Note re Indian Heptatonic Scales

Bill Alves and Haresh Bakshi both provide interesting thumbnail
sketches of early Indian scales. I'm no expert in this area (far from
it), but I would point out one other interesting article on this
topic:

"Early Indian Heptatonic Scales and Recent Diatonic Theory" by John
Clough, Jack Douthett, N. Ramanathan, and Lewis Rowell (In Music
Theory Spectrum 15:1, spring 1993, pp.36-58)

As pointed out by Clough et al, one fascinating aspect of the two
early scales, sa-grama and ma-grama, is that both may be viewed as
second-order maximally even structures. If we forget about the tuning
argument surrounding Indian scales for the moment and view the sruti
as "abstract chromatic steps," we get a rather startling relationship
to diatonic systems.

The usual major and minor triads are second-order maximally even in
the following sense. First, as is well known, the white-note scale
C-D-E-F-G-A-B is first-order maximally even with respect to the
12-note abstract chromatic, i.e., the seven notes are "stretched"
(Gk="diateinein")
as evenly as possible over the 12, yielding the interval pattern 7\12
= 2212221. If we then apply the same principle to the diatonic for
selecting three notes in seven, we get 3\7 = 223 (in some rotation).
Combining the two, you get 3\7\12 = 435, a major triad. Rotate the
original diatonic interval string and you get the other two possible
white note chords, 345 and 336 in the usual places.

(2) (2) (3) (2) (2) (3) (2) (2) (3)
(3\7)
22 12 221 21 22 212 12 21 222
(7\12)
4 3 5 3 4 5 3 3 6
(3\7\12)

White-note seventh chords are likewise second order maximally even
(4\7\12)

Now look at the maximally even 12\22 = any rotation of the basic
pattern 222221222221. Apply the same principle as above and you get a
variety of second-order maximally even scales, two of which are:

sa grama,

2 1 2 2 2 1 2
(7\12)

2 1 2 2 2 2 2 1 2 2 2 2
(12\22)

3 2 4 4 3 2 4
(7\12\22)

and ma grama

2 1 2 2 2 1 2
(7\12)

1 2 2 2 2 2 1 2 2 2 2 2
(12\22)

3 2 4 3 4 2 4
(7\12\22)

There is much more in the article I cited, and any misleading
statements anyone feels I am making here regarding this phenomenon
should be placed at my feet, not those of the authors of the Spectrum
article.

Steve Soderberg

Hi Stephen -- thanks for bringing up the Music Theory Spectrum article, on
which I've conversed with John Clough at length. There are at least two
problems with this interpretation as far as I can see. First, the genesis of
the numbers 7, 12, and 22 is left totally unexplained. Second, the Modern
Indian Gamut, a set of 12 from 22 which supports the ma- and sa- gramas and
their standard alterations, is not a rotation of the pattern 222221222221
but instead 222212222221 (as usual, tetrachordality appears more important
than maximal evenness). On the other hand, I was able to come up with a
different definition of second-order maximal evenness, which results _only_
in the ma- and sa- gramas and their standard alterations, while Clough and
Douthett need to supplement their definition of second-order maximal
evenness with a requirement that there be only one tritone (interval of 11
srutis) in order to get only the ma- and sa- gramas and their standard
alterations.