Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

[vgovindan]

http://youtu.be/MJ3RQVIKErw

[sureshvv]

Excellent lecture.

[SrinathK]

There are a few additional points which I have found (after massive plotting in excel) :

1) The algorithm gives 24 values for 12 tones. The Sa and Pa being fixed, the other 2 values are removed, giving 22. (These 2 values are very highly dissonant against the tonic sa or a tanpura - a wolf note even -- so they cannot be used in practice). That's why you get 22.

2) The 22 sruthi system's main purpose seems to have been to ensure that from any position, a 12 tone scale can be played, something which can't be achieved with a system of just 12 frequencies. See, there are actually only 12 notes all along where sruthi bhedam is viable (The just intonation ones). The problem is -- in sruti bhedam, some notes have to decide whether to be in tune to the tanpura (original shadja) or the new shadja. We can't do that with just 12 tones in the scale, but with 22 tones it becomes possible.

3) The accuracy of sruti bhedam in the 22 sruthi is FAR superior to changing keys in Equal temperament (which was ironically developed for that very purpose). And ET was first made in China

Most of the intervals are spot on, but there are some intervals where there is a mathematical inaccuracy of 1.957 cents (~0.2%), which is too small to actually be noticeable.*

In Equal temperament tuning, the M1 and P have the smallest error (~1.96 cents), but in all other notes, the error is much greater (as much as 15 cents!). This last error is so noticeable on a keyboard, a piano or a harmonium.

ET only wins because of a practical advantage -- it takes a computer algorithm to properly harness the power of all 22 sruthis while playing on a regular keyboard. Even in Dr. Oke's instrument, you would have to adjust those plugs manually if you are changing key so that you can play on the correct frequencies, and if the composition calls for sruthi bhedam while playing, you can't do that easily (If his scales of Hindustani ragas are accepted, then this may be a problem even in ragamalikas). With ET you don't have to do that.

The modern gamaka system BTW has no use for 22 shrutis, nevertheless they are still as relevant as before in the (much unexplored) area of sruthi bhedam and sruthi shuddham in long plain note kaarvais. On a kaarvai, one has to land at those spots otherwise one will sound out of tune (like G3 of Shankarabharanam). This also means that in today's gamaka based system, it is not advisable do a sustained kaarvai on every "note" (like M2 in varaali).

4) 22-shruti system cannot use the 7th harmonic. And for good reason, it creates an unholy mess in sruthi bhedam. No wonder it was called dissonant. At best it could serve as an occasional blue note or in a 4 part chord of 3,4,5,7 harmonic notes (barbershop 7th), but then again, where did we ever use those in Indian music?

[vgovindan]

Dr Oak shows four variations each for R, G, M, D and N
Purna Sruti----pramANa---nyUna--pramANa
S-----------------r1-------r2-----------R1------R2
R2---------------g1-------g2----------G1------G2
G2---------------M1------M2---------m1------ m2
P----------------d1-------d2----------D1------ D2
D2---------------n1-------n2---------N1-------N2

---- SrutyAntara



S 100
r1 105.3497942
r2 106.666666
R1 111.1111
R2 112.5 (Suddha)
g1 118.518518
g2 120
G1 125 (Suddha)
G2 126.5625
M1 133.333333 (Suddha)
M2 135
m1 140.625
m2 142.3828125
P 150
d1 158.024691
d2 160
D1 166.666666 (Suddha)
D2 168.75
n1 177.777777
n2 180
N1 187.5 (Suddha)
N2 189.84375

How do we reconcile this with 72 Melakarta Scheme?

Please also refer to http://www.theveena.com/veena/frets-sruthis.html on 22 Shrutis

[SrinathK]

It's easy. As I explained above, there are only 12 notes all along. But you need 22 tones if you want to sing a 12 tone scale from any point. Leaving aside the old theory for a minute, it is easy to see with some math and excel crunching and modern harmonic theory that no other set of frequencies work so well for all positions.

Now for the exact answer, depending on the harmonic relationships (S-P, S-M and S-G) between various notes in the scale, only one of the 2 frequencies for a particular note is chosen (either N1=187.5 or N2=189.84 for kakali nishadam, but not both at once). This means that there will only be max. 12 tones in use in any particular scale out of the 22. This is how Dr. Oke decides which frequencies to use for Hindustani scales.

http://www.22shruti.com/research_topic_55.asp
http://www.22shruti.com/research_topic_57.asp

BTW, veena frets as of today are in fact close to the ET system of tuning!

Strictly speaking, Melakartha is a much later scheme than the ancient Indian system.

[vgovindan]

Is it possible to delineate (mathematically) minimum distance between two successive notes to obviate dissonance?
Can M also have 3 variations instead of only 2 as in 72 mElakarta scheme? According to Dr Oak, this seems to be possible.

According to book 'gAnAmRta varNa mAlikA' by AS Panchapakesa Iyer, there are a total of 483 janya ragas for each mElakarta - taking into account sampUrNa, shADava, auDava bhEdas. Therefore, he comes up with a figure of 34,776 janya rAgas for all mELakarta rAgas put together. However, Dr Oak says only 5000 rAgas in CM. Is he talking only about sampUrNa rAgas in 22 Sruti scheme?

[SrinathK]

1.957 cents (1 schisma) (on SOME intervals in Sruthi bhedam, most are perfectly in tune) is more than good enough for all practical purposes. Mathematically it is the smallest possible error.

I have played chords in a java appelet using cents to fine tune the intervals. The difference between a pure interval and a schisma is almost too subtle to notice in the case of major thirds, fourths and to some extent fifths (just a slight amplitude fluctuation, can't call it dissonant) to a well trained ear, but you'd never know in other intervals. On a 1m string, the error would be less than 0.2 mm which is really beyond human limits of precision.

On the other hand, ET with more than 11-15 cents of error is sometimes jarringly dissonant and still music has gotten away with that for centuries now.

M would not have 3 variants. Although Dr. Oak uses M1, M2, m1 and m2 for the 4 possible frequencies (both prati and shuddha madhyamam inclusive), only one of them will be used at any time (81/80 is a VERY dissonant interval of around 21 cents).

Also M and m would not come together in a scale unless the it is a dwi-madhyamama and uses both shuddha and prati madhyamams. But even in such a case only 1 frequency for each madhyamam would be valid for that scale.

[SrinathK]

I don't know if Dr. Oak's analysis is applicable to present day Carnatic scales. As far as ASPI's analysis goes, while 34,776 SCALES are possible, evolved ragas are phrase based and not scale based (ragas which are scalar are much more restrictive), so there will be considerable overlap. This analysis also does not cover ragas with multiple scales like Nattakurinji or Basant Bahar or dwi madhyama ragas like Behag, Sindhubhairavi or Hamir Kalyani. It doesn't even cover ragas with vakra phrases in their aro and avarohanam either like Bindumalini or BegaDa or Amrita Behag or even ragas like Darbar or Reetigowla which demand 2 Ga and 2 Ni.

So the melakartha-janya system has it's own limits. Defining ragas to arohanam-avarohanam scales is a severe limitation on many ragas. You want to use dhattu phrases or janta phrases in one raga only to find out that those phrases are critical to another raga. So that suddenly limits your scope of possible phrases. I mean just look at The chittaswaras in the Pancharatna krithis -- where will combinations like that come if ragas are simply seen as arohanam-avarohanam based?

[vgovindan]

Srinath,
Thanks for your detailed responses. Thanks for your time.

[Rsachi]

[vasanthakokilam]

Srinath; See if I got your point about 22 sruthis being useful with sruthi bedam with the following example.

If we shift to Komal Rishab, and play Shuddha Madhyam the ratio comes to 1.0666 * 1.3333 = 1.4221
But prati madhyam is only 1.4062 but Teevartama Madhyam is 1.4238 which is much closer, so that is the one to use than prati madyam.

Did I get that right?

[sureshvv]

Srinath; See if I got your point about 22 sruthis being useful with sruthi bedam with the following example.

If we shift to Komal Rishab, and play Shuddha Madhyam the ratio comes to 1.0666 * 1.3333 = 1.4221
But prati madhyam is only 1.4062 but Teevartama Madhyam is 1.4238 which is much closer, so that is the one to use than prati madyam.

Did I get that right?

Is that the prati madhyam of the equi-tempered scale or the 22 sruti scale?

Anyway I am also not clear on this. Can Srinath explain why 22 is better for graha bedham?

[vgovindan]

An Observation : The nomenclature - ati-kOmaLa, kOmaLa, Suddha and tIvra kOmaLa uniformly for R, G, D, N sound more clear than CM's different nomenclatures -Suddha, catuSruti, shaTSruti for R and D - Sudha, sAdhAraNa, antara for G - Sudhha, kaiSiki and kAkaLi for N. I am sorry if someone feels offended.

However, I am yet to understand as to how the four-fold division - ati-kOmaLa, kOmaLa, Suddha and tIvra - are to be equated with three-fold division of CM.

[vasanthakokilam]

vgv, I agree that the HM nomenclature is quite uniform and is easy to keep track of. For CM I am perfectly content with R1, R2, M1, M2, D1, D2 type scheme, the higher the number, the higher the cousin swara ( e.g. R3 > R2 > R1 )

The so called three-fold division of CM is really only two fold.

Because, R3, G1, D3 and N1 are not separate swarasthanas but aliases as follows

R3 = G2
G1 = R2
D3 = N2
N1 = D2

Why that is so is quite fascinating and it goes back to a very clever trick by Venkatamakhin to accommodate the vivadi ragas. If you have a vivadi raga that goes like 'S R1 R2' that is not acceptable ( how can you have two R s, that defies convention, so was the thinking I suppose ) but you rewrite it as 'S R1 G1' ( because R2 and G1 are aliases and hence substitutable ) then it is fine since the raga uses only one R and one G. But this theoretical device does cause confusion.

[rshankar]

VK - it must take the genius of Apple's autocorrect to declare the vivAdi rAgas as vivid, right??

[vasanthakokilam]

Nice, Ravi. Those who like vivadi ragas will sure agree with Apple!

I noticed it only after you pointed out. I recently switched to Safari from Chrome (Chrome is such a battery hog ) and Safari auto correct is in full swing

[arunk]

An Observation : The nomenclature - ati-kOmaLa, kOmaLa, Suddha and tIvra kOmaLa uniformly for R, G, D, N sound more clear than CM's different nomenclatures -Suddha, catuSruti, shaTSruti for R and D - Sudha, sAdhAraNa, antara for G - Sudhha, kaiSiki and kAkaLi for N.

Our nomenclature is a "remnant" (and I believe confused remnant) of the system that preceded it i.e. the grama-murchana system in which the 22-sruthis played a part, where each swara had a certain allotment of sruthis (and they added to 22). One big aspect of confusion comes in the today''s swaras with the "suddha" prefix (which I think started with ramamatya and continued with venkatamakhin), which if interpreted to mean the positions of the original swaras of Sadja-grama which indeed has been referred to the suddha swaras, is 100% wrong (i.e. today's R1, G1, D1, N1 would not tee up with the ones in sadja-grama which supposedly was closer to kharaharapriya scale).

Yes and to be irreverent, I do agree that our nomenclature is mess in more than one aspect ! (different prefixes for different swaras, incorrectly applied prefixes, you name it, we got'em all )

Arun

[arunk]

Some explanation of chatshruthi, and SatSruthi etc.

Original ri and da swaras of sadja grama had "3 sruthis". It wasn't necessarily the position was 3 sruthi from sa and pa, although that is an interpretation. Early texts like dattilam mentions a swara as containing "3 sruthis', "2 sruthis etc.

In that original system, it is not believed that "sa" held the position of always the tonal centre as it was later. It was at the same status as the other swaras. In both sadja grama and madyama grama, there was only one ri and one ga, and these were revered and so referred to Suddha swaras (pure?). (later texts e.g. by Sarngadeva's time do talk about other variations)

Later (perhaps 11th-14th century?) when grama system merged and fretted veena came into being, and also sa emerged as tonal center, the first position after sa was a ri which was at the position of R1 (e.g. of MMG). This now got suddha-prefix (because it was the first ri position) and got assigned 3 sruthis. Same for D1.

So the next ri position i.e. today's R2 got 4 sruthis possibly because the math does indicate that even in grama system a a full-step i.e. 2 semitones was 4 sruthis, and this ri was a "full step" from Sa. And because of 4 sruthis, it got the chatuSruthi prefix, and the next position R3 - same as G2, got 6 sruthis (4+2, i.e. one half-step from R1), and thus got SatSruthi prefix.

These same applied to dha.

The terms sadharana and kaisiki for nishadham I believe were there in older grama system itself. So is the term antara (as "other" gandhara). So to not be iconoclastic, I guess these terms were adapted and followed.

Arun

[vgovindan]

If you have an avorohana for a vivadi raga like 'S R1 R2' that is not acceptable ( how can you have two R s, that defies convention, so was the thinking I suppose ) but you rewrite it as 'S R1 G1' ( because R2 and G1 are aliases and hence substitutable ) then it is fine since the raga uses only one R and one G.

That is strange! Then what is apasvara? I was under the impression that two variations of same svara sung consecutively produces apasvara. Though Dr Oak explains apasvara as 'besur' or 'asur' pointing to those areas which are the interregnum between two Srutis (svara-sthAna). According to him, when you 'stay' in those prohibited areas, the ears perceive besur - am I right in my understanding? However, I also feel that apasvara is perceived when a wrong svara is used in a rAga - for instance R2 instead of R1.

Next, am I correct in assuming that ati-kOmal and kOmal are two sub divisions of one svara of CM (R1 or G1 or D1 or N1)? Similarly, are Suddha and tIvra are two subdivisions of another svara (R2 or G2 or D2 or N2)?

Arun,
Our posts coincided. I shall study what you said.

[SrinathK]

Earlier I had posted the full excel file in a related thread in the technical discussions sessions, but as I feared, it was WAY too much info at once. So I will explain just one particular case of graha bhedam at one time here.

So first of all, let me shift the shadja to the position of R2 (as per Dr. Oke's convention). I am taking this as an example as this R2 is aligned perfectly to the pa string, therefore it would be easy to do sruthi bhedam on this note. Now take a good look at the image of the chart in the link below :

https://www.dropbox.com/s/73nlz8916zef5 ... e.PNG?dl=0

Column B gives the labels of all the 22 tones and the Column C gives the values of each of them in %.

The yellow cell at 100% is the location of the shadja which, in the 2nd column, is at S (normal). In any column, start on the yellow cell and read downward.

In column G, as you can read at the bottom, the base Sa shifted to R2. Now the yellow cell at 100% has moved to the position of R2 (Cell # G5)

Now when the Sa shifts to R2, you would expect that r1 --> g1, M1 (shuddha madhyamam) --> P, etc. As you can see, The ratio between P and R2 = 150/112.5 = 133.33 which is the value shown in cell G14. Now that R2 is the base, every interval got divided by 112.5 to give you the values in column G.

You can read down to the bottom, then LOOP back up to the top (like a game of snakes) to read the rest of the values.

As you can observe, most of the ratios are perfectly preserved during the graha bhedam and these are shown in the white cells.

But look at the ratios in the blue boxes -- these suffer from a very slight mathematical inaccuracy of 0.2% . This is because 160/112.5 = 142.222 (in fractions = 64/45) which is very close to, but not exactly 142.383 (= 729/512). This is the (almost imperceptible) error I was talking about. This is why some musicologists have quoted 64/45 for the prati madhyamam (but didn't explain why).

The ratios in the red boxes are too inaccurate to achieve any proper alignment. So, they are discarded. These will become APASWARAS. This leaves you with 20 tones. If you look closely in column G, you lost the intervals of 135% (M2) and 180% (n2) and you will not find these values in Column G.

But you can make a perfectly good 12 swara scale with the remaining 20 tones with no problem. The ratios that you need are still there.

Similarly in each graha position, you will sacrifice some of the 22 tones (not the same), and you will have to work with the others. This means that each time you do graha bhedam, you get a scale whose TONAL spectrum and capabilities will be slightly DIFFERENT from the original.

The only position where you can't get all 12 tones is at the dissonant M2 position. There is no panchamam for this (If you want it, you have to go for the 2nd value of Sa from the original 24 tones, the S which was discarded as it is dissonant). It's doubtful if it will be used for graha bhedam at all when the perfectly aligned M1 is available. But then could this have inspired scales devoid of panchamam?

An actual example of this is in Mohanam -- shift S to R2 and you get Madhyamavati. But if you felt that this madhyamavati sounds a bit different, you're not alone. Part of the reason why ragas sound a little different from normal when they appear in shruti bhedam is because their tonal spectrum is indeed slightly different -- you have to use n1 as you don't have n2 for the position of the kaishiki nishadam anymore.

Similarly, you might have observed that the m2 slightly changed from 729/512 to 64/45. In my analysis I found that every interval can slightly shift by this small amount (in different positions of course).

In ancient Indian music, these subtle tonal differences could have been used to give the ragas of that time their individual "flavour". Dr. Oke is still using it. http://www.22shruti.com/research_topic_38.asp and http://www.22shruti.com/research_topic_38.asp

I have read somewhere that a similar device existed in J.S.Bach's time or earlier. This meant that a C Major would sound different from an F major or a D major (because the key change was achieved by graha bhedam) and this is something that predated the popular adoption of Equal temperament. This system however had some limitations because of the necessary additional tones apart from the fundamental 12 and so it gave way to Equal temperament.

Next time I will talk about the limitations of Equal temperament. Also I see that a lot of you are mixing up the ancient Indian system and symbology with today's more modern system -- it's a recipe for all sorts of confusions and it is also a fact as @arunk pointed out, that we still stick to the remanants of an older system that we no longer use. Indeed I agree that the HM terminology has better clarity here. I will only discuss this topic using Dr. Oke's symbology. Amazingly the 22 shruti system is more relevant to modern harmonic theory and applications than to today's CM system!

[vasanthakokilam]

I was under the impression that two variations of same svara sung consecutively produces apasvara.

That is the vivadi sequence. 40 of the 72 melakarthas use one or more of such a sequence in their Aro/Ava definition

Using R2 instead of R1 type mistake is just using the wrong swara for that raga.

The commonly held notion of apaswara is when one does not sing with swarasuddham ( clear example is a long drawn karvai on a plain note at the slightly off swarasthana. But even in passages of flat notes, if someone is not quite hitting the right swarasthana, the apaswara is going to show )

[vasanthakokilam]

Srinath, thanks for the detailed write up.

I think I understand except for this. But this is most probably due to some mistake I am committing.

But look at the ratios in the blue boxes -- these suffer from a very slight mathematical inaccuracy of 0.2% . This is because 160/112.5 = 142.222 (in fractions = 64/45) which is very close to, but not exactly 142.383 (= 729/512). This is the (almost imperceptible) error I was talking about. This is why some musicologists have quoted 64/45 for the prati madhyamam (but didn't explain why).

I am using the following general procedure: To find the equivalent swara on column G, go up by 4 slots on column C.

If I do that, your two blue slots on line 15, 16 match up like this

140.466 -- 135
142.222 -- 140.625
This does not match the correspondence between 142,222 and 142.383 you mention above.
What am I doing wrong?

(Similarly
2 and 3 blue slots match up like this
187.289 -- 180
189.63 -- 187.5

and line 17 and line 4 red slots

148.148 -- 142.383
197.531 -- 189.844
)

[SrinathK]

@vasanthakokilam What am I doing wrong?

Ans :

I am using the following general procedure: To find the equivalent swara on column G, go up by 4 slots on column C.

The error here is that you assume that the shrutis are evenly spaced (they are not). The logic that worked well for MODERN Carnatic swaras and the logic of Equal temperament (Shift everything up by 1 tone or 1 semi tone), but it will not work here.

I'll show how it is done manually :

1) When S shifts to R2, 112.5 becomes the new 100%. 112.5 / 112.5 = 100%

2) Assume r1 will shift to g1. Proof : 118.519/112.5 = 105.35 % (which is the value in cell # G6). So old g1 is the new r1.

3) r2 will shift to g2. 120/112.5 = 106.667%. What I am doing is, I am taking the value of each of the 22 intervals and dividing by the value of R2. The new ratio will tell me how the old interval has changed. It is the result of the calculation that tells me how the intervals have shifted.

4) What happens to P at 150% ? 150/112.5 = 133.33% Old P is now the shuddha madhyamam (M1) for R2 (as expected).

5) Now come to the next interval d1. What happens? 158.025 / 112.5 = 140.466% ~ 140.625% (The value of m1). Now the schisma (microscopic error of 1.957 cents) appears in this interval, so it is labelled in blue. Keep in mind that this error is truly miniscule -- even a mild vibrato is > 20 cents and a good vibrato may span as much as 40-65 cents!

So observe closely. After 133.33%, the next value is 140.466%. The value in between at 135% has vanished! That means M2 (teevra shuddha madhyamam) has been lost in carrying out graha bhedam in the new scale !

Why? Because the gap between M1 and M2 is 1.125%. The gap between P and d1 is 5.35%. The gap between M1 and m1 (prati madhyamam) is 5.469% (1 schisma difference). Therefore the scale will naturally jump from M1 to m1 bypassing M2 altogether. The number of shrutis lost in a graha bhedam is equal to the number of red squares.

6) After d1 comes d2. Since old d1 is aligned to m1, old d2 is aligned to m2 as the gap between both pairs of intervals is the same 1.125%

7) Now what happens to Old D1 ? Do the math and you get : 166.667/112.5 = 148.148% ! This is the value of the lower P' (chyuta panchamam) that appears only in the ORIGINAL 24 tone scale. Now we've already decided we are not going to use this value in the 22 shruti scale, so I labelled the cell red. Note : This does NOT mean we have lost NEW D1 since New P' doesn't correspond to any interval in the 22 shruti scale, so discarding it is no issue.

Note that if you are willing to switch off the tanpura and go full western, there is no adhara shadja reference that is stopping you from using ALL the 24 tones anymore. I have a separate excel sheet for the 24 tone scale where this cell will NOT be marked in red. But later on that ...

8) Come to the last interval at R1. Now the value is 111.111/112.5 x 2 = 197.531 %. I multiplied by 2 since by this point, you will be in the next octave of the original scale. Every time you reach the bottom of the table and loop back to the top you will cross an octave. I did this because it was very convenient to do this in excel.

This value 197.531% is not found anywhere in the 22 shrutis, so it is an invalid interval and gets red-labelled. By this point you should know that R1 and R2 are dissonant.

9) If you want to find the position of new N2, you should try old r2. You get 106.667 x 2 / 112.5 = 189.63% (A schisma is here too, so that square is in blue).

You might have observed that after 177.778 (n1), the next interval on the new scale is 187.289 (N1). There is no new n2 at 180% for the same reasons as before. So all in all, you lost 2 tones, but you got enough possibilities for a 12 tone scale with none of the errors of Equal temperament.

I will discuss Equal temperament for the next post. I know I haven't answered the query as to why 22 tones is better than 12 tone ET yet, but before I come to that, I hope everything is Ok now.

[vasanthakokilam]

Got it, Srinath. Thanks.

[vgovindan]

https://www.youtube.com/watch?v=AHJTqbHhLds

Please listen to 6.0. about Eka-Sruti Rishabham. How is it justified to say that this (Eka-Sruti Rishabham) does not have a Svara-sthAna?

[arunk]

VGV Sir

Think of this "eka-sruthi rishabha" as a label for a particular flavor of ri that has an oscillation with sa and comes in say saveri, and gowlai etc. Do not confuse it with the assignment of first sruthi positions in any paper on 22 sruthis. As Srinath says those positions do not find much use in music practiced today and also do not pay too much attention into the literal meaning of nomenclature in use today because that can be misleading and that is due to terms being "a nod" too earlier system.

This gamakafied ri is called eka-sruthi is because (even though it is not held steady and thus is a curve) people sense it having a lesser pitch compared to the steady R1 which you may encounter in other ragas (this is mentioned in the demo also) and the mayamalavagowla mela ri is unofficially assigned dvi-sruthi (again half step roughly 2 sruthis as per nod to earlier system). So perceived lower pitch (of a curve) than 2-sruthi R1 and thus eka-sruthi. In this regard my earlier post is wrong in that first ri *today* isn't assigned 3 as earlier but ramamatya did assign it 3 sruthis, and I think his first ri is today's r1 (yes confusion reigns supreme )

To reiterate what Srinath says, when dealing with sruthi positions in any 22-sruthi related paper/discussion - safe to ignore similar/same terms in actual use today - they are NOT the same.

Arun

[vgovindan]

Arun,
It is nice of you to have responded to me. Somehow, I feel there is too much of ad-hocism in all these. As Dr Oke (at 7.0) asks in his demo, 'what is this zarasa' (zara is the Urdu word for 'a little') pointing to the current usage.

In the present case also, the demonstrator says there are 22 svaras. Obviously, gamaka seems to take precedence over svaras, while gamakas do not seem to fit into any verifiable definition. That is the problem.

Sorry for my foolish questions.

[arunk]

vgv sir.

Not that you may be thinking this way, but let us not club this one as "current practitioners" (post saint composers) have somehow taken a perfect system and trashed it due to falling values, kali yuga etc.

No the confusion started way before trinity. It started at the very beginning of mela system and who knows perhaps even earlier

There is a time-hole in our music history - the time when old system (Sadja-Grama/Madhyama-Grama, 22-sruthis, no sa as tonic) went away and new system emerged (no gramas, sa as tonic, fretted vina, and early melas). We are not sure how that transformation happened, and I am not sure we know when and where also. We have Sarngadeva (12th century) for the first, then a hole, and then we have ramamatya (16th) (and vidyaranya I think a bit earlier). The latter treatises have remnants of connections to earlier system but they have seemingly logical inconsistencies.

PS: (My 2 cents) The trouble is theory should always reflect practice and in reality, practice is organic and ever-changing which implies theory is bound to become at odds with practice over time unless theory evolves. But in a system where adherence to Sastras is considered literally and figuratively religious, you get dogged attachment to tradition, and you get confusion of wrongly attaching evolved/morphed practices to old theoretical principles. Some guy writes a treatise of theory as he sees it (i.e. based on practice around him, and based on what he thinks is right/wrong which of course is based on religion in our case), and this unleashes an army of protectionists for generations who think everything else must be wrong. Even in pre-ramamatya and supposedly in the time of old system (or perhaps during the transitional phase) there were rumblings about deviations. For example, commentator on Sarngadeva's work (I think 13th century? I forget) has noted that nattai uses a rishaba that has 6-sruthis and this is not mentioned in theory and "how is this right", he also notes what is later known as pratimadhyama (2 sruthis higher than the only madhyama mentioned in theory) which wasnt codified also.

Arun

[vgovindan]

Arun,
My whole approach has been from the point of view of a student of CM. We seem to say that HM has gharAnA system and there is no standardisation. Well, as I dive deep, I find CM is no less - every parampara seems to have their own versions. That is alright till there were private music schools. Now that we have Government sponsored schools for CM, I wonder as to what standardised teaching these students are undergoing. Well, in order to elicit intelligent answers, one should ask intelligent questions. Unfortunately, I am not competent to ask such intelligent questions; there are many intelligent answers from you, Srinath and VK et all, though. Well, someday, somebody will bell the cat.

[arunk]

VGV Sir,

Then I must admit that I have no idea how this tees up with a lec-dem on 22-shruthis - if someone is entertaining the notion that we should standardize today's music based on 22-shruthis, there are no cats and bells, there is only dog-barking and a wrong tree. I will stay off this topic now.

Arun

[vgovindan]

Arun,
I seem to have conveyed my views wrongly. I am neither a proponent of 22 Srutis nor can I say otherwise, as I have admitted that I am not competent. Kindly do not take me wrongly.

[arunk]

VGV Sir -

I did not. I am indeed genuinely puzzled as to how your questions about nomenclature tees with standardization.

Perhaps this may help. As wrong as they are in relationship with 22-shruthi nomenclature, those swara names in use today for our 12 swaras are indeed standardized across the board within CM world. If I say antara gandharam, or chatusruthi rishabham or Satsruthi rishabham, even eka-sruthi rishabham (not a swarasthana, but a label for a particular flavor of ri with gamaka), all in CM world (who has been exposed to all these teems) would know exactly what I mean. There is no confusion. The confusion arises only when you look at how those names came about and try to match it against things of historical value.

Hope that helps (?)

Arun

[vgovindan]

Arun,
Thanks.

[SrinathK]

Earlier, I too was of the opinion that the 22 sruthi postions may have something to do with the higher and lower variations of swaras that are seen in different ragas (I even made a post trying to guess what tones may be used in some ragas sometime before, only to rue it afterwards). Frequency analysis has clearly shown that the 22 shrutis have virtually no role in gamaka oriented phrases. In fact even in the same raga, the gamakas vary slightly from musician to musician and from instrument to instrument and school to school.

The gamaka may very well be the ship invented to explore the sea between the lands of consecutive shrutis, only for it to blur the boundary between land and sea altogether.

To call the rishabha of Saveri as a shruti is really a misleading term. It is in fact a moving phrase, as the video that @vgovindan very correctly shows. It is no note. It is an oscillation labelled as "Ri". More precisely, you are actually shaking the Sa and labelling it Ri -- so "Sa"-"ve"-"Ri" aayiduchu. But alas, horror of horrors, a few seconds later it is called an ekashruti ri, which actually has nothing to do with gamakas !! (Ok just kidding). As @arunk puts it, the problems all start when you try to understand the meaning of these swara labels (which have long since lost their original meaning) and try to fit it into current practices.

I have always felt, that to use the terminology (and the logic) of the 22 shrutis to today's swaras in CM (chatushruti rishabham for e.g.) has no real meaning at all. It is just a label and we use it because everyone else uses it, nothing more. That is an older time, a note based music if I should say. What we have today is PHRASE based music where swaras are merely labels for the phrase sung on them.

While some old authors tried to assign shrutis to the notes of various ragas, in my opinion this is just a symptom or a reaction to the evolution of the art and the rise of gamakas -- a transitory phase of confusion -- an attempt to fit old theory in more modern music. I don't think it is far from the truth to think that those musicians and musicologists of the past were probably just as confused (or way more confused, lacking today's advanced tools) than we are and were trying to reconcile the old ideas with evolving practice of their time and trying to understand vague and ambiguous terms and passages in even older books. We are still doing it occasionally. Fortunately, looking at this topic from the viewpoint of modern physics has clarified matters. Today we are finally at the point where we can say that the old system no longer exists. Many statements in old books are rather ambiguous and confusing -- so rather than going that route, we can turn to modern harmonic theory and then we see many things becoming quite clear almost immediately once we are free of the headache of interpreting vague passages in treatises. It becomes quite clear from simple math and acoustics as to where this number 22 came from in the first place.

I myself tried to identify shrutis in today's raga swaras to find out which was higher and lower only for frequency analysis to tell me it was a totally pointless (and misleading) exercise. I don't have any doubt in my mind that musicians and musicologists of those times were also driven by a similar urge to reconcile theory with practice and whatever statements they made (Like a gamaka being called a shruti) were the result of the ensuing confusion.

We Indians believe music is divine in origin, but our IDEA of "divine" is synonymous with "old" "traditional" "infallible" "rigid" and "eternally perfect" and as soon as we believe these ideas our own intelligence turns off whereas the EXPERIENCED reality that is divine music shows a completely different history of one that is forever evolving, creative and rediscovering more variety and wonders within itself.

Also Dr. Oke is a Hindustani musician first, and the positions of the 22 shrutis are still relevant to the plain notes of HM and the harmonium.

[vasanthakokilam]

Srinath,
For completeness, can you post another table table like the 22 sruthi table with only the ratios for the 12 swarasthanas that are currently in practice. I keep going to different sources and since no one follows a uniform nomenclature of, say R1, R2 and R3 and also whether they really meant by 'r' or 'R' I am not really sure which ones they refer to. ( also, as a sanity check, point out which 12 rows of the 22 row table are today's swarasthanas )

I get what you and Arun are saying. This discussion gives me some clarity. As you two said'chuthusruthi Rishabam' as used today should not be given any connotation of 4th of anything, it is just a label. So is ekasruthi rishabam.

Interestingly, our tradition ( more so in the south than north ) seems to have figured out Psychological Perception of Swaras as the basis of raga definition and aesthetics.Like you oscillate between Note H(igh) and Note L(low), the resulting effect can standing for a swarasthan that is between H and L. We see this Thodi G2 which is a psychological perception since in reality it is an oscillation between M1 and R2.

Similarly Kanada Ga and Darbar Ga. ( what are the high and low notes for these two? ).

What other such swinging notes exist which have this 'Psychological Perceptive Middle' qualtity?

True a lot of HM prayogas use plain notes to great effect ( at least in the way I perceive it as without gamakas ) and the gamakas in predominant use is the slide ( Etra jAru and irakka Jaru ). Though some extreme ones does not sound good to my ears, HM just owns that M to G2 slide and they exploit that to great effect in so many ragas. i guess there is similar such prototypical slide in the uttaranga. Do they have any use for sruthis in effecting the jAru or they start from and in one of the 12 swarasthanas.

Also Dr. Oke is a Hindustani musician first, and the positions of the 22 shrutis are still relevant to the plain notes of HM and the harmonium.

I am not clear on this. I thought Arun was saying that these sruthis were useful at the grama times when there was no fixed tonic. HM is a fixed tonic system and why/how are those sruthis useful for them? Since they use a lot of plain notes, can we actually do a frequency analysis and see if they fall on the sruthis?

I have a question on this whole issue of Swarasthanas, sruthis and swarasthana bedam and I will pose that later after I get a bit more clarity.

[chitravina ravikiran]

I had come across this excellent presentation by Dr Oke a few months back.

In CM, shruti values have to be contextualized a bit differently, since we also use oscillated notes. For centuries, many of these oscillations were confused with shrutis by many CM musicologists, leading to incorrect classification of shrutis (like Begada Ma and Ni, Gowlipantu Ma) which I highlighted in my lec-dem in Music Academy in 2012.

http://www.rasikas.org/forums/viewtopic. ... chitravina

It is extremely important to note is that CM and HM evolved to their present states millennia after 22 shruti theory was proposed. And even the concept of Raga came in much later. Thus retro-fitting 22 shrutis into what we understand as notes in ragas is always bound to have limitations. That said, a blend of scientific & aesthetics can give us a glimpse of most of the values discussed in books, as can be seen in this audio link of that lec-dem.

https://www.dropbox.com/s/11a3gfmmeofju ... ravina.zip

I also proposed that a distinction be made between apa-swara and apa-shruti.

[arunk]

I am not clear on this. I thought Arun was saying that these sruthis were useful at the grama times when there was no fixed tonic.

vk - I said more useful in grama times, but I did not say because there is no fixed tonic then

I simply said the terms chatusruthi, Satsruthi etc. labels for swaras (particularly ri) came later and that was perhaps because sa as tonic came about by then.

Arun

[SrinathK]

@vk, I don't think there was ever a time when Indian music didn't have a tonic Sa. It wouldn't be Indian music otherwise.

And it's an honour to have Ravikiran sir here to contribute to this thread. That's way beyond anything I expected. In the future, when I do get some time and privacy, I'd sincerely want to learn music from him.

Now in the old system of Indian music, there seems to be a precise definition for a wrong note. If the error was any greater than the schisma of 1.957 cents, you can be mathematically defined as out of shruti. You can't do that in today's Carnatic Music -- one's ears are still the best judge.

@vk, Since you asked for the most popular plain notes in today's music, I'll tell you what. I can certainly say that the just intonation intervals like Sa at 100%, Pa at 150%, G at 125%, M at 133.33% and R at 112.5% as being universal as they are the easiest to align by ear. Others, not so much. I know that R1 at 111.11% is not so popular and I don't think anybody uses M2 at 135% or G2 at 126.56% as they are all hard to hit (R2, M2 and G1 are much easier to align to as they are natural harmonics) and almost no one has the precision to hit such small differences accurately (The difference between R1 and R2 is just 21.5 cents and even a small vibrato is larger than that, the slightest quiver in your voice would blur the gap, what to speak of gamakas). As for all the other notes, I cannot truly say which ones aren't in practice.

But there is a major problem in the question (something I realized only now). See, most of us only use plain notes in our basic classes (where we still can't sing any note very well) and before you know it you are in the world of gamakas. Quite a few of us would have used the keyboard or harmonium to guide our plain note singing ensuring that the only plain notes we ever saw were those of 12-tone equal temperament. and our tonal ear for plain notes never quite developed beyond that point.

In fact for the semitones like r, d and N (as per Dr. Oke's convention), the human ear, even among expert musicians is not a consistent barometer. However if you want to know what are the places where shruti bhedam is carried out, I'll tell you that they are all on the just intonation positions. My excel sheet therefore contains the analysis of graha bhedam on the original 24 tone scale as well as the 22 shruti scale on ALL positions and the data tells me clearly that the 22 shrutis give you enough material to do graha bhedam with an accuracy far superior to that of Equal temperament.

There are some other intervals like 25/18, 36/25 which are not shrutis despite their use in just intonation, because they don't pass the graha bhedam test. You lose too many tones at these points to be able to sing a full scale.

[arunk]

@vk, I don't think there was ever a time when Indian music didn't have a tonic Sa. It wouldn't be Indian music otherwise.

Srinath as befuddling it may seem, actually there is no evidence among the multiple texts (Natyasastra, Dattilam, Braddesi Sangitaratnakara etc.) to support the theory that Sa was the beginning of every "scale" (using it loosely) or raga. Sa was first swara yes and it was so with Sadja grama. But not with madhyama-grama which was equally important - for it ma was first - its murchana was ma-pa-da-ni-sa-ri-ga-ma. Note that it seas not that "ma" of sadja grama was held as "sa" in Sruthi bedham.

Now there is some text that tells how one can derive madhyama grama from sadja grama with a concept similar to sruthi bedham but again it was not plain sruthi bedham as the two grams did not have that relationship in terms of precise spacing of sruthis. So you do have to do a tonal shift of sorts but then you have to adjust on top. This can be (and has been) construed that they know about derivation of scales following sruthi bedham like techniques, but the texts do not say explicitly so, and they most definitely do not say you move "sa" to that new position, and you use "Sa" as starting swara.

So while they may have certainly had a tonal centre for melodies like we do today (in fact who knows, it could have been multiple like with WM), there is not much evidence to support that they always assigned that tonal centre as "Sa", nor did they express the scale/murchana as every raga with sa as starting and indispensable swara. They expressed it starting from ri or ga or ma etc. - they even had ones which didn't have the sa swara.

Arun

[SrinathK]

Before I continue to discuss ET (I want to do so, but the topic is interesting enough where it is), I want to share where I got the inspiration to do all this. It all started in Music club when we had a discussion over Shruti Bhedam and there was an argument that it is invalid as it is harmonically incompatible and there were others both for and against (most regurgitated from the days of the GNB saga when he introduced it in concerts) came out. Some part of that history came out as well in the process and I could gather that it had been a controversial matter with furious debates and GNB had taken quite a bit of hurtful flak for it. But there was no real concrete proof -- same old "traditionalists" views on "innovation" is never going to satisfy me.

That's when I wanted to dig deeper into the matter. Along the way, stumbled across the 22 shruti topic eventually leading me to Dr. Oke's site. That's when I realized that the 24 tones arise naturally DUE to the process of graha bhedam from Sa. A lot of reading and excel crunching later, I finally have the proof that while people were right in saying that 12 tones aren't good enough for graha bhedam, the ancient Indian system of 24 tones are perfect for the job. 22 of them are usable in practice when a tanpura (or any pitch reference) plays in the background. And if you can do it with rigidly fixed plain tones, there are no issues with gamakas also. Q.E.D.

@ Arunk, I am not too aware of what's there in old books. My goal was to find a mathematical basis for graha bhedam and somehow I stumbled on the conclusion that the ancient Indian music system was in fact the best possible fit. I couldn't quite get what was said in old books at times (e.g. I don't get the particulars of the procedure you mention. I think I know what it is, but I need a clearer description. There are many such confusions and I know that even now most of us aren't too sure of what those authors meant). So I turned to modern harmonic theory and physics to get the answers. It did free my mind from the headaches of interpreting old texts and conflicting views and allow the math to explain itself.

Having read your post, I think our visualization of shifting sa to a new location (co-ordinate transformation) is a tool we came up with to make the process easier to understand, possibly sometime after Sa as the first swara became the dominant tradition. Probably... as we really have very little documentation of the evolution of our music.

I felt a bit sad for GNB when I finished my number crunching. He had the right idea all along -- but the result of my excel sheet told me that the truth was more complex than what I had thought.

[vgovindan]

Kalidasa - said to belong to 5th Century AD has written a stotra on Syamala Devi -
http://sanskritdocuments.org/doc_devii/ ... ml?lang=sa

He says -
सरिगमपधनिरतां तां वीणासङ्क्रान्तहस्तां ताम् ।

This topic is being discussed in a parallel thread -
http://www.rasikas.org/forums/viewtopic.php?f=33&t=21623

Please also refer to Dr Oke's further research on Shadja and Madhyama grama -
http://www.22shruti.com/research_topic_29.asp

[arunk]

Yes VGV Sir. This makes sense - both Natya Sastra and Dattilam pre-date kalidasa. Sadja grama was the main one - and its order of was indeed sa, ri, ga, ma, pa, dha and ni.

Arun

[sureshvv]

I had come across this excellent presentation by Dr Oke a few months back.

In CM, shruti values have to be contextualized a bit differently, since we also use oscillated notes.

But Dr. Oke clearly defines "shruti" as a note where you stay on. I don't think he considers oscillated notes as "shruti". IIRC, he uses the term "a-sur" to denote those frequencies that are glided over.

I also proposed that a distinction be made between apa-swara and apa-shruti.

Is this part covered in the dropbox link? Can you explain?

[vgovindan]

Derivation of 22 Srutis - S Vidyasankar -
http://www.carnaticcorner.com/articles/22_srutis.htm

[SrinathK]

@vgovindan There are a few things that this article doesn't tell you.

1) What is the difference between 64/45 and 729/512 for m2?
Ans : 1.957 cents. The schisma error of 0.16%. In my earlier post I stated that 64/45 is 142.22% and 729/512 is 142.38% so that's why that cell was blue marked. 64/45 might get a blue label in my excel, but these 2 values are actually not distinct shrutis.

2) 40/27 (148.15%) in the 24 tone scale is the lower value of P (which I labelled as P' in my 24 tone chart). Since @arunk was talking about madhyama grama, I should mention that the position of M2 (135%) does not have a panchamam within the 22 shruti scale and it is the only position where a you don't get 12 tones to make a full scale with. You have only 11. If you want a 12 tone scale at this point, you are forced to use Sa at 100% and the interval between them will be 148.15%. You see why M1 is always preferred as it is a natural harmonic, while no one uses M2 these days even on plain notes?

This picture should make it clear where it comes from (See the circled parts) : https://www.dropbox.com/s/vnth80lnrnza2 ... 2.JPG?dl=0

I hypothesize that this was what the ancient Indians were seeking in some manner, they wanted at least 12 tones from all 22 positions (it is only possible on 21 out of 22 positions) -- at least I do -- it makes sense in modern context. It is also likely that the concept of relative pitch (where Sa is the first note whether you are in 1-kattai, 2 kattai, 5 kattai whatever) was still not developed or that the musicians of those times were trying for some absolute pitch reference for Sa (like Western music keeps the A note at 440 Hz). It is also possible that madhyama grama was meant for women and their higher pitched voices (in which case M2 would be SORT similar to women setting their shruti to F or G today). It is also likely that this attempt to standardize the pitches from Sa onward is the reason why till today we talk about shruti as a pitch reference -- "You are singing in 6.5 kattai ? Your shruti is very high!" Perhaps that was why the tones were referred to as shrutis? Were S-R-G-M-P-D-N originally an attempt to create something similar to the fixed pitches of C-D-E-F-G-A-B that are used today? I don't know if any of this is even correct. But it's worth a thought.

One thing I can be sure of. Any attempt to standardize pitches (shrutis) for the swaras in old times would have surely failed quickly. You see, in western music, even after several CENTURIES of evolution and performances, it wasn't until 1955 that the A was first fixed firmly at 440 Hz (and it is still not followed everywhere - e.g. Modern baroque music). Ancient Indian musicians without any technological developments (and the lack of Newtonian physics) would have had a harrowing time finding an absolute pitch for various notes. The human ear cannot identify a frequency with such precision -- what is called absolute pitch is really a perfect ability to imitate the tone in one's head (something which is easier to do as a child). Indian music has been predominantly vocal based since antiquity and every voice is different. I really doubt that without the aid of a tuning fork at least, whether one could tune a violin or a tanpura to the same pitch every time, even today. It would be very difficult in ancient times to find a common standard of pitch with each musician putting for their own ways. Just consider how recently A was set to 440 Hz.

So it would be only a matter of time before this experiment on absolute pitch would be abandoned and relative pitch would take over. Eventually each person would sing in whatever pitch that suited them and that would have been the case ever since.

3) The 24 tone scale on the other hand, does not have the problem in Point #2 as we can use the higher Sa (S') at 101.25% to align perfectly with M2. But really, you cannot sing that note with the tanpura or a pitch reference in the background -- it is far too dissonant.

So why only 22 out of these 24 tones are shrutis? As I said, S' and P' are dissonant Pythogorean commas. Another reason is that the 22 shruti scale is SYMMETRIC in both ascent and descent (This you can read on http://www.22shruti.com/. The 24 tone scale is NOT symmetrical in arohanam and avarohanam even though it allows you to play a 12 tone scale from all 24 positions . S to S' is a change of 1.25 % but Upper Sa to N2 is a change of 5.35%. Therefore our S' and P' might have been something like a "blue note" 7th harmonic in barbershop music at best, but that's that.

4) Once again I see the same issue of people trying to identify shrutis in modern Carnatic ragas. Guess this article was before the era of computer analysis.

5) One final thing -- you know the article talks about people arguing for 48 and 44 shrutis? You know why? Well, I found out where that comes from. You see, some people would love to count all the blue squares in my excel sheet (the ones with the error of 0.16%) and take them as different shrutis altogether. It has no basis as per the old Indian system and it is seeking more theoretical perfection than what even MATHEMATICS would permit . If you do this for ALL 22 shrutis on various graha bhedam positions, you will get 44. Do the same thing in the original 24 tone scale and you get 48. B-)

But clearly this idea is an arbitrary one. At one point we should accept that the laws of numbers will not allow you to get arbitrary precision on all the tones at all the positions. The only question is therefore, how good is good enough. I'd say the 22 shrutis and the original 24 tones are more than good enough.

[SrinathK]

I sense an opening in the discussion so @sureshvv, here's your answer. Wikipedia has a very good article on 12 tone ET -- the most common and popular form. http://en.wikipedia.org/wiki/Equal_temperament

Generally, vocal string and wind sections prefer to tune to Just Intonation (JI) while keyboards and pianos are tuned to 12-tone ET. 12-tone ET won out because of the ability to tune the 12-tone keyboard, piano and other fixed pitch instruments easily, allowing harmonic freedom to to play in all 12 keys at the cost of musical imperfection on every note. (Again the original problem has always been to play a 12 tone scale from AT LEAST 12 positions in the octave, mind you.

The frequency ratio between any 2 notes is constant. It is equal to 2^(1/12) = The twelfth root of 2 = 1.059463%. This is defined as the semitone or a half step. Two semitones make a whole tone or whole step. It is precisely this reason why you can play on any key since the interval between 2 semitones is always going to be constant.

At this point I should introduce the concept of cents. The size of an interval in cents = log2 (f2/f1)*1200. In my case f1 is always 100 and f2 is always the value of the interval I am taking. As per 12 tone ET, the spacing between any 2 semitones by this formula will always be 100 cents.

So let's compare the difference between 12 tone ET and 22 shruti side by side. The picture below is self explanatory. The 22 shruti system gave me only one error, the least possible as per mathematics of 1.957 cents and most of the intervals in graha bhedam were perfect. But ET? Just see the numbers in that last column :

https://www.dropbox.com/s/jjhp8nwdhjdpz ... S.JPG?dl=0

You can see that the errors vary from -19 cents to +17 cents and everything in between. These errors are HUGE, except for M1 and P, they would all be red labelled in my excel sheet. A major third (G1) is off by more than +13 cents. There are no perfect intervals in 12-tone ET. (I admit after having put in the formulas in the spreadsheet today morning, I am surprised at how large the errors are. I knew that 12 ET was not perfect but this was much more than I expected)

The nature of ET means that these errors will persist on EVERY position, in every key, in that same order.

The impact of these errors is quite obvious in this video : https://www.youtube.com/watch?v=p_CBjW2zt1A The error in all these chords is the 13 cents in the major third. I can only imagine what 19 cents sounds like.

Unfortunately since the values of C-D-E-F-G-A-B have already been fixed by Equal temperament, I believe the 22 shruti harmonium wouldn't go along so well with a Radel shruti box or a tanpura app when you suddenly change the pitch to another key (Now-a-days all our kattai values are all ET), unless you were REALLY good at adjusting those knobs. It's a marvelous instrument though. I wonder how much it would cost.

[vasanthakokilam]

I think I can use some help at a fundamental (axiomatic) level. I realize this is how it is done from the times of Pythagorus but the issue still remains.

Let us use Dr. Oke link for this purpose: http://www.22shruti.com/research_topic_22.asp

The rationale for S, P and G outlined in step 1 makes a lot of sense. They are produced by successive modes of vibration of a string as the energy dissipates. And with respect to the higher and higher Shadjam, these harmonics becomes the G and P of a scale with that Shadjam.

Let us consider these sruthis/swaras/ratios as axioms derived from nature.

Ma can be derived from the above Axioms. Instead of building up, you build down from the Tara Shadja. That is, ask the question "what lower frequency from which you can have a Pa relationship to the tara shadja?'. That happens to be at a ratio of 1.3333 from the lower shadja.

That is,
if x is the shadja, then the above question can be modeled mathematically

If 1.5 y = 2x ,
what is y's relationship to x?

Solving for y, we get the ratio.
y = (2/1.5) x = 4/3x

Straightforward enough.

It is a bit convoluted derivation but I can appeal to the spirit of the axioms and conclude that the brain can make up consonant intervals in the upward as well as downward direction from the Shadjam using just those two natural ratios that gave rise to G and P. Let us raise the downward ratio formation to the level of an 'Axiomatic Operation'.

So Ma derivation is cool since it uses only the Pancham ratio and only Shadjam as the base. The above extended operational axiom is the moral authority for that derivation.

But what is the musical rationale for deriving 'Nishad is the 'Gandhar' of 'Pancham', Rishaba is the Pancha of the Pancham etc. Just because they are the natural intervals why should they be natural/consonant with respect to the original Shadja? That seems to be jump in logic without providing any justification as to why that is so. Dr. Oke's argument like 'We know musically that Nishad is the 'Gandhar' of 'Pancham', or, musically Pancham:Nishad ratio/bhava/relationship has to be that of Shadja:Gandhar' is not satisfactory. It sounds like circular reasoning. I do grant that the math is sound otherwise once you get past this problem. My reluctance with this applies to the derivation of the rest of the ratios as well.

Btw, once the various sruthi ratios are derived using whatever means ( ignoring the above point and also accepting Dr. Oke's argument for the sruthi ratios ) I do like Srinath's assertion/finding that these derivation of ratios work for sruthi bedam. That logic is sound because it is all about shifting the Shadjam but I have to get past the above issue for the rationale for deriving the fundamental ratios themselves beyond S, G, M and P.

[SrinathK]

But @vasanthakokilam, what other relationships can be there?

You can derive 6 harmonics naturally :
1st : 100 - S (Fundamental)
2nd : 200 - S upper (octave)
3rd : 300/2 = 150 - P
4th : 400/3 = 133.33 - M1
5th : 500/4 = 125 - G1
6th : 600/4 = 120 - g2

That's it. 7th is a dissonant harmonic w.r.t all the other intervals. Beyond this the higher harmonics have virtually no energy at all

But you could try :

9th Harmonic : 900/8 = 112.5 - R2
10th Harmonic : 1000/9 = 111.11 - R1

The eleventh is again dissonant, no one goes any further. In fact the next possible ratio that is valid is only on the 8th harmonic (or should I say 16th) : 8x2/15 = 106.667 = r2 (Note : octaves don't apply. 8/15 is simply 16/15 but in the lower octave)

You can't go further. You can repeat the exercise on P and you will get d2, D1, D2, n2, and N1 in addition.

As for all the other relationships, they HAVE to necessarily harmonically connected either perfectly or to the minimum precision possible by the math. The S-G relationship (major third) generates the remaining frequencies http://www.22shruti.com/research_topic_55.asp

If you look closely, every one of the tones has at least one major third (S-G) relationship, 1 S-P relationship and 1 S-M relationship (if you draw the arrows backwards)

Another way is to take all the tones generated from natural harmonics and apply the cycle of fifths on them. The 2 are equivalent. Therefore it is only natural that other tones don't fit into the graha-bhedam or the symmetric nature of the scale.

There are only 3 prime numbers : 2 (representing the octave), 3 (representing the fifth) and 5 (representing the major third) in all the intervals. 4 and 6 are just multiples of 2 and 3. The Pythogorean tuning used only 2 and 3 and was found to be incomplete, so the Indians went a step further to 5 and completed a closed circuit of intervals (as the figure in topic 55 will show).

The next prime number would be 7 (the 7th Harmonic), but being a prime number, intervals containing multiples of 7 in the fractions would not be compatible with these 22 in graha bhedam. The 7th harmonic would be consonant with the other harmonics in some chords but that would be it.The nearest value would be 175% (700/4) and graha bhedam at that position gives all sorts of crazy values in my sheet that have no connection to the others.

In fact, I have found that trying to extend the 22 / 24 tone scale to accommodate 7-intervals using the same logic the Indians did on the Pythogorean scale could result in the 22 shrutis ballooning to a truly absurd number (probably past the 3 digits mark!) . Most of these intervals would be red flagged in graha bhedam and it would be a monstrosity to attempt all of them.

Since the 22 shrutis is good enough to give a closed set of intervals (and whose precision is already beyond most people's limits), we have stopped here and decided to call it a day. PHEW!

[vasanthakokilam]

You can't go further. You can repeat the exercise on P and you will get d2, D1, D2, n2, and N1 in addition.

Srinath, I guess I drank Dr. Oke's 'naturally derived' cool aid a bit too much. As you wrote, the first 6 and 8th and 9th can be construed natural. I interpret natural as:'They actually exist on a vibrating string at whatever minuscule energy level'. The rest do not exist in that natural fashion if the fundamental is interpreted as Sa. (Right? if not, please tell me ).

Of course, one can ask 'what else will you do?'. Fair enough and fine as well. But elevating it to a natural ratio requires some suspension of our axiom. ( the axiom bring: Our brain treats 2 and 2 power N as essentially the same and all the other harmonics heard in reference to those fundamentals are consonant and hence good as swarasthanas ).

To be sure, I am not too big a stickler. Here is how minimalistic my axiomatic base is. If someone says that the reason why those non-natural whole number ratios derived from P sound good is, the musical phrases surrounding those notes really cause P to be the tonal center and in turn cause the brain to interpret them with respect to P. I can buy that in the generic musical sense and not with respect to any scale or raga or shadja grama or key etc. In abstract terms, it is possible to construct good sounding music without reference to any such building blocks and later interpret the music in terms of such frameworks. If the framework does not fit, so be it, it is not the fault of the good sounding music. So goes the reasoning.

I do not know how much of this is valid and more significantly whether these are misconceptions on my part.

[SrinathK]

@vk, The 7th is in fact a natural harmonic, but 7 being a prime number doesn't mix with fractions made of 2,3 and 5. I mean the Pythogorean scale was a 3 limit tuning system (using only 3 and 2 in it's intervals), but this wasn't enough. Meanwhile the Indians added 5 and that caused the number to blow up to 24 from which 22 were used. I have tried to see what happens to my excel when 7 is introduced and the results make me fear that it would cause the no. of shrutis to explode to insane values.

I'm not sure I understood the rest of what you wrote. Actually not all the 22 shrutis can be called consonances except the ones that occur on natural harmonics. M2 at 135% for one is very strongly dissonant (in theory it would be equivalent to the 27th (!!!) harmonic, but since when have strings vibrated more than 6-8 harmonics?). Interestingly, leaving aside all prime numbers above 5 and their multiples, once you get to the 27th harmonic, one in all 12 notes of the chromatic scale should belong somewhere in the scale (though I have to check this out more rigourously as to which ones).

The various methods used to link the tones together are in fact equivalent.

it is possible to construct good sounding music without reference to any such building blocks and later interpret the music in terms of such frameworks

Gamakas yes! But attempts to interpret them as related to fixed intervals will (and have) failed. This is the very scenario that exists today. However, in long karvais (sustained notes) with no gamaka, one has to land where the math says, or else one will sound out of tune (this also means that not every note of every raga can be sung / played as a sustained one whimsically and that there will always be some svaras which exist only as phrases and not notes). And then there is graha bhedam and the possibilities it brings, in which context this ancient subject is still very relevant to today's Carnatic music, and even modern harmony.