Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Miscellaneous topics on Carnatic music
SrinathK
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#51 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

@vasanthakokilam, I think you're reading too much into Dr. Oke's research topic #22. I'd suggest to look at topic #55 instead.

http://www.22shruti.com/images/rational ... raagas.png

I would like to add that my previous analysis on the harmonics and their intervals was a little premature. You see today I just made an excel sheet and cranked up the harmonics further. If you take the 1st 27 harmonics which are multiples of 2, 3 and 5 only (i.e. Harmonics # 1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30,32,36,40,45,48,50,54,60 and 64), you get 18 out of 22 intervals right there in the first 27 rows. You even get the dissonant S' and P' of the 24 tone scale in the next couple of steps.

You will get all 22 if you are willing to go further (few harmonics after this are pure multiples of 3 and 5, so you'll skip scores of them at once). By the 11th row on the table (Harmonic number 15), you already have both n2 and N1. One more step gives you n1. By the time you get to the 10th harmonic, you already have both D1 and D2. So yup, you can derive most of them quite easily from Sa. r1,

And if you're wondering, it's easy to plot it in a table in excel (with the right formula, the whole table can be generated from a single cell).

Now the issue of consonance. My excel table not only gave me all the ratios, it also gave me which of the harmonics (and how many) are aligned to which of these ratios. The more the harmonics connected to that interval, the better the consonance. The unison (1:1) is consonant with all the harmonics and the octave with every even harmonic (and it decreases progressively from there with each subsequent harmonic less and less consonant).

In practice I don't see a string generating more than 5 or 6 harmonics at all before it loses too much energy. Even on my violin strings, beyond the 5th harmonic (it's easy to generate the natural harmonics by lightly touching the string) I can't get the string to do anything beyond the 5th harmonic at the position of G1, there's simply not enough vibrating length and when the string becomes too short, it becomes much too stiff to vibrate properly. (Deflection is inversely proportional to the 3rd power of the length, so your chances of getting higher harmonics diminish VERY rapidly).

So I see another practical rationale for sticking to just 5-limit intervals and 22 shrutis.

On a side note I even understood the reason why it is recommended that the portion of the string behind the bridge of the violin should be optimally 1/6th of the length between the bridge and the nut for best tone (in practice this will vary from instrument to instrument and string type, but not by much).

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#52 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

So yes, I think I've shared all I've found on this, so I might as well call it a day. So I'm summing it all down below. The list below gives the values of all the 24 tones and the selection of 22 shrutis from them based on dissonance and scale symmetry (Symmetrical means every descending ratio is the inverse of the corresponding ascending ratio). This is to also reconcile with other suggested values for the shrutis given by different authors and clarify what really happened. Finally we also have a mathematical proof of shruti bhedam's validity and it's possibilities.

1 S 1/1
2 S' 81/80 --> Discarded for both producing asymmetry in scale and dissonance.
3 r1 256/243
4 r2 16/15
5 R1 10/9
6 R2 9/8
7 g1 32/27
8 g2 6/5
9 G1 5/4
10 G2 81/64
11 M1 4/3
12 M2 27/20 --> Highly dissonant, yet retained.*
13 m1 45/32 --> ~Indistinguishable from 1024/729 (this appears naturally in graha bhedam and differs by 1 schisma = 1.957 cents from m1).
14 m2 729/512 --> ~Indistinguishable from 64/45 (same as above). (In theory 64/45 produces a little fewer blue cells in graha bhedam :lol:)
15 P' 40/27 ---> Discarded for dissonance, although it is necessary to make the scale symmetrical in both ascent and descent.*
16 P 3/2
17 d1 128/81
18 d2 8/5
19 D1 5/3
20 D2 27/16
21 n1 16/9
22 n2 9/5
23 N1 15/8
24 N2 243/128

*There are some theories that S being the fundamental note (and not a derived ratio) should not be taken as a separate shruti at all and that the 22 shrutis only mean the 22 derived ratios. In which case P' at 40/27 would also be taken as a shruti despite it's dissonance. Their logic is that M2 and P' both are inverses of each other and both are connected to the 27th Harmonic so it doesn't make sense to avoid one while retaining the other. This would also make the 22 shruti scale perfectly symmetric and it would allow M2 to have 12 tones in scale (with the caevat that only P' will be available) and it would explain why Bharata Muni had talked about P being dissonant in Madhyama grama, he might have used P', but discarded S'. The jury is still out on that one. :twisted:

It is likely that before the days of the tanpura (which is a recent 16th century invention) and it's s-P-S tuning, people may have used some device to play only the Shadja (S) and align to that, which might explain why P' may have been considered. Or maybe they just used veenas. But we really don't know.

*In today's practice, M2 may be abandoned when singing in M1(madhyama shruti) owing to the resulting dissonance with the M string. P' may be abandoned in normal singing (s-P-S-S) owint to the dissonance with the P string. This may be why Dr. Oke chose to abandon P' out of the 24 tones generated. In all cases S' at 80/81 is always dissonant regardless of whatever tuning of the tanpura so it is always abandoned.
Last edited by SrinathK on 23 Feb 2015, 13:48, edited 3 times in total.

vasanthakokilam
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#53 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vasanthakokilam »

Awesome, Srinath. Thanks.

Your calculations show that the sruthis can be 'naturally' derived even if they can not be realized as such with real vibrating strings and that the same can be 'second-order' derived with re-starting the derivation from a first-order derived swara (e.g. Pa ) that use only the lower and practically feasible harmonics.

That is comforting.

I can even interpret it as the 'anahata' in the first order becomes perceivable in the second order.
The more the harmonics connected to that interval, the better the consonance.
This kind of consonance principle naturally appeals to me.

That picture by Dr. Oke in topic 55 is well done and it illustrates the main point clearly.

And I think I now know why you wrote earlier in this thread that sruthis are even more important for western chord based music and they can achieve much more natural music instead of resorting to this unnatural twelfth root of two approximation (right?). If I am not wrong, your reference to Dr. Oke's topic 55 serves to reinfornce that point. Given the G and P ratio relationship among the sruthis, there is a ready recipe for chordifying something with three notes that are un-compromised 100:125 or 100:150 ratio apart. (now I recall that youtube video where Dr. Oke shows that by playing the correct three sruthi combination simultaneously and compare it to the ET chords.)

Can you help me with one fundamental thing? I may have asked this before. Dr. Oke's contention is that Deskhar should use G2 (81/64) and not G1 because of that sruthi relationship. For me to know which ones are slight departures ( sharper or flatter ) from what CM uses, what is the mapping from CM's common numbering of the "Nominal' swarasthanas ( e.g. R1&R2, G2&G3, M1&M2, D1&D2, N2&N3 ) to the four varieties of each swara in the 22 sruthi system.

btw, does the sruthi level distinction ( and the picture in topic 55 ) shed any light on assigning parent raga to the janna raga? For example, Bhoop is Janya of Yeman/Kalyani and Deskhar that of Bilaval ( the HM notion )?

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#54 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

@vasanthakokilam, I edited my table a little bit and added 2 footnotes as there are a few different opinions out there on this topic that have not all been resolved. The reason is that the tanpura with it's s-P-S tuning is a 16th century invention and the 22 shruti era is much older.

Now for your queries. Yes, for chords they would work very well. But see, on a REGULAR keyboard, it would take a computer to quickly decide which of the 2 values for any note would be more suitable. E.g. In this 22 sruthi scale, G1 does not have a perfect third above it (25/16 is not in the scale), but G2 does. So the computer would have to think fast here. (Pulling knobs isn't something you would have time for when scales are changing quickly). This is the only practical limitation. Also WCM does not use a tonic or a tanpura, so the concept of "Key change" is strictly speaking, not equivalent to the Indian concept of "Graha bhedam" -- Graha bhedam sacrifices some of the 22 tones so that at least 12 are still available, but WCM has given up on that a very long time ago.

If a western keyboard as it exists today changed key, they would still be able to play all the 22 tones since there is no tanpura or tonic reference to red flag the intervals. All the intervals would shift upwards. I'd say that WCM has evolved too far along the lines of 12-tone scales and it would take a computer to calculate all the frequencies in a different key at the turn of a dial so that one can keep playing. Such tech didn't exist back in the day when the keyboard was first invented and that's why 12-ET won out despite it's imperfections.

In today's CM? The 1st few natural harmonic notes should be more popular as they are easier to it and most people still do it while checking shruti or singing kaarvais. Unfortunately beyond that I cannot say which of the shrutis are used in today's notes or whether such a mapping is even applicable between the 22 tone scale and the present 12 svara scale.

The reason is that most of us abandoned our plain notes after geethams and we rarely, if ever use gamaka-less notes from varnams on. Even the way we were taught those saralis and janTas is quite suspect you know. (I had used a harmonium to get my plain notes "right" -- and my master had told me to stop it after a point, now I know why). After that my plain note practice was always by ear (and the accuracy of the ear is questionable beyond the 1st 6 harmonic notes, even for expert musicians).

For your third question, In CM, this concept of selection of shrutis for a raga may not be applicable to the modern melakartha-janya system at all. The 22 shruti system is much, much older and as I see it, most ragas were derived and existed independently of each other. Nearly all janya ragas we know were never really derived from the sampoorna scales. It is an artifact of modern CM grammar (and the efforts of recent musicologists) that they have been clubbed into this "parent-derived" raga category. In fact this system has severe limitations. It doesn't address ragas with bhashanga swaras, or ragas with multiple arohanam-avarohanams or purely phrase based ones, even dwi madhyama ones and the MA has several times argued as to which mela to put ragas like ATaana and Sourashtramam under. This has even led to a trend to favour more scalar ragas among the ones that arose in the 20th century, deriving them from Aro-Avaro instead of individual phrases.

About HM I don't know that much about HM ragas in the first place. For the most part, I think they too evolved independently (though there was some old system of parent-child ragas in the past).

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#55 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vasanthakokilam »

Srinath,

>>The reason is that most of us abandoned our plain notes after geethams and we rarely, if ever use gamaka-less notes from varnams on.

There are unmistakably plain note usage in many CM ragas and compositions. Even if not in compositions they are so
in alpanas. I am not even thinking of what is actually practiced today but what are the nominal CM ratios for the 12 swarasthanas. I am hoping that the answer is something other than the ET ratio.

>So the computer would have to think fast here. (Pulling knobs isn't something you would have time for when scales are changing quickly).

It is not a problem for the computer to do that of course. it is more a human computer interface problem for the musician. And given how incredibly diverse and creative some of the interfaces are with smartphone and tablet applications, someone can come up an interface that is workable. I can not visualize now what that will be, but it will be fun to look at what the interface needs to do and work on that with some creative abandon ( thinking outside the box).

>In CM, this concept of selection of shrutis for a raga may not be applicable to the modern melakartha-janya system at all. The 22 shruti
>system is much, much older and as I see it, most ragas were derived and existed independently of each other.

Quite true. But consider this. The assignment of Janya ragas in CM is devoid of any musical aesthetics. The answer to why Mohanam should be a janna of 28 is, it is so because that is the lowest melakarta number among the potential ones. That is good having an answer so we can fill out the charts.

I was just thinking out loud that given how Dr. Oke assigns specific sruthis to ragas (based on that consonant relationship in topic 55) if that can act as a tie breaker. Just for illustration, if Kalyani and shankarabharanam(or harikambhoji) are assigned different G sruthis ( not saying that is necessarily the case, just for illustration ) and if Mohanam uses the Ga sruthi of Kalyani, then there is a rational basis to put Mohanam under Kalyani.

(In HM, Bhoop is from Yaman that and Deshkar is from Bilaval and their rationale seem to be based on subtle musicality. Not that I understand that well)

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#56 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

Here's one other thing. There are many equivalent methods of deriving these intervals (some of which we discussed above). Bharata Muni has given a procedure where 2 veenas are tuned and the Pa string of one veena is lowered to P' and the derivation of the shrutis goes from there. A few points is made how successively detuning the 2nd veena to the next lower shrutis causes different swaras to merge together at the same pitch (something which my excel sheet has derived in it's entirety).

That reminded me of something I found out today. In that pic on topic 55 that I posted, one can notice something interesting. You realize that one can add P' on the very left of the first row (to the left of R1) and simply apply the cycle of fifths and proceed right. That will take you through the whole sequence all the way to S' (which is discarded), leaving M2 as the last note, with all the tones derived in the process.

@vasanthakokilam,
Just for illustration, if Kalyani and shankarabharanam(or harikambhoji) are assigned different G sruthis ( not saying that is necessarily the case, just for illustration ) and if Mohanam uses the Ga sruthi of Kalyani, then there is a rational basis to put Mohanam under Kalyani
Well, this is one place where HM and CM turn and go their separate ways. Maybe in an ancient gamakaless era, your wish might have been granted. :)
Last edited by SrinathK on 28 Feb 2015, 18:58, edited 1 time in total.

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#57 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

And last, while the 22 shruti system was mainly a melodic system and probably not intended to be used as a harmonic one, it still has a lot of harmonic capability. Now you can observe that G1 doesn't have a major third of it's own in this scale (5/4 * 5/4 = 25/16, augmented fifth, not in the scale) -- you would have to shift to G2 instead. That's the price you pay for sacrificing tones in graha bhedam.

Which brings us to another interesting observation. Out of the 22 / 24 tones, any intervals which are red flagged (forbidden) while carrying out graha bhedam on one of the variations (e.g. G1) are available in the other (G2). This means that while you lose tones while shifting Sa to G1 or r1, you get them back in G2 or r2. Could this have been used as a dynamic solution to address the issues in graha bhedam or resolve intervals, especially considering that the precision between 2 shrutis is smaller than a vibrato and very difficult to attain by most people? This could be done on a regular keyboard in fact, if the computer AI was smart enough to decide between G1 and G2 or r1 and r2 by itself. An world class instrumentalist knows the fine art how to adjust the intervals on the fly so that they sound in tune (or as intended).

This is the main reason for my statement that 22 microtones is good enough for doing graha bhedam with 12 svaras.

(An even finer dynamic variation such as a 64/45 + 729/512 combination on m2 would remove even the blue cells, but that's asking for too much).

Fretless instruments could allow you to make any amount of adjustment and use any number of tones. But in keyboard or fretted instruments, they are stuck to a set of tones so that mandates a fixed number of pitches -- not too large, not too small, but enough to play whatever the music demands. In this we can say that our music is originally veena based while western music is orginally keyboard based.

I may speculate that 3 or 4 part chords and more complex harmonies were not envisaged by the raga oriented ancient Indians -- they would have sought out enough precision to allow for proper combinations of 2 notes at most within the scale at all points. And 22 shrutis gives you as much precision as the math would allow. From the purely melodic point of view, that could do it.

But assuming they might have tried at least 3 part chords. One observes something very interesting.
1) Positions for S where a just minor triad chord is possible (100:120:150) : S, r1, R1, R2, G1, M1, m1, P, d1, D1, n1, N1 (P' at 40/27 can also do it)
2) Positions for S a just major triad chord is possible (100:125:150) : S, r2, R2, G2, M1, M2, m2, P, d2, D2, n2, N2

What it means is that doing graha bhedam and shifting S at these positions will leave you with the intervals to play a major triad or a minor triad.

@vk, I wonder if this would answer your question as to which plain notes would be most preferable to modern CM. There have always been 12 swaras all along.

No need to use equal temperament (and the solution is way more precise, mostly perfect). It goes to say that the tonal character of major and minor scales would be markedly different in a way that 12-ET could never do.

At this point you wonder if ancient India and it's interaction with the Greeks and other Europeans might have influenced the evolution of their music systems. I mean, in Indian music, owing to it's marked dissonance, P' is rejected as it would be badly out tune against a Pa shruti string of a veena or a tanpura. But had the 22 shrutis been decided based upon the ability to play major and minor scales, I believe M2 would have been the one rejected in the 22 shruti scale in shadja grama and instead delegated to a separate category as the origin of madhyama grama.

In modern CM, R1, M2 would be really unpopular in plain notes strictly from a melodic point of view and nobody would use P' and S'. From the viewpoint of major and minor scales, R2 and M1 can handle both themselves so R1 and M2 would be unpopular even from that angle. So I would say the most popular plain notes today would be : S, r1, r2, R2, G1, G2, M1, P, d1, d2, D1, D2, n1, N1 and N2 (choose your 12 wisely :lol: )

You wonder what else you are going to find on this subject. Time and again as you want to bring the curtains down on this (overly geeky) discussion that would involve googling a lot of words, more discoveries emerge. :mrgreen:

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#58 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vasanthakokilam »

But assuming they might have tried at least 3 part chords. One observes something very interesting.
1) Positions for S where a just minor triad chord is possible (100:120:150) : S, r1, R1, R2, G1, M1, m1, P, d1, D1, n1, N1 (P' at 40/27 can also do it)
2) Positions for S a just major triad chord is possible (100:125:150) : S, r2, R2, G2, M1, M2, m2, P, d2, D2, n2, N2

What it means is that doing graha bhedam and shifting S at these positions will leave you with the intervals to play a major triad or a minor triad.

@vk, I wonder if this would answer your question as to which plain notes would be most preferable to modern CM. There have always been 12 swaras all along.
Srinath: Thanks for pointing this out. Quite interesting. I keep thinking about your point that western music with their need for such three note chords needs the 22 sruthi system more than CM. It will be interesting to get the take of Western Music theoreticians who discuss tuning. In such discussions, one should not mix the practicality of dynamically and easily tuning to these various ratios and whether musicians and rasikas have that fine a musical sense, fidelity and resolution. (They are important practical considerations alright) From a music perspective the 22 sruthi scheme is better for harmony is a great finding. I wonder if they acknowledge that.This in one sense amounts to turning the common perception on its head (e.g. at a broad level, the common perception being Indian is JI and Western is ET )

Back to CM ratios,

What I want to establish as baseline is this: the 22 sruthi system uses a nomenclature r1,r2,R1,R2. CM uses R1, R2, G2, G3 etc. What is the mapping for playing plain and steady single notes. We need something definite to characterize the CM tuning.

Is the following a good one to go by?


S - 1

R1 - 16/15
R2 - 9/8

G2 - 6/5
G3 - 5/4

M1 - 4/3
M2 - 64/45

P - 3/2

D1 - 8/5
D2 - 5/3

N2 - 9/5
N3 - 15/8

S - 2

(We can acknowledge that R2, N2 and M2 may have an alternate ratio that is published in some literature but still considered CM)

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#59 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

@vasanthakokilam, For a non-shruti bhedam case, the ones you've specified would be the easiest to tune to, with one correction.
Is the following a good one to go by?

S - 1
R1 - 16/15
R2 - 10/9
G2 - 6/5
G3 - 5/4
M1 - 4/3
M2 - 64/45
P - 3/2
D1 - 8/5
D2 - 5/3
N2 - 9/5
N3 - 15/8
S - 2
Make R2 as 9/8, that is the natural harmonic in tune with the P string. Otherwise all is well. 10/9 is quite a dissonant interval in fact.

M2 @ 64/45 is pretty much equal to m2 = 729/512 (or you could call it the Just Intonation variation). Depending on which value you use, the other will appear in graha bhedam at the blue cells). In fact the JI value is slightly juster than the Pythogorean value (fewer blue cells in my sheet).

While observing that, I should point out that these are all Just Intonation Ratios.

So your final solution (w.r.t to the 22 shruti values) :

S - 1
R1 - 16/15
R2 - 9/8
G2 - 6/5
G3 - 5/4
M1 - 4/3
M2 - 64/45
P - 3/2
D1 - 8/5
D2 - 5/3
N2 - 9/5
N3 - 15/8
S - 2
Last edited by SrinathK on 09 Mar 2015, 00:15, edited 1 time in total.

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#60 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vasanthakokilam »

Thanks Srinath. Updated my post above with 9/8 for R2

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#61 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vgovindan »

Srinath,
The LCM of the denominators of the twelve values is 360. And the LCM of the denominators of all the 22 values given in Dr Oke's paper is 622080 which is 360 * 2^6 * 3^3
Is there any significance?

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#62 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

Very interesting. Unfortunately I think this is only mathematically significant, since all the numbers in all the ratios are made up of 3,2 and 5 multiples only.

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#63 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vasanthakokilam »

I was playing around with Arun's Swarasthana iOS app (btw, all iOSers should get it, it is packed with so many features that the price you pay is a steal).

One thing I noticed is that its swarasuddham feature detects all these than-thonri swaras if I play it the tambura sound. Hope that is what I am seeing there, Arun can confirm. (as an aside, thanthoNi (than-thonri, tamil for swayambu) is a scolding word but a bit amusing that it originates from a deep concept )

Anyway, for example, it shows the G3 when tambura is playing is Sa ( with Pa sound turned off in Radel ). In addition It shows another fairly bright horizontal line that is just above Sa and much lower than R1. Like that there are a few such things. I will post a screen shot later. If Arun reads this, I am sure he will have something to share on this.

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#64 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vgovindan »

Significance of 360 and 9 -

https://youtu.be/inWnhZp_A-M

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#65 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vasanthakokilam »

Just a status report. As part of the effort to increase my swaragyanam, I play the tanbura (Radel) in the background whenever possible. It is set to C#, mono, no Panchamam string.

The swaras I hear in that around are just incredible.

Today, I clearly heard G3 P D1.. P P for a while. Interestingly it is not always there.

Isn't D1 an unlikely swara to hear as a swayambu swara? While I hear G3, P and M1 a lot of times, for D1 today is the first time. In fact the reason I even got curious about is, the tanpura sound started sounding like bowli (with that G3-P-D1..-P sequence).

Listening to just tambura is so revealing, both the fact I hear a lot more swayambu with repeated listening and also the fact they disappear once in a while.

Update: G3 P D1.. P P is back. I did not hear it for a few minutes.

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#66 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by arunk »

vasanthakokilam wrote:I was playing around with Arun's Swarasthana iOS app (btw, all iOSers should get it, it is packed with so many features that the price you pay is a steal).
Anyway, for example, it shows the G3 when tambura is playing is Sa ( with Pa sound turned off in Radel ). In addition It shows another fairly bright horizontal line that is just above Sa and much lower than R1. Like that there are a few such things. I will post a screen shot later. If Arun reads this, I am sure he will have something to share on this.
One at G3 is quite possible. The one below R1, I would say is interference (?)

Arun

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#67 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by arunk »

Not sure what the math is for hearing M1, because this is how I see it:

For hearing ma (M1), at least one of the "ma" swaras i.e. M1, M1' (tara-stayi), M1'' (ati-tara-stayi), M1''' ..., must be in the harmonics of Sa (i.e. emitted by tampura).

These are at frequencies M1, 2*M1, 4*M1, 8*M1 etc.

For this to present in harmonics of Sa (S, 2S, 3S, 4S), M1 must be of relation x/2^y (2-power-y) of Sa (the 2^y comes because of 2M1, 4M1, 8M1 as above). But M1 is 4/3, and denominator 3 is not a integral power of 2. So not sure how we can hear any "ma" (i.e. M1, M1', M1'') in pitch+harmonics of Sa. This is true for any swara.

For example, G3 has 5/4, and P has 3/2, and R2 has 9/8 and they all fit this math. If say tampura is at 120Hz, then its pitch+harmonics are 120, 240, 360, 480, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920, 2040, 2160, 2280, 2400, 2520, 2640, 2760, 2880, 3000, 3120, 3240 ..... etc..

The "ga" (G3) swara (across all stayis) is at 150 (g), 300 (g'), 600 (g''), 1200 (g'''') etc. So first occurrence of a g3 in pitch+harmonics of Sa emitted by a tampura is 1200Hz (10S).

The "pa" swara (across all stayis) is 180 (p), 360 (p''), 720 (p''') .... First occurrence is is 720Hz (6S)

The "ri" (R2) swara is 135 (r), 270 (r'), 540 (r'''), 1080 (r'''). First occurrence is 1080 (9S)

I have heard D2, which I guess could be 27/16, and that "dha" swara is 202.5, 405, 810, 1620, 3240, ... - First occurrence in harmonics of sa is 3240 (27S). Maybe that was what I was hearing.

For Ma (120*4/3 = 160), I guess the first match would at 160*120 i.e. 19200Hz ( :-) ??)

Arun

SrinathK
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#68 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

@arunk, What one would actually hear is a superposition of harmonics & resonances from all 4 strings. So no wonder musicians seek Sa, then Pa then antara gandhara G while aligning to the pitch, but not M1. Most notes we think we hear are actually "phantom notes" which our brain fills up. M1 when sung would align with several harmonics in the tanpura which in turn would give our own ears the feedback as to the alignment of that note.

I'd actually advise using a real tanpura on this test as the pitch variations of a real instrument can be quite different. I have a theory that there is also a resonance effect involved when singing any note aligned with a real tanpura that could subtly alter the vibration of the strings which is why musicians with sensitive ears complain that the electronic gadgets don't give the same level of feedback as a real instrument.

arunk
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#69 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by arunk »

my response was mainly based on vk mentioning hearing m1 with just sa, and no pa.

(also had a math error in previous post, for pa, first match is 360Hz - 3S, which of course is what it should be).

> M1 when sung would align with several harmonics in the tanpura which in turn would give our own ears the feedback as to the alignment of that note
I would be interested in the math that identifies the m1 component in the harmonics of pa`, sa, pa, sa'.

Arun

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#70 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vasanthakokilam »

Thanks Arun and Srinath.
Isn't D1 an unlikely swara to hear as a swayambu swara? While I hear G3, P and M1 a lot of times,
To clarify, I did not hear M1 in this particular experiment (with just the Sa and no Pa ). My M1 reference was about hearing M1 at other times. Sorry for not being clear there. I do not recall that exact occasion, let us assume I had the P note on that time. Does it make M1 still possible? I will pay attention to it. May be that was just my misidentification.

How about D1, with only the Sa string? Is that just a phantom one? If it is, that is incredible in itself because I heard that so vividly for quite some time.

BTW, in a real tambura what are the strings tuned to and in what sequence?

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#71 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by ganesh_mourthy »

Sa Pa Sa ( manthra shadja) Sa (manthra shadja)

vasanthakokilam
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#72 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vasanthakokilam »

Thanks G_M.

Arun, what are the ratios does your Swarasthana iOS app use for the various swaras in the SwaraSuddham feature?

Just as a related aside, I am quite dismayed that all the flutes I have except may be one do not align perfectly with what your program shows. If S, R2 and G3 are perfect, M1 shows up lower. Someone who was with me could hear it and when I rotated the flute outward a bit both your program and that person said it is just right. So there is indeed a problem with the hole position. But that seems to be an issue only on long karvai notes. Anyway, it is still quite disappointing. I need to go flute shopping with your app in hand.

This is probably common knowledge but the blowing intensity in addition to changing the amplitude changes the frequency as well. Your app shows it changes by as much as a quarter tone fairly easily and if I try hard even by a semi tone.

And third which is my problem, if I start off with perfect alignment to S, after playing for a minute or two and I come back to S, it is not quite aligned. Not by much but people with sensitive ears can hear it. I can probably adjust it so it matches the tambura sound but that is not instantaneous.

Anything coming out of my flute that sounds like music is in itself a miracle given all these issues. :)

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#73 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by arunk »

Hmm this is kind of opening a can of worms. First, you are implying (implicitly) there is a single authoritative position for every sthana. That is true for P, M1 and G3 for sure. But not for others - I don't think one should be arriving decision about hold position based on sync against the app! Anyway, for other swaras, the app uses "one of the" ratios thrown around (e.g. I actually use 10/9 for R1, 5/3 for D2, 16/9 for N2, 15/8 for N3, 256/243 for R1 and 243/128 for N3). So why specifically these? It was a subjective decision made by me based on what I thought sounded right (or close enough) for the lessons done for some of the main/common scales .

Also, one more correction from my earlier post
> For Ma (120*4/3 = 160), I guess the first match would at 160*120 i.e. 19200Hz ( :-) ??)

This is wrong as well I think. 19200Hz (160th harmonic for Sa of 120Hz) would not be M1 swara. Quiz - which swara would it be?

BTW, with pa (mantra) and pa (tara) also, as per above math (i.e. assuming that is right ;-) ) , I am not sure we can get a harmonic that would match the "ma" swara.

Arun

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#74 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

No, you in fact do not get M1 directly. What happens is that we sing a M1 and check the (inverse) Sa-Pa relationship between M1 and S as well as the 9/8 relationship between M1 at 4/3 and P at 3/2. Also the fundamental S corresponds to the lower P of M1. The next closest match is the 4th harmonic at 400 % of the fundamental, which corresponds to the panchamam 1 octave above the relative panchamam of M1, i.e. the Sa above the mel Sa. Alignment with a tanpura is also done checking the relative intervals with the tanpura strings on these kind of notes.

On a side note, I wonder how well tuned are most flutes in the market. A few of my friends got some cheaper flutes in college and while playing with them I found that not a single one of them was without some tuning problem or the other.

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#75 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vasanthakokilam »

Arun, no can of worms wrt to your great app, I only plan to use it for S, P, G3 and M1 when buying a flute. Srinath, I checked 5 of my flutes and only one of them, a lower pitched one was correct for M1.
First, you are implying (implicitly) there is a single authoritative position for every sthana. That is true for P, M1 and G3 for sure. But not for others
this is quite disconcerting, isn't it? What do the instrument makers use? I guess they go by ear but is there a consensus? Do these ratios change according to school? How can Violinists tune their instrument if they can not agree with what the vocalist uses. Hope the situation is not as dire as I am imagining.

Is the problem only on the mathematical ratios and not on the actual sound produced as perceived by our ears. Again, I hear Srinath saying that even small differences are 'hearable' by sensitive musicians. So we can not leave all this to chance. ( I am not saying they all need to know what the actual ratios are but at least they need to musically agree on the tone of each swara as precise as possible )

I am not even thinking of raga specific sharps and flats we will leave it to the artists to make those adjustments.

There is a table Srinath provided a few posts above.

Can we classify them as follows? This is all with respect to CM

a) The swaras for which there is consensus on their ratios
b) The swaras for which there is some debate but they are very close to matter
c) The swaras for which the ratios are different enough to matter.

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#76 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

What do the instrument makers use? I guess they go by ear but is there a consensus? Do these ratios change according to school? How can Violinists tune their instrument if they can not agree with what the vocalist uses
Tuning a violin to s-p-S-P is no problem. We take the vocalist's pitch whatever it is and fine tune accordingly. There is a fine adjustment button on all those electronic tanpuras and fine tuners on violins. The violin is fretless so we don't have a problem. As for veenas and flutes, I cannot say. Somewhere on this very forum I remember reading an observation that veena frets were very close to 12-ET. As for flutes, quality control is a big question mark in the cheap models beginners like my college mates used. I haven't found any other source on the net with relevance to Carnatic music specifically going so nitty-gritty into these frequencies and their analysis as we have on this very thread. I think this is a first.

Yes schools differ. My ears tell me that the Parur school as an example use a lot of the HM positions in their notes which give that flavour in their raga renditions. It even varies from instrument to instrument - veenas go one way, flutes in another, while vocals are only as good as their swara gnanam and sruthi shuddam will permit.
I am not even thinking of raga specific sharps and flats we will leave it to the artists to make those adjustments
We are out of note territory and into phrase territory as soon as we talk of ragas. CM really doesn't use the plain notes that much you see.

And ears despite being marvelous instruments when trained, are really not that precise either and they are NOT helped in anyway by the kind of (mostly dreadful) acoustics we can (barely) afford. And pray, how many people have you come across who will peek into frequencies up to the decimals to check their alignment? How many instrument makers you think know these values and try to tune accordingly? These 24 pitches are well known for centuries now, but to put them into the context of plain notes in Indian music and present them in this manner like that 22shruti website is recent science.

<The above rant is to point out the contributing factors to shruti shuddham issues in CM that aren't going to go away any time soon>

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#77 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by vgovindan »

Two interesting articles on Harmonics - music and everywhere -
http://ray.tomes.biz/story.htm
http://cyclesresearchinstitute.org/cycl ... nics.shtml

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#78 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

I'm cross referencing another post here :- viewtopic.php?f=2&t=31566
SrinathK wrote: 23 Jul 2018, 17:34
In the Madhyama Grama Pancama should be made deficient in one Sruti. The difference which occurs in Pancama when it is raised or lowered, by a Sruti and when consequential slackness3 or tenseness [of strings] occurs, will indicate a typical (pramana) Sruti. We shall explain the system of these [Srutis]. The two Vinas with beams (danda) and strings of similar measure, and with similar adjustment of the latter in the Sadja Grama should be made [ready]. [Then] one of these should be tuned in the Madhyama Grama by lowering Pancama [by one of Sruti]. The same (Vina) by adding one Sruti ( lit. due to adding of one Sruti ) to Pancama will be tuned in the Sadja Grama. This is the meaning of decreasing a Sruti (lit. thus a Sruti is decreased). Again due to the decrease of a Sruti in another [Vina] Gandhara and Nisada will merge with Dhaivata and Rsbha respectively, when there is an interval of two Srutis between them. Again due to the decrease of a Sruti in another (Vina) Rsbha and Dhaivata will merge with Sadja and Pancama respectively when there is an interval of [three] Srutis. Similarly the same [one] Sruti being again decreased Pancama, Madhyama and Sadja will merge with Madhyama, Gandhara and Nisada respectively when there is an in interval of four Srutis between them. Thus according this system of Srutis, [each of] the two Gramas should be taken as consisting of twentytwo Srutis.
It is theoretically possible if there's a veena with 22 precisely spaced frets per octave (for which the exact positions of the 22 intervals on a string need to be known). You need two of them, and tune one to the next lowest note played on the other.

Let's call these intervals as S,r1,r2,R1,R2,g1,g2,G1,G2,M1,M2,m1,m2,P, d1,d2,D1,D2,n1,n2,N1,N2 (the 22) and finally top S. Each of the 12 notes we know got 2 values from the algorithm that derived the shrutis, and the ratio between the higher and the lower is 81/80. S and P got frozen, the experiment will not work otherwise, so there's our 22.

Now here's the mistake in the ubertext - implying that the veena is being tuned down by 1 pramana shruti (80/81) every time. No, It isn't. The shrutis follow a certain sequence of ratios up and down the scale and they can only be decreased or increased according to that order.

I am not going to show the math in this post (a simple matter of ratio adjustment). I will if anyone's interested in a subsequent post.

1) First you tune the second veeNa's P to m2.
2) Then when you tune the P down one more shruti to m1, then g2 merges with R2, g1 merges with R1, n2 and n1 merge with D1 and D2 respectively. So far so good.
3) Then when you go another step further, down to M2, then R1 merges with S, and D1 will merge with P. Check. Note, d1 and r1 cannot merge with P and S.
4) Then when the P is decreased down to M1, then P and M1 have merged. Meanwhile, M2 merges with g2 (not G2), M1 merged with g1. S merges with n1 (NOT N1 as so many have wrongly understood!!). Check. Therefore by virtue of detuning by 4 shrutis, there is a 4 shruti gap between n1 and S.

Some people have wrongly interpreted this as a 4 shruti gap between N2 and S (and they also implied that S has 4 values), but the math clearly shows that it's between n1 and S.

The whole confusion here is that no one has bothered to clarify specifically which interval merged with what! They didn't say WHICH gandhara merged with which rishabha, when each of them has 4 possibilities. We don't even know if bharata muni himself used this kind of labelling scheme - if he didn't, then what he wrote up there won't make any sense and it's up to the math to tell us what's really going on.

And this is what has baffled musicologists till date and caused all the confusion. Otherwise the debate on 22 shrutis would have been closed from the math long ago.

So what we need here are 2 identically tuned veenas with 22 frets, such that the intervals fall at these locations : http://www.22shruti.com/research_topic_33.asp

Then one veena plays m2, m1, M2 and M1 and the other veena tunes the P string successively downward to these intervals. And then the other strings of the second veena are tuned to match the reduced P. This is definitely doable.

The whole idea of arbitrarily tuning down a veena by several pramaNa shrutis was absurd enough, whoever said that definitely hasn't tuned strings in their life. If you check the revised ratios with that assumption, it guarantees that no interval will ever sync with the first veena again.

And the idea of tuning down the second veena arbitrarily by ear is ridiculous. I mean, that is an interval of 21.5 cents, so you need to be precise to 0.5 cents, which is 1/200th of the gap between two semitones. The only way to determine the Pythogorean comma properly is to actually hear it on a piano tuned to 5ths, or play R2 on both the S and lower P strings of a modern veena and hear it for oneself.

But the above solution is doable and the only mathematically possible one, provided they can figure out how to get the frets to work. If they managed to get a harmonium to do it, I think it is possible to create a custom veena as well.
Last edited by SrinathK on 24 Jul 2018, 10:37, edited 1 time in total.

SrinathK
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#79 Re: Lec-Demo on 22 Shrutis - by Dr Vidyadhar Oke

Post by SrinathK »

And this labelling scheme of Dr. Oke for the 22 intervals has one more benefit - it tells you why these names chatushruti rishabham, shatshruti rishabham and their equivalent dhaivatams exist - a vestigial reminder of the 22 shruti system.

For R2 is 4 shrutis from S, g2 is 6 shrutis away from S (this is the note that we use as shatshruti rishabham). Similarly D2 and n2 are 4 and 6 shrutis above P.

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