus corresponds to pancham. Fourth harmonic, 400 Hz (200 Hz X 2) is the sa of taar saptak. 5th harmonic, 500 Hz lies in the taar saptak. This is ga of taar saptak. The ratio of ga to sa of taar saptak is 500/400 or 5/4. This gives us ga of madhy saptak as 250 Hz and 125 Hz in mandra saptak. Thus the fractions 2/1, 3/2 ang 5/4 decide the position of Sa, pa and ga. How the other swars are fixed?

Now, if you play mandra pa which is sa x 3/2, it’s second harmonic will be sa X 3/2 x 2 i.e sa x 3 which is pa of madhya saptak. It’s 3rd harmonic will be sa x 3/2 x 3/2 i.e. sa x 9/4. This is greater than 2 x sa. This is Re of madhya saptak. Half of this Re i.e. sa x 9/8 is the re of madhya saptak. If we now look at Sa of madhya saptak, it is the 2nd harmonic of sa of mandra saptak. What is the corresponding swar in mandra saptak whose 3rd harmonic (i.e. pa) is Sa of madhya saptak? Let’s say it is ‘n’ so that n x 3/2 = 2. This gives n = 4/3. That’s ma of mandra saptak. This gives us the positions of swars in poorvang.

Madhyam, as the name suggests, is the middle swar of a saptak. If we consider ma as sa then dha is ‘ga’ of ‘ma’ or the third swar from ma. Therefore dha is 4/3 x 5/4 = 5/3 i.e dha is sa X 5/3. Similarly dha is also the ma (4th swar) of ga. It is therefore also given by 5/4 x 4/3 = 5/3. This is consistant. Finally the third harmonic of ga i.e. pa of ga or ni is obtained by 5/4 x 3/2 = 15/8. On the other hand ni is the ma of ma or 4/3 x 4/3 = 16/9. So we have two nishads. This choice of ni assumes that uttarang is symmetrical with poorvang. But that is not the case. Poorvang and uttarang are different. Therefore 15/8 seems a proper choice.

We will now consider swarantar or intervals between swars. Look at the table below,

Swarantar means distance or frequency ratio between adjacent swars. Previous swar x swarantar = next swar. The swarantars are,

Note that the intervals differ. Also note that the poorvang and uttarang are not symmetrical. THIS IS IN SHARP CONTRAST TO THE EQUAL TEMPERED SCALE. There is nothing like "Poorvang" and "Uttarang" in ET scale. in ET scale sa-re-ga-ma is similar to pa-dha-ni-sa. All swars are equally spaced. So FUSION MUSIC is all CONFUSION! Ustad Vilayat Khan, while commenting on ‘east meets west’, once remarked in his interview that ‘it is like mating a cat with a dog to get some weird animal’. You can’t play raagdari, in it’s pure form, on a keyboard. Do not allow anyone to take you for a ride.

You may ask ‘did our ancestors know Physics?’ Surely they didn’t. So what? They were living in a world that was free of noise. They were able to hear all subtleties of musical sounds. Even today if you listen to music in a calm and quite place you hear sounds that you would have missed otherwise. Moreover our ancestors were great experimentalists. They were constantly experimenting with their music and musical instruments. Music and musical instruments have evolved over centuries. Bharat Muni (about 300 BC) mentions Dhruva or Achal Veena and Chal Veena and experiments with them. Classic example is Tanpura which is a unique feature of our classical music. If pa and sa are played and you listen carefully you can hear ga (antar gandhar). We now know that if you play two swars of frequencies f1 and f2 together a third swar (f1 + f2)/2 is also produced. pa is 3/2 times sa. So (1 + 3/2)/2 i.e 5/4 times sa i.e. ga is produced. The great experimenters already knew this through their experiments. The tradition still continues. Pt. Ramnarayan and Ustad Vilayat Khan modified Sarangee and Sitar respectively to suit their style of playing keeping the tradition intact.

Tanpura

Tanpura accompanying a vocalist or instrumentalist is not just a ‘drone’. It plays a pivotal role. Tanpura strings are normally tuned to pa ,Sa, Sa and sa, or to ma ,Sa, Sa and sa A properly tuned good quality tanpura is very rich in harmanics that give rise to other swars of saptak. Let’s see how.

As explained in the discussion on the ‘natural harmonic scale’ a saptak is based on harmonics. Strings on the tanpura emit swars that contain the the swars to which the string is tuned and their integral multiples or harmonics. For the sake of simplicity if we assume that the mandra sa has a frequency of 120 Hz then mandra ma and pa will have frequencies 160 Hz and 180 Hz respectively while madhya Sa will have a frequency of 240 Hz. If we enumerate the frequencies of first seven harmonics we get the following frequencies for the four tanpura strings. Table 3 enlists the frequencies of shuddha swars in mandra, Madhya, taar and ati aar saptaks. This table will help you to identify a swar that corresponds to a particular harmonic.

Table 1 - Harmonics for pa, Sa, Sa, sa tuning

* ma is slightly komal.

Table 1 shows that along with sa, pa and Sa harmonics of re ga, ma ni – komal. ni are also present.

Table 2 - Harmonics for ma, Sa, Sa, sa tuning

Table 2 shows that along with sa, ma and Sa harmonics of ga – komal , ga, pa. ni – komal are also present.

Below is a table of frequencies for your reference.

Table 3 - Table of frequencies for shuddha swars.

Number of harmonics produced in a tanpura depends on various factors like wood quality, wood seasoning, quality of joints, shape of the ghodi (bridge), uniformity of wires etc. More the number of harmonics richer is the naad (timbre).

Above discussion on harmonic nature of saptak and it’s ‘naturalness’ should be evident. You will also appreciate how tanpura provides a ‘surel’ reference for the artist. Tanpura plays a major role in Indian classical music.

I am not a musician, musicologist or an artist. So correct me if I am wrong.

Prof. Atul Phadke
Dept Of Physics
S. P. College, Pune