5-limit in India?

John Chalmers wrote:

> A 4th century CE (AD) date for the Indian 22 shruti scale certainly allows
> for Greek influence. Alexander the Great reached the Indus in the 4rd
> century BCE and a Greek state lasted in Khorasmia until Roman times. Greek
> astronomy clearly influenced Indian astronomy (the Paulisa Siddhanta,
> Romaka Siddhanta, Surya S. etc. ) at an early date, thought to be around
> 4th century CE by Neugebauer). Since the chief music theorists of the
> Hellenistic period were also astronomers, I see no problem with their being
> a Greek influence on music as well, or at least on music theory.

I'd have thought it was more likely that the influence was in the
other direction. It's the Greeks (Pythagoras, Plato) who were in the
habit of fitting musical scales to mathematical ideas, without worrying
about music as an art. Ideas about music could have been spread by
traders, but it would have taken a long time for them to be taken up
as common practice. 700 years, in fact.

Paul Erlich wrote:

>I'm not sure whether you were trying to say that the specification comma=1,
>diaschisma=0, leads unambiguously to 22 tones while your previous
>specification comma=diesis could lead to 12, 22, 34, or 46. If this is what
>you were saying, can you guide us through the math?

No, I'm suggesting that the 22 note sruti scale is best interpreted
as a tempered scale with diaschisma=0. The number of notes you need
is the smallest number that make all the distinctions you require.
Once you decide that commas are important, 22tet is the first ET that
includes them which is no worse an approximation than 12tet. A 4th
Century Indian theorist trying to fit observed scales to a set of
roughly equal intervals could well have hit upon 22 steps. There are
other choices that could have been made, but 22 is a historical fact.

In the same way, medieval Arabic music makes best sense with Skhismatic,
and Renaissance European music with Syntonic, temperament.

> I believe that the sa- and ma- gramas were tuned by ear (with 5-limit
> intervals) by musicians before theorists decided to quantify them in
> terms of a 22-tone scale.

That's my suspicion as well. The 22 note scale seems to be a
solution to problems in intonation that arose through the use of
5-limit scales. It makes absolute sense to me as a 20th Century
European theorist.

> The Pythagorean series was more important in ancient musical
> theory than it is now; for example the medieval Arabs used a 17-tone
> Pythagorean series to quantify some apparently 5-limit scales.

The Pythagorean series was important in cultures that didn't
recognise the syntonic comma. Is it right that Chinese music
theory mentions 12 note Pythagorean scales? The Medieval Arabs
and Europeans took up Pythagorean ideas because they placed a
high value on Greek learning. I don't see that this applies to
India. The 22 note scale could have arisen from 11 gramas, but
the gramas don't look like Pythagorean scales. If the sruti
scale is influenced by Pythagorean ideas, why no mention of
frequency ratios?

We've decided behind the scenes that the sruti scale looks like
this in doubly positive temperament:

r q r q r q r q r q r q r r q r q r q r q r
S r R g G M m P d D n N S'

The basis here is:

(r) = ( 8 -5)H
(q) (-11.5 6)

The two notes that can't be altered by q are Sa and Pa. Sa is
the tonic of Indian scales, and so is an obvious choice to keep
fixed. The fifth from this can also be kept perfect. The 22
notes are generated from cycles of fifths about each of these
points. This is the obvious 22 note scale to use for Indian
music, as I understand it.

My own belief, for which I have no evidence, is that, if 5-limit
harmony has a single origin, it was in a culture that performed
choral music. When a large number of people sing together,
harmony becomes important. If 5-limit intervals are considered
desirable, they will be easy enough to recognise. I know of no
ancient Greek or Indian choral music, so the chances are that they
both learnt it from a third people. As choral music leaves no
instruments to be dug up by archaeologists, we'll probably never
know who this people was. The existence of the 22 note scale,
and it's persistence -- at least in theory -- to the present day,
suggests that 5-limit scales were well established in India by the
4th Century AD. From the Greeks, we know that they had been around
for at least 700 years by then. That's all I know.



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Subject: Tuning of the 2nd sruti
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