The ass that can sing in all gramas [was: [tuning] Shruti Researching]

Dear Haresh, dear tuners,

I started reading this list about a month ago, looking for clues to
evaluate an idea I have about the 22 srutis--an essay in historical
reconstruction, if you will, and quite different from your intent;
actually, I will argue that you are trying to give meaning to a word
that has been misunderstood for centuries because the underlying
practice is no longer there. I have to confess that my sources are not
necessarily reliable (the internet :o}[not only]), that I have a hard
time telling apart the Northern and Southern traditions and generally
don't know much about Indian music..., but if I only faintly consider
making a fool of myself with my theories, this seems the right time to
do so. Here we go:

Assumption #1 The concept of the 22 srutis is more ancient and very
likely different from their interpretation in terms of just and
pythagorean values (which were probably imported from Arabian
theorists--likely with a change in actual musical practice--in the time
of the moguls at the latest).

Assumption #2 The srutis define a grid on which to place the 7 svaras.
If you come up with scales that are modes of each other (maybe with a
comma difference here or there) such a fine grid seems like overkill. So
it is reasonable to assume that there were scale genera with distinct
interval sizes, and srutis were used to relate quite different types of
scales. The unequivocal equation of 4 sruti=9/8, 3 sruti=10/9, and 2
sruti=16/15 that you find is probably too simple and inaccurate; it may,
on the other hand, be assumed that srutis of different sizes are taken
from a harmonic series, where *one* grouping of four sruti equals 9/8
&c. And finally, if you want to define a grid, you don't go by your
voice, you take an instrument. This instrument was the vina.

Assumption #3 (Concerning some gestalt aspects of interval cognition, a
topic I couldn't find anything on. Glad about hints!) Going by my
totally subjective personal experience, there is no succesive interval
in the continuum that stands out like 9/8. From there I can extrapolate
10/9 to find 5/4, and I can divide it into 17/16 and 18/17. To openly
generalize, I posit that there is something in the ear/brain connection
that can adjust ("tune") smaller units ("intervals") so that they yield
a larger unit that is "simpler" (is this a definition of "harmonic"?).
The 9/8 is a natural starting point.

The vina has four strings, two pairs an octave apart that are tuned to a
4/3.
The problem: How to place your frets for a specific type of music.

Let us assume you already have determined the places of 4/3 and 3/2. You
divide the 9/8 between them into four parts by ear (=with all the
accuracy you need); the possibly idealized intervals are 33/32, 34/33,
35/34, and 36/35.
Then you extrapolate the intervals downwards: 32/31, 31/30. These give
you 16/15.
Extrapolate further. After 3 times, you have 10/9: 28/27, 29/28, 30/29
(i.e., 30/27).
You also have a 4/3: 28/27, 29/28, ... 35/34, 36/35, i.e., 36/27.

The nine intervals on the lowest open string are (28, 29, 30, ... 35,
36)/27. Sharing a possible fret, they correspond exactly to these nine
intervals in the 4/3 string: (112, 116, 120, ... 140, 144)/81. Another
9/8 is needed to complete the octave. Again: divide 4 times (36/32). On
the higher string, four more intervals are added: (33, 34, 35, 36)/18.
9+9+4=22: the sruti.

The interesting part

Three scale genera or grama were reported and identified by their sruti
steps.

The least problematic is Shadja Grama, 4-3-2-4-4-3-2, which yields a
well known sequence when started right in the middle (the fifth sruti)
of the lower tetrachord:
9/8, 5/4, 4/3, 3/2, 27/16. 15/8
is available on the pair of strings tuned an octave higher. (All 22
sruti from this starting point are
_1/1_, 33/32, 17/16, 35/32, _9/8_; 7/6, 29/24, _5/4_, 31/24, _4/3_,
11/8, 17/12, 35/24, _3/2_, 99/64, 51/32, 105/64, _27/16_;; 7/4, 29/16,
_15/8,_ 31/16. _2/1_.)
[";": string 4/3 up, ";;": string 3/2 up]

Madhyama Grama (4-3-4-2-4-3-2) is said to start on Pa (the fifth degree)
which makes many commentators rotate the scale they determined in the
first place. No need to do this here: You start counting from the 4/3
string which makes the lowest available tone a 3/2. Extinct for quite
some time; I don't remember and can't find whether it was still used in
the Mogul era.
8/7, 5/4, 45/32, 3/2, 12/7, 15/8
selected from
_1/1_, 29/28, 15/14, 31/28, _8/7_, 33/28, 17/14, _5/4_, 9/7 297/224,
153/112, _45/32_, 81/56;; _3/2_, 87/56, 45/28, 93/56, _12/7_, 99/56,
51/28, _15/8_, 27/14, _2/1_.
The same count of four sruti that gave 9/8 in Shadja Grama gives 8/7
here, showing the inadequacy of 22-ET to approximate the sruti.

Lastly, there is Gandara Grama (4-3-2-4-3-3-3), described as either
nonexistent--apparently there is a fable, where an ass declares himself
able to sing in all three gramas, not knowing that Gandara Grama is a
mere theoretical construction to please some god--or as being extinct
for a long time, having a high tessitura and being kept secret by hunter
societies. With this setup of sruti, however, you find a scale
reminiscent (except for the seventh harmonic being lowered to allow for
another 9/8-10/9 grouping) of alphorns, jaws harps and musical bows if
you start from the top of the higher tetrachord and does have a high
tessitura on the vina:
9/8, 5/4, 3/2, 5/3, 11/6
taken from
_1/1_, 33/32, 17/16, 35/32, _9/8_;; 7/6, 29/24, _5/4_, 31/24, _4/3_,
11/8, 17/12, 35/24, _3/2_; 14/9, 29/18, _5/3_, 31/18, 16/9, _11/6_,
17/9, 35/18, _2/1_.

I take the fact that there are often more than seven "good" but unused
intervals available from a given starting note as a strong indication
that the purpose of the sruti was indeed to regulate the seven svaras
themselves for a given style, not to account for a certain number of
ornamental notes. The ornaments themselves are a different matter--very
much to learn for a singer, if he wants to stick to the generating
pattern of the sruti on the vina, and probably regarded as superfluous
by the vina player himself as soon as he is done with fretting his
instrument. (Were there female musicians?)

Ready to be murdered

Klaus Schmirler