Challenge to Bharata Muni

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uday_shankar
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#1 Challenge to Bharata Muni

Post by uday_shankar »

I've long held the theory that Bharata Muni just winged his chapter on the dhruva vina chala vina experiment to demo the 22 shrutis.

Here's a translation of the relevant chapter (28) of Bharata's NatyaShastra... I have highlighted the first line that gives Bharata away as a mere theoretician with no practical idea of how strings are tuned, but somehow he decided to throw in this non-experiment into his treatise. There are many other subsequent booboos...

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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.

THE CHALLENGE

"Dear Bharata Muni, take in a lungful of air and now breathe out exactly 1% of the air in you lung and then hold your breath. The moment you do that to within +/- 0.01%, I will immediately lower the panchama of the chala vina by exactly one pramana shruti"

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#2 Re: Challenge to Bharata Muni

Post by SrinathK »

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, 10:36, edited 5 times in total.

uday_shankar
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#3 Re: Challenge to Bharata Muni

Post by uday_shankar »

Spare your analysis, these are.not fretted Vinas. They are like the yazh or harp.

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#4 Re: Challenge to Bharata Muni

Post by SrinathK »

Oh well, do you have a harp or two with 22 strings to spare? :mrgreen: :D You could do this and get all the strings aligned without working out a single ratio once S is fixed. You could even do this experiment with 2 harps without the need to retune. Just tune as per the procedure below.

http://www.22shruti.com/research_topic_31.asp

Did Bharatamuni have a harp like that? I don't think he did. So I guess you will have to make a 22 shruti harp for the chitravenu now... :mrgreen: (BTW, I love the instrument). No I am serious. It will sound divine.

The harmonic possibilities of such a harp are far superior to anything tuned in Equal temparament.
Spare your analysis, these are.not fretted Vinas
But in my consolation, I think my math wasn't wrong. :ugeek:
Last edited by SrinathK on 23 Jul 2018, 18:37, edited 2 times in total.

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#5 Re: Challenge to Bharata Muni

Post by uday_shankar »

Srinath,
Paraphrasing what Einstein told Lemaitre, one of the first to postulate expanding universe based on GTR, "your maths is all right but your physics all wrong" perhaps we should tell you in this instance, "your maths is all right but your practical utility very little"... Which is not to say I don't enjoy your math or flights of imagination :). Keep it coming or the forum will be a very boring place

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#6 Re: Challenge to Bharata Muni

Post by SrinathK »

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.
And 0.5 cents is 1/2400 of the gap between 2 octaves. The maximum sensitivity of the human ear is 1 cent.

So that means is our poor Bharata muni needs to exhale exactly 1.79% of the air in his lungs, with a tolerance +/- 0.04%. :D

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#7 Re: Challenge to Bharata Muni

Post by SrinathK »

And this labelling scheme has one more benefit - it tells you why these names chatushruti rishabham, shatshruti rishabham and their equivalent dhaivatams exist - a vestigial reminder to these intervals.

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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#8 Re: Challenge to Bharata Muni

Post by uday_shankar »

SrinathK wrote: 23 Jul 2018, 17:53So I guess you will have to make a 22 shruti harp for the chitravenu now... :mrgreen:
I do tune using consonance principles, according to the raga I'm playing and many of the 22 shruti intervals do occur. More interestingly, I have chords which are tuned per natural harmonics... never hear those anywhere. I don't like many of the shruti values, and end up not using them, such as:
16/15 (too high for my idea of shuddha rishabam, prefer a lower value usually)
10/9 (what a horrible rishabam, don't like it in any context)
6/5 (too high for my sadharana gandharam, prefer a lower value usually)
27/20 (too low for my prati madhyamam, prefer a higher value usually)
8/5 (too high for my shuddha dhaivata, prefer a lower value usually)
16/9 (what a horrible nishada, don't like it in any context)

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#9 Re: Challenge to Bharata Muni

Post by SrinathK »

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.
There is another issue with Bharata's wording here. He seems to suggest tuning the P string down by 1 pramANa shruti or 1 Pythagorean comma to put the veena in madhyama grama, and then continuing to keep tuning it downward 3 more times, and he'll get a gap of 4 shrutis between n and S. As it turns out, this is wrong.

There is a harmonic problem with Bharata's madhyama grAma - the fifth interval is reduced by 1 syntonic comma to 40/27 - a diminished fifth, extremely dissonant. http://www.22shruti.com/research_topic_29.asp

Bharata knew this very well. That's why he insists on using a dimished fifth in his madhyama grAma - I suspect he had to because probably shadja was a fixed note of absolute pitch like A440 is today in Western music, so madhyama grAma had to be calibrated to that. There's a theory he wanted to use shadja grama for male voices and madhyama grama for female voices, and the difference between them is usually between 4-5 keys.

I have no idea why he could not have simply sacrificed R1 instead, not only is it dissonant, it's also very convenient to dump it. Why for the love of god alter the perfect fifth?

If we listen to Bharata's words and tune down the P string by 1 pramANa shruti to P' (40/27) and then tune down 3 more times (as he seems to suggest) successively to m2, m1 and M2, there is a problem. Instead of S merging with n1 at the end, it merges with n2 and S happens to be only 3 shrutis above n2, and not 4. Unless he thought of including P' as an extra shruti (this has confused some people).

I myself think that what they were trying to suggest is to use P in shadja grama and P' for madhyama grama, all other intervals were the same. From there the only way for his logic to work in both shadja and madhyama gramas then is to tune the veena's P string down successively to m2, then m1, then M2 and then M1 as I described it in Post #2. That's 4 shrutis down and then S merges with n1, and the math is correct.

The Natyashastra therefore blundered in describing the procedure properly by describing a gap of 4 shrutis when he actually only tuned his string down by 3. Tuning the P down to madhyama grama P' doesn't count as an extra step.

I also have no idea what he meant when he said this. Although after reading Dr. Oke's website, I think it could have been explained in a much less ambiguous way.
[quote : Chatush Chatush Chatush Cha Eva Shadja Madhyama Panchamou Dve Dve NishĀda GĀndharou Tri Tri Rishabha Dhaivatou]

Enough confusion and complexity has been created by these authors in trying to describe everything verbose without figures or math. But that's the best you could do when technology was restricted to palm leaves and teaching methods were mainly oral.
Last edited by SrinathK on 25 Jul 2018, 08:34, edited 8 times in total.

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#10 Re: Challenge to Bharata Muni

Post by SrinathK »

uday_shankar wrote: 23 Jul 2018, 19:46
SrinathK wrote: 23 Jul 2018, 17:53So I guess you will have to make a 22 shruti harp for the chitravenu now... :mrgreen:
I do tune using consonance principles, according to the raga I'm playing and many of the 22 shruti intervals do occur. More interestingly, I have chords which are tuned per natural harmonics... never hear those anywhere. I don't like many of the shruti values, and end up not using them, such as:
16/15 (too high for my idea of shuddha rishabam, prefer a lower value usually)
10/9 (what a horrible rishabam, don't like it in any context)
6/5 (too high for my sadharana gandharam, prefer a lower value usually)
27/20 (too low for my prati madhyamam, prefer a higher value usually)
8/5 (too high for my shuddha dhaivata, prefer a lower value usually)
16/9 (what a horrible nishada, don't like it in any context)
True, some of these intervals are dissonant with a tambura - on a melodic scale, they can be dispensed with. There is one use for them, it's in playing chords where the relative consonance is more important. And that's where we get far better chords than ET. And it looks like Dr. Oke's milked this concept into getting different shrutis for his raga arohanams and avarohanams.

For me the R1 (111.11 % or 10/9) and M2 (135 % or 27/20) are the worst offenders when played against a tambura, the syntonic comma was never more evident. Even the G2 at 81/64 isn't that bad.

These 2 positions also give the least number of valid intervals in the scale when you try doing graha bhedam on them.

In fact the M2 is the only one without a valid panchama for itself (a perfect 3/2) - it gives only 11 tones (an ancestor of panchama - less scales?). If you want a panchama you have no choice but to use a diminished 5th (40/27) using S. So there is in fact a theroetical, mathematical basis for madhyama grama from this 22 sruthi scale, at least.

R1 in turn is the only one without a valid madhyama. It doesn't gel well with the P at 3/2 either.

Here's something interesting. http://www.22shruti.com/research_topic_42.asp

It's not just me. Take a look at the only 2 positions that don't feature in ancient scales - the R1 and the M2.

And while we're talking veenas, the good doctor is at work on this : http://www.22shruti.com/22_shruti_veena.asp
Last edited by SrinathK on 25 Jul 2018, 08:34, edited 1 time in total.

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#11 Re: Challenge to Bharata Muni

Post by Sivaramakrishnan »

Srinath Ji,
My sincere plea is that you must endeavour to present a paper in the ensuing Season preferably at the morning lecdem series of Music academy. The topic deserves deliberation among experts.

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#12 Re: Challenge to Bharata Muni

Post by shankarank »

SrinathK wrote: 24 Jul 2018, 12:25
In fact the M2 is the only one without a valid panchama for itself (a perfect 3/2) - it gives only 11 tones (an ancestor of panchama - less scales?). If you want a panchama you have no choice but to use a diminished 5th (40/27) using S. So there is in fact a theroetical, mathematical basis for madhyama grama from this 22 sruthi scale, at least.

R1 in turn is the only one without a valid madhyama. It doesn't gel well with the P at 3/2 either.
There is a Prf. SRJ Lec Dem at parthasAraty svAmi sabha, where he discussses the generation of kalyANI through the cycle of fifths and asks rhetorically "how do people quote some lakshaNA kara who claimed kalyANI is not suitable for tAyA and AlapA??" Adding " one should not take lakshaNA karas words too literally as true (appaTTAM)". He goes on to state that dvimadyama rAgAs must have been known ? And demonstrates the new cyUta pancama.

That recording was either in Sangeethapriya or somewhere - cannot locate it.

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#13 Re: Challenge to Bharata Muni

Post by SrinathK »

I hope though that we are not comparing gamaka based music to plain tone music. Kalyani as it is today has no relation to a scale derived from a microtonal system.

Also while Bharata muni's check points for verifying shrutis by checking the tuning of 2 Veena's are fine, when I think about what I wrote in post #9, there is more than one way to interpret his mistake.

If his words are taken literally, the Natyashashtra seems to have gone from P to P' and then if his next checkpoint is to be believed, skipped m2 (yes, he skipped m2) and gone directly to m1, then M2 and finally M1 and declared that there are 4 shrutis between n1 and S. He lost a valid interval there.

This paragraph is misleading on the correct sequence of shrutis from P to M1 and by extension the rest of the scale. The correct order of pitch reduction factors in descending from upper S or P is 243/256, 80/81, 24/25 and again 80/81. If 3/2 of P is multIplied successively by these 4 factors, you get 4/3 of M1. Along the way you get m2, m1 and M2 as well.

But if Bharata' procedure is to be believed, he swapped the first 2 factors in trying to describe his tuning check. When applied in this wrong order, the whole 22 shruti scale goes out of the window and you get weird intervals like 160/81 for N2, lose N1, get a new syntonic comma below d1 and another below M1 (320/243), skip G2 and what not , and 2 panchamams. Weird huh? Imagine the dissonance.

Even the Carnatic corner article on the 22 shrutis which is excellently written fell for this blunder and thought P' was a valid shruti. However this is only the diminished P of madhyama grama.

For a moment there I thought Bharata had come up with a totally different 22 shrutis of intervals many of which have no meaning in any system of tuning.

Fortunately we have many other checks which are in fact correct and tally with the math.

To answer the question as to how 2 Veena's or harps can be tuned down by 1 pramana shruti, one can tune down until G2 of the 2nd harp coincides with the G1 of the first and the dissonance vanishes. This kind of micro adjustment simply cannot be done without cross checks.
Last edited by SrinathK on 25 Jul 2018, 08:39, edited 1 time in total.

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#14 Re: Challenge to Bharata Muni

Post by uday_shankar »

SrinathK wrote: 24 Jul 2018, 22:35Pythagorean comma
I think you're repeatedly and erroneously using the term Pythagorean comma to mean pramana shruti.

Didymus comma = Synotic comma = Ptolemaic comma = pramana shruti = 81/80

The Pythagorean comma is a completely different ratio and comes about because cycles of pure fifths never quite meet up with cycle of pure fourths. Specifically it is the ratio:

(3/2)^5 / (4/3)^7 = 3^12/2^19 = 531441/ 524288 = 1.0XXX

i.e., 5 cycles of fifths come closest to 7 cycles of fourths but far from close enough to wink that they are the same note :). Of course, the problem was famously "solved" with equal temperament where these two were adjusted to be equal... The fifth took a small hit and went a little flat and fourth took a little bump and went a little sharp so they land at the exact same pitch five and seven cycles later respectively. Minimal deviation in any given octave (definitely an order of magnitude lower than the average male CM singer of yesteryears' apashruti). I think ET is a splendid invention by CPE Bach and others. The real flaw of further mutual adjustment that make up the ET, as far as Indian music is concerned is the extra deviation in the other intervals, most egregiously the major third (G3).

Indian tuning systems never talked about about cycles of fourths or fifths afaik. It was all about pramana shruti.

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#15 Re: Challenge to Bharata Muni

Post by SrinathK »

Yes I was making an error. It should be syntonic comma. I edited my posts now.

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#16 Re: Challenge to Bharata Muni

Post by uday_shankar »

Sivaramakrishnan wrote: 24 Jul 2018, 17:24deliberation among experts.
SRK, who are these "experts" ? Anybody who does anything practically ? Or is it more theory and numbers and hand waving ?

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#17 Re: Challenge to Bharata Muni

Post by Sivaramakrishnan »

uday_shankar wrote: 26 Jul 2018, 18:22
Sivaramakrishnan wrote: 24 Jul 2018, 17:24deliberation among experts.
SRK, who are these "experts" ? Anybody who does anything practically ? Or is it more theory and numbers and hand waving ?
Uday,
Experts: those who have a say in the topic / related areas.
I consider you as one.
Need not be a member of the experts committee for deliberating.

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#18 Re: Challenge to Bharata Muni

Post by SrinathK »

Practically everything that needs research on this topic is done.

You know what we need? An intelligent keyboard that can itself decide which combinations of notes give the best possible melodic or harmonic combinations. And it can also play in keys of C, D, F#, B flat, as needed, by setting the shadja frequency accordingly, so that key changes don't upset it and stop it from playing modern music. (That is one limitation of the harmonium, as the key is fixed, and while you can get 12 tones out of many positions, you won't always get the set you need - singing in another shruti means a new harmonium).

The chords played with the 22 shruti scale frequencies are substantially more consonant than the ones played on 12-ET. https://www.youtube.com/watch?v=p_CBjW2zt1A

Melodically, the tambura is a much later invention than the harps and yazhs, and all of these 22 will not gel well with it.

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#19 Re: Challenge to Bharata Muni

Post by uday_shankar »

SrinathK wrote: 27 Jul 2018, 12:52The chords played with the 22 shruti scale frequencies are substantially more consonant than the ones played on 12-ET.
Srinath, this is not profound news. The basis of chords is natural consonant intervals and the ET was created for the strength of expanding polyphony and harmony.

As a matter of practice, I tune three sets of chords on the chitravenu according to the rAga used (you should go check some clips) according to natural consonance. We don't need a side by side demo of natural vs equal tempered to know the difference. Anybody who has practised listening and tuning knows what the difference is. This still does not need 22 shruti theory... a major chord is 1 5/4 3/2 and a minor chord is 1 6/5 3/2. Now the intervals 3/2 (pa) and 5/4 (G3) are precisely tunable by listening to harmonics, i.e., consonance. Those are all the intervals you need to tune any fundamental chords or for that matter all notes of a rAga. For example, a minor chord can be tuned by coming down to 4/5 of the the fifth (3/2), whereby you get 6/5. For my ear, 6/5 is a little too sharp, so I bring it down a tad (can't guarantee it to be 32/27 precisely, but close) but keep it consistent with all other G2's. End of story. I play a completely consonant zither, scale wise and chord wise.

Without a multi string zither / harp that is used in some context, all of this talk is nonsense because it has no impact on anything being practiced. One can generate any frequency to any precision with a digital device and if we're going get excited about it merely on the basis of a priori knowledge that something is mathematically precise, heaven help us. I can create a robot cyclist that can coast through the velodrome better than any living, breathing person. So the real joy is in the journey, of experiencing the creation of true consonance approximations, sometimes with spectacular precision at other times not.

Rapid key modulation, the basis of all harmony and polyphony, is practically impossible without something like equal temperament or mean tone temperament.

Finally, for somebody in Carnatic music to be talking about precision of notes is in itself nonsense. It has been proven beyond all doubts that Carnatic music is only about the melodic shapes and not about precision of swarasthanas. So even equal temperament works as good as anything else.

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#20 Re: Challenge to Bharata Muni

Post by SrinathK »

Speaking of experts, listen to the 22 shruti harmonium at work :

http://www.22shruti.com/homepage_22_shr ... ultani.asp
http://www.22shruti.com/homepage_22_shr ... eshkar.asp
http://www.22shruti.com/homepage_22_shr ... _marwa.asp

And because the tones of 12ET are sandwiched between two shrutis for a given note, it can actually play equal temperament also, if one knows how to pull out the knobs part way and match it against a pitch pipe :

http://www.22shruti.com/22_shruti_harmo ... ordian.asp

uday_shankar
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#21 Re: Challenge to Bharata Muni

Post by uday_shankar »

I like Dr Oke's 22 shruti harmonium... in a saner world it would be widely adopted and become the de-facto harmonium for Hindustani music.

My friend Warren Senders tells me that artists with fixed shrutis like D# etc, used to custom tune their 12 tone harmoniums to natural intervals so they minimize discordance with the drones.

With Dr. Oke's 22 shruti harmonium, that is prebuilt and offers a lot more options for any given rAga. You can just push or pull knobs and develop a set of notes/shrutis for the raga you feel is most appropriate.

Over the last decade or so, harmonium tuners have started using the digital chromatic tuners for filing their reeds so inadvertently they may be corrupting the intonations which they did with some "traditional" ear training.

I use a shruti box (i.e., with bellows) when "performing" (i.e., once in a blue moon) which I had made and get regularly checked at Kannan Musicals on Mint Street, in Sowcarpet. The owner there is one of the few experts left in Chennai who can precisely file the reeds of a shruti box. And he told me how that shruti box panchama is slightly sharper than the harmonium panchama, without any knowledge of ratios or temperament !

So yeah, Dr. Oke's harmonium is cool and would be nice if widely adopted, but still doesn't answer my original challenge to Bharata :p.

SrinathK
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#22 Re: Challenge to Bharata Muni

Post by SrinathK »

There is one way - take R1 on one harp and tune the P string of the other to a perfect 4th above it, or use the R1 string on the 1st harp. It will tune to the P' from the original 24 tones. The rest of the detunings can be done by unison.

It is also possible to create madhyama grama by simply using M2 as shadja and retuning every other string except the original S strings. That automatically creates a diminished 5th.

It might be that this text is simply a thought experiment as a check for the sruthi values, but apart from that one glaring procedural slip in going from P' to m2, it does work out.

22 shrutis aside, how did they tune these yazhs (which had only a limited number of strings) in the first place?

uday_shankar
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#23 Re: Challenge to Bharata Muni

Post by uday_shankar »

Srinath,
Possible. Just keep in mind that a super intelligent person like you can "explain" and "rationalize" everything, find ways out in theory, but at the end of the day we have to take a hard look at equipment, procedure, experimental feasibility, repeatability, daily practice, etc...

As for tuning yazh's, that's not hard. Santoor players, sitarists, sarodists, chitravina players (and chitravenu players) routinely use some technique to tune their strings to the scale of ragas. Similarly veena fret setters. No musician or veena tech. I know has had any specific "22 shruti" training.

So, take any common rAga, here's how:

Shankarabharanam
S -
R2 - panchama consonance with panchama, 9/8
G3 - consonant third with S (listen to harmonics and/or beats, it is precisely tunable) 5/4
M1 - consonance with S 4/3
P - consonance with S 3/2
D2 - consonance with G3 (this gives 5/3, which I prefer, but if you want 27/16 you can use consonance with R2)
N3 - consonant third with P (listen to harmonics) 15/8

Slightly sharper notes can be tuned by ear... for example, if we think G3 could be slightly sharper in a raga like Kalyani, we can first get to 5/4 and then sharpen it to preference and correspondingly sharpen N3 too (mutual consonance). If your judgment is good, perhaps you might hit 81/64 precisely (probably a little higher in reality!).

For ragas, with R1 like Saveri, I used to tune them as a third down from M1 (4/3 X 4/5 = 16/15) and pull them further down a little since I am more comfortable with something like 90 cents (even less than the equal temperament of100 cents) than the "standard" value of 16/15.

By some set of mutual and successive precisely tuned consonances, strings in a couple octaves can be brought into tune. Needs good ear, otherwise a complete failure. Novice chitravina players or those without the ear discrimination are advised to simply stuff the sympathetic strings with cloth and not worry about them.

NOTE: I've used beats above, which is commonly understood to mean the beating when there are small differences in frequency but there's a distinct "feel" one can get when you get a perfect panchama or G3 (5/4). In fact such a "feel" is possible even for S-D2 or 1 to 5/3, I have no idea why. I can tune 5/3 just by "feel" with just shadja... I think maybe because shadja has 5/4 embedded in it as the fifth harmonic and by deep listening you can kind of tune to the consonance of that...

shankarank
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#24 Re: Challenge to Bharata Muni

Post by shankarank »

uday_shankar wrote: 29 Jul 2018, 17:45 NOTE: I've used beats above, which is commonly understood to mean the beating when there are small differences in frequency but there's a distinct "feel" one can get when you get a perfect panchama or G3 (5/4). In fact such a "feel" is possible even for S-D2 or 1 to 5/3, I have no idea why. I can tune 5/3 just by "feel" with just shadja... I think maybe because shadja has 5/4 embedded in it as the fifth harmonic and by deep listening you can kind of tune to the consonance of that...
Isn't that what consonance is? Cannot understand why you asked that question? Just in case I misunderstood.

You can look at it two ways : The frequency of the beats is the difference in frequency. so 3/2 - 1 = 1/2 - the frequency of the lower ShaDja.

5/4 - 1 = 1/4 - the frequency of the lower, lower shaDja.

I learnt the consonance from M. Subramaniam's rasikas disk that I bought from Sashikiran.

Reasonance is between 1, 2 , 3. But after that the best superpositions, somewhat pleasing to the ear is then, between the lowest mutual prime integer ratios like 5/3, 7/4, 9/5, 6/5 . You will likely hit those when jumping from 'sa' assuming your system is optimizing consonance. But it may not, as it may be influenced by a rAgA feel and training.

Other reasoning behind it is , for any signal, harmonics attenuate ( reduce in amplitude) faster as the terms ( of Fourier) progress from 2, 3, 4 and so on. So an 81/64 in relation to 'sa' will be the consonance caused by the 81st harmonic of sa with 64th harmonic of 'ga' - highly attenuated. You are actually sensing 81/64 because it is closer to 5/4.

Only other way is relation to 9/8 -> 27/16 -> 81/64 assuming your system has graha bhedammed to 9/8!! The latter is too far fetched!! More so because the first basic consonance is between 27/16 and 81/32 and you are back deriving the lower "ga" within the middle octave.

May be if somebody is executing da-ga'-ri' kind of phrase switching octaves this is possible.

All said, then the issue seems to go back to the tone. A good tonal voice is able to produce rich harmonics with lesser attenuation at least for first few terms ( in Fourier expansion!) , so you can feel the consonance better?

uday_shankar
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#25 Re: Challenge to Bharata Muni

Post by uday_shankar »

shankarank,

Agree with all your points and analysis. Just that "beating" is generally understood to be of a slower kind, i.e., when the frequencies are much closer together. However, when we understand from the frequency domain, consonance becomes crystal clear and the way you state consonance is exactly the way I explain it to kids (right or wrong !).

And yes, for any kind of musical perception, consonance, etc it is all about the first few harmonics.
A good tonal voice is able to produce rich harmonics with lesser attenuation at least for first few terms ( in Fourier expansion!) , so you can feel the consonance better?
Absolutely. With the additional caveat that keeping things flexible and elastic reduces inharmonicity. So we need rich harmonic content AND they should not inharmonic.

While we're at this, I'd like to point out that you won't find the problem of inharmonicity and untunability discussed much anywhere, but that's the crux of the problem with many tonal and shruti issues. Inharmonicity happens when the spatial component of the solution of the wave equation suffers a distortion for physical reasons and spills into the temporal part. I have investigated this with voice, strings and of course most significantly with wind. I am working on a paper on flute overblow inharmonicity (in my spare time!) and the optimization problem in the context of chitravenu. You can't avoid it, but you can minimize it or take it to regions where it doesn't matter much, such as lower frequencies. This means, that you choose the bore of the flute smaller, reducing the inharmonicity in higher fundamentals, for example.

Inharmonicity plays tricks with the ear. Inharmonicity of piano strings has received wide research attention. Longer string span, for simple physical reasons, are less inharmonic - there's a formula of some questionable merit for calculations using an "inharmoncity coefficient" for strings. Let's call this C. Let the theoretical frequency of any harmonic be

fh = n*ff where ff is the frequency of the fundanmental and n the harmonic number.

Then the actually generated "harmonic" frequency i.e., the inharmonic one, is realated to the true one by:

fi = fh*sqrt(1 + C*n^2).

So this kind of gives you an idea of how the inharmonicity behaves as n goes up... the square term means the rate increases, albeit somewhat smothered by the root. Trivially, as C->0 , fi = fh, i.e there is no inharmonicity. When n is small, i.e., the "first few harmonics", fi comes closer to fh.... and so goes the blah blah. The formula for C itself is hard to type but involves the usual suspects - cross-section, tension, length, Young's modulus of the string material, etc..

For normal length piano strings (i.e., upright, baby grand, etc) often piano tuners stretch the octave according to the empirically derived Railsback curve. Perhaps with modern audio acquisition systems and analytical tools, a better measure can be generated in real-time or maybe is being done already.

For tambura, some tamburas can never be tuned (short span, thick strings, bad bridge) to satisfaction.. Notice how long the necks of those tamburas used by Gundecha brothers are ?

Imagine the inharmonic state of a Carnatic voice abused for decades singing brigas meant for nagasvaram ? Or spent too much time practising chromatic passages (vivadhi melas!)... Even unison consonance becomes difficult for such a voice because of the power of inharmonicity :(.

Circling back to the notion of the "first few harmonics" and the optimization problem, it becomes clear why we MUST have good attenuation of higher harmonics :

1. We need rich harmonic content because we can feel consonances better
2. There will always be some level of inharmonicity
3. Inharmonicity increases with harmonic number
4. So an optimal voice/tone is one rich in "first few" harmonics, minimal inharmonicity
5. Good attenuation of higher harmonics is necessary (per point 3 above) ????

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