probable 3==5 and possible 3==7 bridges in Indian tuning

I started writing a private response to Paul Erlich that I decided
would actually be better suited out here in the open, and while
writing it also stumbled upon an idea that is relevant to a
recent discussion here.

Dan Stearns wrote [TD 671.16]:

> Is there any historical, musical precedent for 7-limit extended
> Pythagorean similes?

Paul Erlich [TD 671.17] pointed him to Margo Schulter's posting
http://www.ixpres.com/interval/td/schulter/septimal.htm
and Margo herself [TD 672.8] replied that Jacobus of Liego (1325)
described a Pythagorean 'large whole-tone' or _tonus maior_ of
~228 cents (2^-22 * 3^14), disparagingly, and that her references
to 'septimal schismas' (which form bridges from the 3-limit to
the 7-limit, and which I write as 3==7) appear in conjunction with
her own 20th-century 'Xeno-Gothic' tuning scheme, based on an
interpretation of the vague tuning system outlined by Marchettus
of Padua (1318). -See my webpage on Marchetto at
http://www.ixpres.com/interval/monzo/marchet/marchet.htm

But I have a hunch that there may have been use of the 3==7 bridge
many, many centures ago. Read on...

In a private email to me, regarding corrections to my webpages,
Paul Erlich wrote:

> Back at http://www.ixpres.com/interval/td/erlich/srutipblock.htm,
> you write,
>
>> [me, monz]
>> Of course we tuning theorists know that this is nonsense: the
>> difference is not the syntonic comma as Lentz says it is, but
>> rather the Pythagorean comma.
>
> [Paul]
> Rather than being nonsense, Lentz' comment correctly applies to
> the srutis after the schismatic substitution is performed to make
> 5-limit ratios, doesn't it?

Yes, it does correctly apply *after* the schismatic substitution,
but...

Have you read Lentz? I'm under the impression you have not.
I suppose I should flesh out this part of the webpage a bit
to give a clearer explanation. I'll make a first attempt at
it here. But be warned that the last time I looked at Lentz's
book was last year - I'm recalling it here to the best of my
ability, but there's certainly a chance that it's a bit inaccurate.
Anyone who has a copy handy is invited to comment further on what
I say here; by all means, please correct any errors.

Lentz explains that the tuning would have been physically
arrived at by 2 series of '4ths' and '5ths' above the reference
pitch: one series being 3^(1...11) and the other being 3^(-1...-10),
and (usually) alternating '4ths' and '5ths' to keep all tones
within one '8ve'. This gives a system of 22 tones, with a 3/2
between the reference pitch and its '5th', and a Pythagorean
comma between the other 10 pairs of chromatic notes.

*Then*, after explaining this, he blithely states (in different
language than that which I use here) that his table of the
resulting ratios has been simplified by changing the larger-exponent
Pythagorean ratios into their comma-equivalent 5-limit JI cousins.

IIRC, The actual tuning doesn't really penetrate much further into
Lentz's discussion of Indian music after that; he points out
the difference sizes of 'semitones', but doesn't give any
kind of specific accounting of their uses in any specific
scales or modes.

Therefore, it's not clear to me whether he really means to
describe the Pythagorean system resulting from the prescribed
tuning method or the 5-limit system given in his table. He
generally speaks of the ratios in terms of the latter, but a
feeling keeps gnawing away at me that his real focus should have
been the Pythagorean system he originally described. That's why
I say that the real difference between the paired srutis is the
Pythagorean (and not the syntonic) comma.

As I state in my webpage, altho there is apparently no evidence,
I think it's entirely possible that musicians in ancient India used
an extended system tuned by ear in Pythagorean ratios, and then
used the concept of schismatic substitution in order to incorporate
Pythagorean ratios which closely approximated 5-limit ones, to
provide subset scales in pseudo-5-limit-JI. But the actual
ratios still would have been Pythagorean.

Of course, I'm also willing to allow that over a (relatively
short?) period of time, good musicians could have gotten to the
point where they altered the tuning of those schismatic
substitutions by ear so that they really *were* 5-limit ratios.

Then, going back to my reference to the recent Tuning List
postings, there's also the observation that the interval between
3^-7 (= ~1086 cents) and 3^7 (= ~114 cents) is ~ 973 cents,
pretty darn close to a 'harmonic 7th' of ~969 cents. With
this sound available in their tuning system, I have no problem
extrapolating the data to reach the assumption that the ancient
Indian musicians also could have used 7-limit (or Pythagorean
pseudo-7-limit) ratios in their music.

-monz

Joseph L. Monzo San Diego monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
| 'I had broken thru the lattice barrier...' |
| -Erv Wilson |
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