It should be noted that the mathematics behind the solution is outlined here is already well known. For example, it finds explicit mention in books like Rangaramanuja Iyengar's History of South Indian Music, and Vidya Shankar's Art and Science of Carnatic Music I am also fairly certain that books of Bhagylakshmi and Sambamurthy mention core aspects of this solution. However, in the first two books mentioned above, there is no detailed explanation of the solution particularly in correlation with the vina experiment. I feel that doing so is important as it elucidates how the solution adds up mathematically, and how it is a plausible "behind the scenes" explanation for an experiment which obviously seems to have been done emperically ("playing it by the ear") without awareness for the math behind it.
There is perhaps a general sentiment in these books (and Rangaramanuja Iyengar states this explicitly) that carnatic music has evolved past the gramas of Bharata's time. Hence, these authors seem very well aware that the 22 "popular ratios" prevalent today are not the same as Bharata's. They seemed to have tried to reach beyond it to explain why he divided the octave a specific way, or how the ancient Indians might have arrived at the positions for the seven svaras in the grama etc. And that probably led to application of the cycle of fourths/fifths model to carnatic music. It also looks like many of the popular ratios may be a result of the cycle of fourths/fifths model with a dose of Bharata's divisions as explained later.
However, all this has sprung forth myriads of ratios leading to discrepancies among the lists among these authors. Given all this, one can understand why some people are skeptical or dismissive of this whole concept of division of octave into 22 sruthis starting from Bharata. It is also important to separate the theory that confines itself to only Bharata from the theory that goes much beyond Bharata.
I do not want to comment much on the validity/invalidity of the cycle of fourths/fifths or any other models with respect to their applicability to carnatic music here. I also do not want to speculate much on how the ancients might have arrived at the positions for the different svaras in their gramas and why they did it in a specific way. I want to concentrate only on explaining how Bharata might have divided the octave, and whether such a division can be carried out empirically. In the Conclusion section, I do offer that some of the popular ratios mentioned in the books may have been arrived as a result of incorrect application of Bharata's division of sruthis.
It is also very important to note that the mathematical solution presented here is not the only possible solution. There are several solutions that are mathematically possible. This is also explained in the Conclusion section.
Note that Bharata does not give any explicit indication about whether a svara is fixed at a certain sruthi. In fact he mentions that svaras contain sruthis (i.e. sruthis are explained in terms of svaras) and says shadja has 4 sruthis etc. However, if we look at the details of how the svaras transition in pitch during the experiment, we see that the svara (perhaps typically) occupies the position at the end of its sruthis. For example, in the experiment, when rishaba is lowered by 3 sruthis, it is supposed to merge with shadja. This means that shadja and rishaba are separated by 3 sruthis. Since Bharata also says that that shadja has 4 sruthis and rishaba has 3 sruthis, we can combine the two and deduce that the svara starts at a position immediately following all its sruthis. In other words, there are 3 sruthis between shadja and rishaba, 2 between rishaba and gandhara and so on.
Dattila in his Dattilam is much more explicit corroborating this as he say effectively that In shadja-grama, commencing from the sound accepted as shadja (shadjatvenagrahitah), the third higher sound (or sruti) is rishaba; this is beyond doubt .
Based on the first two assumptions and the spacing of svaras in shadja grama as noted by Bharata, we obtain the following ratios:
shadja grama | |
svara | Ratio |
shadja | 1 |
madhyama | 4/3 |
panchama | 3/2 |
nishada (9 away from madhyama) | (4/3) * (4/3) = 16/9 |
gandhara (13 before nishada) | (16/9) * 1/(3/2) = (16/9) * (2/3) = 32/27 |
Note above that even though Bharata does not include madhyama-nishada as samvadin, we have still assumed that the ratio between nishada and madhyama is 4/3.
I am actually puzzled at how Bharata or his contemporaries chose the pitch for rishaba and dhaivata as they do not have consonant relationship (i.e 4/3 or 3/2) with other svaras in shadja grama. The only other svara they are consonant with is the madhyama grama panchama, but that itself is not consonant with anything else. So I do not have a rock-solid feel for the positions/ratios of the triplet {rishaba,dhaivata,madhyama-grama panchama} when compared to the feel I have about the positions/ratios of the others. Hence, I have tried to make an educated guess for the ratio for the rishaba here. I then use that to determine the ratios for the other two svaras in the triplet (Note: more on this at the end of the Conclusion section).
There are 3 popular candidates for rishaba - 16/15, 9/8 and 10/9. Out of this, we can immediately eliminate 9/8, since it is consonant with the shadja grama panchama ( (9/8) * (4/3) = 3/2 ). Our rishaba must not be so. So we are left with 16/15 and 10/9. Taking 16/15 will compromise the mathematics behind the division of sruthis as shown below:
If rishaba is 16/15, shadja-rishaba interval = 3 sruthis = 16/15 rishaba-gandhara interval = 2 sruthis = (32/27) * 1/(16/15) = (32/27) * (15/16) = 10/9 But 10/9 (2 sruthis) is greater than 16/15 (3 sruthis)!
Now, this of course assumes something which seems obvious but is not explicitly stated by Bharata: a 3-sruthi interval must be greater than a 2-sruthi interval. However, this relationship must hold true in order for the result(s) of each step of the experiment to match what Bharata says. In the experiment, gandhara merges with rishaba in step #2 (i.e. after two reductions in sruthi), but rishaba merges with shadja in step #3 (after three reductions in sruthi). This usually means that that the rishaba-gandhara interval is lesser than the shadja-rishaba interval, the opposite of the condition we arrive if rishaba is chosen as 16/15. It is indeed possible to presume a model which involve unequal reductions for different strings at each step, that does not cause this mathematical contradiction. However, as will be apparent later, an equal correction model is not only simple, it is also intuitive enough for someone to achieve it empirically "playing it by the ear".
Hence we can eliminate 16/15 and are left with 10/9. With this ratio, it can be confirmed that the mathematics will not be compromised as with the case of 16/15. Note also that 10/9 (and for that matter 16/15) is not consonant with 3/2 (shadja grama panchama) as required.
Based on the assumption that the rishaba is 10/9, we obtain the ratios for the dhaivata (separated from rishaba by 13 sruthis) as (10/9) * (3/2) = 5/3, and the madhyama grama panchama (separated from the rishaba by 9 sruthis) as (10/9) * (4/3) = 40/27.
Hence, our full list of ratios are:
shadja grama | |||
svara | Ratio (shadja grama) | Ratio with prior svara | Sruthi interval |
shadja | 1 | - | - |
rishaba | 10/9 | 10/9 | 3 |
gandhara | 32/27 | 16/15 | 2 |
madhyama | 4/3 | 9/8 | 4 |
panchama | 3/2 | 9/8 | 4 |
dhaivata | 5/3 | 10/9 | 3 |
nishada | 16/9 | 16/15 | 2 |
shadja | 2 | 9/8 | 4 |
madhyama grama | |||
shadja | 1 | - | - |
rishaba | 10/9 | 10/9 | 3 |
gandhara | 32/27 | 16/15 | 2 |
madhyama | 4/3 | 9/8 | 4 |
panchama | 40/27 | 10/9 | 3 |
dhaivata | 5/3 | 9/8 | 4 |
nishada | 16/9 | 16/15 | 2 |
shadja | 2 | 9/8 | 4 |
The experiment then involves the following steps
This supposedly demonstrate the sruthis within the svaras in the shadja grama, which of course add up to 22.
So assuming people knew about shadja grama intimately in terms of how individual strings sound, which is presumed here to be rooted in the familarity with basic consonance relationships, it looks like one can achieve all the steps just playing it by the ear. All that is then apparent is the number of "hops" the individual svaras took to arrive at their predecessor. If we add them all up, it is 22.
The solution presented here does not imply all sruthis to be equal, or equal within a svara. The only implications from the experiment we gather are:
In other word, the quantification (if one can deem it so) of the reductions in sruthi is in terms of what it takes to match the pitches of pairs of strings such that they either become the same or have a consonant relationship. This is completely within the realm for someone playing by the ear. It seems like a very practical approach.
The only thing the experiment implies that in each step, that all strings be reduced in sruthi by the same interval (ratio). This is because otherwise they will not remain in shadja-grama (or "in tune")!
dhruva vina: 1 10/9 32/27 4/3 3/2 5/3 16/9 2 <-sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ chala vina: 1 10/9 32/27 4/3 3/2 5/3 16/9 2 <-sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ Fig-1Step 1: Part (a) - lowering the panchama
dhruva vina: 1 10/9 32/27 4/3 3/2 5/3 16/9 2 <-sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ chala vina (in madhyama grama now) 1 10/9 32/27 4/3 40/27 5/3 16/9 2 <-sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ Fig-2Step 1: Part (b) - lowering other strings of chala vina
dhruva vina: 1 10/9 32/27 4/3 3/2 5/3 16/9 2 <-sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ chala vina: (back to shadja grama but 1 sruthi less than dhruva vina) 80/81 800/729 2560/2187 320/243 40/27 400/243 1280/729 160/81 <-sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ Fig-3
Now obviously all the above ratios are quite intimidating and might seem nonsensical. But it is extremely important to note that they are so only because we are trying to express the pitch of the chala vina in terms of the shadjam of another vina, the dhruva vina, which is slightly out of tune with respect to it because of a slight reduction in sruthi (81/80 is less than 22 cents). With such a small reduction, the ratios better be as complex as above as the two vinas are not going to sound nice together. In other words, they will not have a simple harmonic relationship between them! Also more importantly, it can be confirmed that within the chala vina, the svaras with seemingly nonsensical ratios are related to each other by the same ratios as in the dhruva vina - as it should be in order for it to be in shadja grama. (Note: One can actually do the math and verify this at the end of every step.)
Step 2
Explanation: This step involves lowering all strings of chala vina by one more
sruthi such that the gandhara and nishada converge with rishaba and dhaivatha
of the dhruva vina. In the case of the gandhara, our "destination pitch" is the
ratio for the rishaba of dhruva vina, which is 10/9. The "source pitch" is the ratio
for the pitch of gandhara of the chala vina after step 1, which is 2560/2187.
Hence, this step involves a reduction in sruthi of (10/9) * 1/(2560/2187) = 243/256.
This is the same for nishada, where the "destination pitch" is 5/3 and "source pitch" is 1280/729, and (5/3) * 1/(1280/729) does equal the reduction 243/256.(All ratios are of course expressed in terms of shadja of the dhruva vina)
dhruva vina: 1 10/9 32/27 4/3 3/2 5/3 16/9 2 <-sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ chala vina: (in shadja grama but 2 sruthi less than dhruva vina) 15/16 25/24 10/9 5/4 45/32 25/16 5/3 15/8 <-sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ Fig-4 Notes: 5/4 - this is considered to be antara gandhara, a version of gandhara mentioned by Bharata (as two sruthis higher than the "regular" gandhara as observed here) 45/32 - this is considered by some to be the ratio for prati madhyama 15/8 - this is considered to be kakali nishada, a version of nishada mentioned Bharata (as two sruthis higher than the "regular" gandhara as observed here)Step 3
This is the same for dhaivata, where the "destination pitch" is 3/2 (panchama of dhruva vina, and the "source pitch" is 25/16, and (3/2) * 1/(25/16) does equal the reduction 24/25. (All ratios are of course expressed in terms of shadja of the dhruva vina)
dhruva vina: 1 10/9 32/27 4/3 3/2 5/3 16/9 2 <-sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ chala vina: (in shadja grama but 3 sruthis less than dhruva vina) 9/10 1 16/15 6/5 27/20 3/2 8/5 9/5 sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ Fig-5 Notes: 16/15 - One of the "popular ratios", considered by some as dvisruthi rishaba. 6/5 - One of the "popular ratios", considered by some as sadharana gandhara. 27/20 - One of the "popular ratios", considered by some as tIvra sudhdha madhyama. 8/5 - One of the "popular ratios", considered by some as dvisruthi dhaivata. 9/5 - One of the "popular ratios", considered by some as kaisiki nishada.Step 4
This is the same for panchama, where the "destination pitch" is 4/3 (madhyama of dhruva vina), and the "source pitch" is 45/32, and (4/3) * 1/(45/32) does equal the reduction 81/80. (All ratios are of course expressed in terms of shadja of the dhruva vina)
dhruva vina: 1 10/9 32/27 4/3 3/2 5/3 16/9 2 <-sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ chala vina: (in shadja grama but 4 sruthis less than dhruva vina) 8/9 80/81 256/243 32/27 4/3 40/27 128/81 16/9 sa <-ri <-ga <-ma <-pa <-da <-ni <-sa | | | | | | | | ^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^---^ Fig-6
We can now state Bharata's division of sruthis within a svara as follows:
The 3-sruthi interval (from lower-pitch to higher-pitch) is divided into three intervals, with the intervals having the ratios 25/24, 256/243 and 81/80.
The 2-sruthi interval (from lower-pitch to higher-pitch) is divided into two intervals, with the intervals having the ratios 256/243 and 81/80.
Note that in order to see a consistent structure here, one must read the intervals from right to left and not left to right.
Note that in our calculations, several of the popular ratios do not figure (e.g. 9/8 for the sruthi known as chatusruthi rishaba). We also run into very complex ratios but I maintain that those ratios make sense within the context of the experiment. As explained earlier, many of the popular ratios were perhaps arrived with the presumption that the cycle of fourths-fifths as the underlying model. The intent might also have been to use that model to arrive at ratios which included all the "simple" ones, which were obviously presumed to be pleasing enough to be part of music. Hence, the end result is a list of ratios which is not the same as Bharata's.
Note that some of the popular ratios can be accomodated but within hypothetical new gramas. For example, an hypothetical grama with shadja-4, rishaba-4, gandhara-2, madhyama-3, etc. would include 9/8 (rishaba). A similar case can at least in theory be applied for the sruthi ekasruthi-rishaba reputed to be part of the raga gowLa. This sruthi is assigned the intimidating ratio 256/243. If we presume that the rishaba of the mela mayamalavagowla is 2 sruthis from shadja (i.e. like a new grama), it will have the ratio 16/15, which is close to 100 cents. The rishaba of mayamalavagowla is indeed close 100 cents. Now, if we apply the previously stated division for the 2-sruthi interval, a reduction of 1-sruthi (80/81) does yield the ratio 256/243, the one assigned to ekasruthi rishaba. So the derivation of the ratio seems to consistent with the method of divisions that Bharata used.
However, the interpretation that such an intervening sruthi is singable and is part of the music (as in gowLa), is definitely new and not stated in Bharata. Since this division follows the structure of Bharata's division, it also then presumes that the reason Bharata divided the sruthis among a svara a certain way is because the result yields sruthis that are singable and part of the music. In Bharata's case, the first reduction in sruthi definitely has something to do with this since it is intended to reduce the shadja grama panchama to the madhyama grama panchama i.e. a sruthi that is part of the music (albeit in a different grama). But the size of this reduction completely determines the size every subsequent reduction! So while the size of the first reduction was "planned" (and that too only for one string), the size of the rest of them are dictated by the first and the destination pitch for certain strings (e.g. gandhara string of chala vina must coincide with rishaba of dhruva vina after step-2). Hence, it is my opinion that the magnitudes of the ratios for each division were perhaps not carefully designed to have specific values such that they yield singable intervals every step of the way. They seem to be just a consequence of the difference in pitches between the panchama of the shadja grama and madhyama grama.
It also interesting to note that Dattila in Dattilam, mentions that only some of the sruthis are deemed singable and hence elevated to svaras. This does indicate that in those days that they did not think all 22 sruthis were singable intervals like it is interpreted today. Music has definitely evolved to include more intervals and melas, the modern-day equivalent of gramas. These new intervals theoretically can indeed be divided based on Bharata's scheme, but there does not seem a strong enough rationale then to claim that an intervening sruthi with a ratio like 256/243 yielded via such a division is singable and is indeed sung today. One could argue that it is based on an inaccurate interpretation of Bharata's division to express a pitch that is perhaps perceived as lower than typical mayamalavagowla rishaba (the gowla rishaba is sung as a pitch inflexion from shadja).
It must also be noted that the madhyama grama panchama, the "pivot" pitch that is at the heart of the structure of Bharata's division is no longer in use today! So one can also argue that why should today's music use Bharata's reduction of 80/81 (as with the case of the ekasruthi rishaba). We should also consider that 80/81 is tied ultimately to the ratio 10/9 presumed for the rishaba, which need not be the only possible ratio as explained below.
However, the results will be mathematically consistent with the experiment like our experiment above. We can only speculate as to which solution out of this set Bharata and his contempories may have chosen. The exact nature of the intervals for the rishaba, dhaivata and the madhyama grama panchama of their music still eludes us. They cannot be arrived from the other svaras based on simple consonant relationships. So one should always keep in mind that the "popular" division of sruthis has a major fundamental assumption with regards to the ratio of 10/9 for the rishaba. That assumption is definitely weaker than the assumption for the ratios for madhyama and pancama.