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Indian Classical music intonation chart
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cpiasminc
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August 14, 2012 - 6:40 pm
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Hey, all...

It's been a while since I've been back here, and actually active (I've only had a few chances to lurk now and then).  Actually, because of traveling on account of work (my most recent paper was published in two journals), I've hardly even been home much lately, and not having much time for anything (especially not sleep) when I am, and that also means it's been over a month since I last even got a chance to practice for more than 10 minutes at a stretch.  But then, it's not as if the music died...  just that work is eating up all my time for now and wiping me out in the process.  fainting-1344

Anyway, I did say that I'd elucidate some more information about Indian classical music as I went along, so I did some work on paper after seeing the thread on the fretted violin, and I started figuring out how a fret layout for Carnatic music (just intonation and all) would look. Here are my results.

 

First, I'll go over the fret positioning for equal temperament just for comparison.  This is assuming a playing length of 328 mm, and it covers 1 octave range in half steps.  The positions below are in mm (distance from the nut, so 0 would be the open string)

0, 18.409, 35.785, 52.186, 67.666, 82.278, 96.069, 109.086, 121.373, 132.97, 143.916, 154.248, 164

Equal Temperament positioning curve        Equal Temperament cumulative sum

Equal temperament fret positioning on line

The charts above show a positioning curve, and a cumulative sum curve (both to show how close it is to the "ideal" log-scale and/or pseudo-catenary curvatures)...  which of course, equal temperament would get you as close as you can conceivably ever get without having infinitely many intervals.  The final chart puts the positions on the number line to visualize to scale how far apart the frets would be.

The intonation used in Carnatic and Hindustani music is actually surprisingly close, and while there are technically 22 pitches rather than 12, many are only a Pythagorean comma (81/80) off from each other, and many are otherwise treated as enharmonic equivalents, and in practice you end up only using 12 (dwadasa swarasthanas).  In most of the resources out there, people will actually use 16 (shodasa swarasthanas), but this is inclusive only of enharmonic equivalents, so it would still be assuming 12 unique pitches.  There is some occasional disagreement over which 12 should be the default, because you will occasionally use different variants out of the 22 for very special cases (mostly related to the history of songs in these particular scales), but I plotted the positions that I feel cover the majority of ragas in almost all cases, and elucidate the general feeling of the scales (what we call "bhaavam") in the most generic way.

0, 16.6562, 36.44444, 51.25, 68.8395, 82, 94.7556, 109.3333, 120.438, 133.63, 143.5, 155.226, 164

Carnatic temperament positioning curve        Carnatic temperament cumulative sum curve

Carnatic temperament fret position on number line

You can see a little more irregularity and not quite so "smooth" look to the position curve, but the cumulative sum curve is pretty similar, although it has a more hyperboloidal look to me.  By in large, this implies that the overall sound difference is not huge, but it can be noticeable given the right circumstances.  In the fret position line, you see a lot more instances of bunched-pairs rather than regularly shrinking spacing between frets.

So to give you an idea of the actual frequency ratios I used to get this, here they are.  I'm labeling them according to their 16-pitch labels (because you'll find most resources use the 16-pitch system), but also giving the enharmonic equivalences --

S(tonic) = 1.0
R1 = 256 / 243
R2 = G1 = 9/8
G2 = R3 = 32/27
G3 = 81/64
M1 = 4/3 (just perfect 4th)
M2 = 45/32
P = 3/2 (just perfect 5th)
D1 = 128/81    (notably also equal to R1 * P, as well as inversion of G3)
D2 = N1 = 27/16
N2 = D3 = 16/9
N3 = 243/128
S(octave) = 2.0

Just so you know, the letters refer to the swara syllable names (our solfeggio counterpart).  S -> Sa (tonic), R -> Ri (2nd), G -> Ga (3rd), M -> Ma (4th), P -> Pa (5th), D -> Dha (6th), N -> Ni (7th).

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Fiddlestix
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August 14, 2012 - 7:36 pm
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Huh..... ?dunno

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Kevin M.
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August 14, 2012 - 8:12 pm
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This was very interesting. The cumulative sum curve is odd that it is so similar I would have expected it to flatten out sooner.

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KindaScratchy
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August 14, 2012 - 8:20 pm
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You lost me at "First, I'll go over the fret positioning for equal temperament just for comparison." The rest looks way too much like math.

dunnodazeddroolingrofl

It sounds fascinating, though, and I'm sure others will eat it up. Looking forward to following the ensuing conversation (well, I'll try to follow it, anyway).

When the work's all done and the sun's settin' low,

I pull out my fiddle and I rosin up the bow.

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Fiddlestix
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August 15, 2012 - 1:05 am
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Is this a system I need to know if I wanted to play Indian music or can I just wing it and play by ear ? 

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Fiddlerman
Fort Lauderdale
August 15, 2012 - 8:17 am
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KindaScratchy said
You lost me at "First, I'll go over the fret positioning for equal temperament just for comparison." The rest looks way too much like math.

LOL - Me too. blurry_drunk-2127lumpy-2134violin-1260

"The richest person is not the one who has the most,
but the one who needs the least."

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cpiasminc
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August 15, 2012 - 12:31 pm
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Kevin M. said
This was very interesting. The cumulative sum curve is odd that it is so similar I would have expected it to flatten out sooner.

Yeah, the use of the cumulative sum is something I learned from my psychoacoustics professor way back.  At the face of it, it makes no sense to really add together the distances of fret positions, but he said that it is just a conveniently peculiar comparative tool to see how much of a difference is really audible.  Why it has that significance is still a mystery to me.  I see the analogy to numerical quadrature and all, but I'm not aware of any "calculus of fretted instruments," so it's hard to place what that quantity would really "mean" per se.

It would be darn cool if there was something like that, though.  Is it obvious that I'm a geek?

Fiddlestix said
Is this a system I need to know if I wanted to play Indian music or can I just wing it and play by ear ? 

You can wing it for the most part.  I really just did the math of calculating fingering/fret positions out of sheer geeky interest, and felt like sharing since it is violin-related.  The point of the frequency ratio table at the end was less about being able to play the music as it is about being able to understand the other resources out there on the Web -- e.g. if you look up specific ragas on wikipedia or something, you'll see the scales mentioned with those interval labels I mentioned.

The main thing that would be different from what you're used to in the setup is that the string tuning is different -- rather than 1 fifth apart, it's actually in pairs of fifths with each pair an octave apart.  i.e. instead of GDAE, it's GDGD or AEAE or CGCG (viola) or AbEbAbEb , etc.  This is so that a particular fingering position equals a particular note regardless of whether you play on the G string or the A string.  This is partly necessary when you don't have a fixed frequency ratio between notes (equal temperament gives you that), but it's also generally necessary in Indian music because it's a system where the tonic and the fifth are the only "definitive" notes.  Every other note has the potential to be some sort of wave or oscillation or something that slides from an indeterminate position to the actual pitch.

About the only pitch that stands out as noticeably different is the minor 2nd interval from the open string (the very first fret position) -- it does seem as if 2 mm difference isn't big, but it's actually noticeable to me (it is ~10% error in positioning, after all), but part of it is also the psychoacoustic effect of the space between notes.  For instance, a raga like Vakulabharanam (equiv. to the Phrygian Dominant scale) has a minor 2nd and a major 3rd, but in a scale like this one you would notice the lower 2nd in the Indian intonation specifically because that gap between the 2nd and 3rd feels bigger.  But as you go higher up, the space between fingering positions is so small, that the differences between the intonations are hard to perceive, and the "error" relative to equal tempered positions is less than 1% up there.  This is why a lot of people in recent years will play Carnatic or Hindustani music using equal tempered equivalents (in recent years, it's a little more common, but you can go back to George Harrison or Yehudi Menuhin for older examples).  You also have guys like Jon B. Higgins who trained and performed in several systems and even he points out that there's not *that* big a difference in the actual intonation between Western, Indian, Arab, and classical Chinese music -- it's really all about the handling.

Of course, I'd expect the difference to be much bigger for a bigger instrument, which is why we may have Carnatic violinists, and only 1 Carnatic violist that I'm aware of, but no Carnatic cellists or anything similar.

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Kevin M.
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August 15, 2012 - 1:20 pm
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Just a quick question. Since you tune your violin to GDGD do you still use the same strings for a violin tuned GDAE or are there different strings? If you use the same strings I wonder how strings made for GDGD would effect the sound.

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cpiasminc
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August 16, 2012 - 5:44 pm
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Kevin M. said
Just a quick question. Since you tune your violin to GDGD do you still use the same strings for a violin tuned GDAE or are there different strings? If you use the same strings I wonder how strings made for GDGD would effect the sound.

Well, I definitely use normal strings, and I've not heard of anybody making alternate strings for the violin at any point.  Another point is that if I am supposed to play pakhavadhyam (i.e. as an accompanist), then I'd tune to the voice of the person I'm accompanying, so it's a bit of a wash to try and find specialized strings.  e.g. If I'm playing pakhavadhyam for my dad (who sings in C#), for instance, I'd either tune to G#C#G#C# and play one string lower, treating the D string as a tonic string rather than as a fifth string, or I'd alternatively tune to G#D#G#D# and play normally at a point in line with the fifth drone of the tambura rather than the tonic drone (which is the only interval we'd consider harmonically acceptable in Carnatic music).

One thing I do find is that, depending on the quality of the strings you have, if you go too far out of the "traditional" tuning for which it's made, the sound suffers greatly.  On my ultra-cheap steel strings that came with my violin, if I tune to G#-C#-G#-C#, the lower strings sound mostly okay, but the E string especially starts to sound really flabby and hollow (think of the sound of a violin without a sound post).  The A string also starts to go there, but at that level you can adjust for it by bowing a little closer to the bridge.  The lower I tune, the less leeway I have to play softly before the sound is so empty and weak.

I can only imagine that specially-made strings for the particular tuning would be a little more room for forgiveness.  For instance, a string made specifically to be tuned to a high D instead of an E would have just that little bit more mass, so it would project a little more cleanly even if I play softly.  But with what I have, I can't even begin to consider going much lower, or it just sounds like I'm playing a violin while sitting inside a metal tube.

In general, because the G and D strings are thicker, they can take a little up-tensing, so if I'm playing for myself, it's a little easier to tune to AEAE, and all the strings sound a lot better at almost any level of pressure and bowing speed (though slides and oscillations still sound horrible with soft bowing).  Even at AbEbAbEb, it's generally fine.  I think better strings would take me farther.

I imagine if M.D. Ramanathan was still alive, and I intended to tune to the tonic drone in the same octave for him...  I'd probably need strings with depleted uranium cores just to get the necessary mass.  tongue

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Kevin M.
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August 16, 2012 - 6:35 pm
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cpiasminc said

 

I imagine if M.D. Ramanathan was still alive, and I intended to tune to the tonic drone in the same octave for him...  I'd probably need strings with depleted uranium cores just to get the necessary mass.  tongue

 

roflol

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Fiddlerman
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August 18, 2012 - 10:07 am
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I think Kevin has a good point. Maybe even buy high tension strings instead. That would probably even everything out.

"The richest person is not the one who has the most,
but the one who needs the least."

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