A controversial article about the 12-tone Pythagorean scale in India

Hi tuners.

I've found an article here:
http://www.anaphoria.com/kolin.PDF

BTW: Although it was saved as graphics, I eventually managed to "very roughly" convert it to plain text for my personal needs. :-D

The author essentially claims that Bharata was describing the 12-tone chromatic scale in strict Pythagorean tuning -- and that you get the list of all the "shrutis" by listing all the intervals in each interval class.

However, should this be true, then all of the modes used in indian Ragas would have to be reproducible in the 12-tone Pythagorean chromatic scale, which they certainly aren't. If you look in manuels "modename.html", you find many modes which, after converting to Pythagorean notation, contain doubly-augmented primes and similar stuff. Another story is, of course, that I have absolutely no idea where Manuel found the Indian modes.

Any suggestions/comments/ideas about this contradiction?

Thanks in advance.

Petr

--- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@...> wrote:
> http://www.anaphoria.com/kolin.PDF
> ... strict Pythagorean tuning ... should this be true, then all of the modes used in indian Ragas would have to be reproducible in the 12-tone Pythagorean chromatic scale ... Any suggestions/comments/ideas about this contradiction?

As I understand it, neither Bharata nor Kolinski limits it to 12 notes - rather it is a chain of 22 of the Pythagorean interval 3/2; if the origin note (Sa) is taken to be C in Western notation, then the chain runs from Ebb to E#. I believe that it stops there because B# would be too close to C (=Sa) and Abb would be too close to G (=Pa). Caveat: this is just from my reading, no practical experience.

See also post #72714 which gives links to another proposal (Sambamurthy) as well as Kolinski. Also, I just found:
http://www.22sruti.com/FAQ.html
(see the last paragraph for the ratios)

Steve M.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@> wrote:
> > http://www.anaphoria.com/kolin.PDF
> > ... strict Pythagorean tuning ... should this be true, then all of the modes used in indian Ragas would have to be reproducible in the 12-tone Pythagorean chromatic scale ... Any suggestions/comments/ideas about this contradiction?
>
> As I understand it, neither Bharata nor Kolinski limits it to 12 notes - rather it is a chain of 22 of the Pythagorean interval 3/2; if the origin note (Sa) is taken to be C in Western notation, then the chain runs from Ebb to E#. I believe that it stops there because B# would be too close to C (=Sa) and Abb would be too close to G (=Pa). Caveat: this is just from my reading, no practical experience.
>
> See also post #72714 which gives links to another proposal (Sambamurthy) as well as Kolinski. Also, I just found:
> http://www.22sruti.com/FAQ.html
> (see the last paragraph for the ratios)

Actually I recall that the chain of 22 x 3/2 came from here:
http://www.themusicmagazine.com/kvranganint2.html

I have not read Kolinski carefully - but his main table seems to have the same power of 3/2 in more than one row, which looks wrong.

Steve M.

Steve wrote:

> As I understand it, neither Bharata nor Kolinski limits it to 12 notes -

Not sure about Bharata but Kolinski apparently does, even in the introductory paragraphs. He picks a couple of reasons why he's doing so and later he demonstrates it by constructing the Pythagorean chain in a quite unusual way -- he tunes 6 rising fifths from F on one instrument and 6 falling fifths from F on another, then (to get rid of the Pythagorean comma) he lowers both E and B by one Pyth. comma, which results in a chain from Eb to A with Cb being playable on both instruments.

> rather it is a chain of 22 of the Pythagorean interval 3/2; if the origin > note (Sa)
> is taken to be C in Western notation, then the chain runs from Ebb to E#.

I would also expect something like this but Kolinski never goes that far. To the question of 22 shrutis, his answer is that these aren't actually there from one common starting degree but rather they come out if you list all the intervals (from each degree) which are smaller than an octave. It takes a bit of time to accept a claim like this but it seems to make some sense.

> See also post #72714 which gives links to another proposal (Sambamurthy) > as well as Kolinski.

Thanks, I'll find if I'm able to "crack down" Sambamurthy.

> Also, I just found:
> http://www.22sruti.com/FAQ.html

I don't buy that at all. On one hand, this guy seems to offer many possible variants of each Shruti; on the other hand, he excludes intervals which have very good approximations even in Pythagorean tuning (like 16/15) and includes others that would be terribly difficult to include in a chain of fifths unless you mistune them heavily (like 12/11).

> Actually I recall that the chain of 22 x 3/2 came from here:
> http://www.themusicmagazine.com/kvranganint2.html

Right, this fits perfectly to my way of thinking -- a chain of fifths from one view, 5-limit approximations from another.

> I have not read Kolinski carefully - but his main table seems to have the > same power of 3/2
> in more than one row, which looks wrong.

I'm not sure about his results either. He sais that if you list all of the intervals contained in the scale from each degree (in each interval class), you get the 22 shrutis. Later, he states that you exclude "the 1224 interval" since it's larger than one octave. This only makes me confused because how can I arrive at a 1224 interval if the 12-tone scale doesn't include the comma? Moreover, if I list all the intervals from each degree and in each class, I have to exclude the diminished sixth to get the usual 22 shrutis, for which I've read another explanation somewhere else saying something like "this is done because the octave and the fifth are included only in their pure forms without any commatic alterations".

Petr

>I wrote:
>
>> he lowers both E and B by one Pyth. comma, which results in a chain from >> Eb to A with Cb being playable on both instruments.
>
> Goodness, I meant Fb to A, of course. :-D
>
> Petr

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
> Actually I recall that the chain of 22 x 3/2 came from here:
> http://www.themusicmagazine.com/kvranganint2.html

I mean 21 intervals giving 22 notes, of course.

> I have not read Kolinski carefully - but his main table seems to have the same power of 3/2 in more than one row, which looks wrong.

My problem with the table on page 9 is that he has stated some of the ratios back-to-front (see e.g. rows 11 and 12 - if (3/2)^6:2^3 is 612 cents then (3/2)^6:2^4 is *-588* cents.

His derivation is difficult to follow (e.g. although he mentions a limit of 12 notes, and a "projection onto" 12 notes, he then starts with 13 notes (top of page 8) and extends it to 18 notes ... anyway I think the idea is that his scale is not tempered (I think he means it does not use octave equivalence so Cb and B are 1224cents apart at this stage) and not limited to 12 notes.

In effect, I think he is using a chain of 21 x 3/2 and octave equivalence.

S.

Steve wrote:

> His derivation is difficult to follow (e.g. although he mentions a limit > of 12
> notes, and a "projection onto" 12 notes, he then starts with 13 notes (top > of
> page 8) and extends it to 18 notes ... anyway I think the idea is that his > scale
> is not tempered (I think he means it does not use octave equivalence so Cb > and B
> are 1224cents apart at this stage) and not limited to 12 notes.

#1. Not sure where he's got 18 but let me emphasize what is obvious from reading the actual text, without delving into the tables any great deal.
Bharata demonstrates tuning the scales using two "vinas". Kolinski explains it by saying that the ancient vina, unlike the modern one, apparently didn't have a fingerboard, which means that a 7-stringed vina could play no more than 7 different pitches.
For the first vina, he starts the procedure with F (since Bharata started it with "Ma" and Kolinski chooses C as "Sa") and tunes 6 "falling fourths or rising fifths" from that. Then, for the other vina, he starts with F again but this time he tunes 6 "rising fourths or falling fifths" from that. If the starting tone was, let's say, an F3, then the pitches of the first vina would become "F3-C3-G3-D3-A3-E3-B3" and the pitches of the second vina would be "F3-Bb3-Eb3-Ab3-Db3-Gb3-Cb3". This gives him a scale of 13 tones in an octave, ranging from Cb1 to B1, where the interval between Cb1 and B1 is ~1224 cents. Finally, he gets rid of the Cb-B interval by lowering both E and B by the Pyth. comma -- i.e. first he changes the B of one vina to match a pure octave of the Cb of the other, then he changes the E to a falling fifth from that. This means that the strings on one vina are then tuned to "F-C-G-D-A-Fb-Cb" and the strings on the other are tuned to "Cb-Gb-Db-Ab-Eb-Bb-F". Although Kolinski never confirms it, I think that Bharata's reason for retuning E to Fb was to approximate the 5/4 major third above C (which is about 2 cents away). Interestingly enough, this is where Kolinski's paper ends, leaving us with a 12-tone Pythagorean scale from Fb to A. This may seem surprising at the first glance but we shall recall Kolinski's introduction to the matter where he says that the 22 shrutis are not contained in the scale from one single degree but rather as a list of all the intervals from each degree in each interval class. Again, this may seem very surprising and "courageous" of Kolinski but it makes perfect sense in the context of what precedes and what follows.

#2. I think I should thank you a lot for the "Indian music magazine" link. Varadarangan seems to say, which would support Kolinski's view, that not only the ratios should be strictly Pythagorean but also that when applying the intervals to music, the consecutive swarastanas (intervals in a chromatic scale) should always be 90 or 114 cents, which allows achieving other intervals by shifting the 1/1 to other pitches of the scale, while 12-equal doesn't make this possible because of the equally sized semitones of 100 cents. Inspite of that, unlike Kolinski, Varadarangan doesn't seem to be so strict about the entire scale having only 12 tones but he rather offers the possibility to begin with a 12-tone scale from a single degree and then make a 12-tone scale from each of its degrees. Although he finds some flaws in Sambamurthy's explanation, both Varadarangan and Sambamurthy say that there are 7 swaras in an octave, which means 12 swarastanas, which means 22 shrutis.
My conclusion is that maybe the old Indian tuning only contained the 12 swarastanas (which supports Kolinski's claim that the shrutis are not in the scale all from a single degree) but as time went on, musicians wanted to make it possible to use different degrees as the tonic, which eventually made it necessary that the all of the 22 shrutis were contained in the scale from a single degree.
Now, the only thing that doesn't fit into all this is Manuel's "22-tone modes of Indian shruti scale". There I find modes spanning more than 11 Pythagorean fifths. My personal explanation for this is that after making it possible to shift the tonic of the chromatic scale, people started making modes (in the resulting 22-tone scale) which would be impossible to achieve with 12 tones only.

Petr