Doubly Positive systems

"Doubly Positive," is a term from Bosanquet's General Theory of the Octave
and refers to regular cyclic systems whose fifths are sharper than 12-tone
equal temperament and of which the difference between 12 fifths and 7
octaves (B#-C) is 2 degrees. Another way of expressing this is that the
difference between the 5-fifths semitone (B-C) and the 7-fifths semitone
(C-C#) is 2 degrees. The doubly positive systems are 10, 22, 34, 46, 58,
70.....10+M*12 where M-2 is the number of degrees between C and C#. For,
22-tet M is 3
degrees and 22-tet is described by Wilson as a doubly-positive, ternary
system.
Other ternary systems are 29, 36, 43 and 50-tets.

--John



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Bill Alves wrote:

> The first known Indian treatise to describe the octave divided into 22
> srutis is the Natyasastra of Bharata. Estimates of its date range from the
> 5th century BC to the 4th century AD. I tend to trust Jairazbhoy, who
> estimates that it comes from the "first few centuries AD." Bharata
> discusses two grama, or scale-types: Sa-grama -- 4 3 2 4 4 3 2 -- and
> Ma-grama -- 4 3 4 2 4 3 2. He does not discuss their precise tuning.

Thanks, 4th century is very early. I'm convinced that these
scales are an attempt to describe diatonic scales with a major
tone as 4 steps, a minor tone as 3 steps and a diatonic semitone
as 2 steps. I think it is quite plausible that an Indian musician
could have set up a scale with 22 perceptually equal steps, and
compared it with tunings in use at the time. That there are two
different tones in the scales means this can't be an attempt to
describe a negative meantone. It could well imply triadic harmony
using small integer ratios.

Incidentally, I had Ma-grama as 4 3 2 4 3 4 2. Apologies to
anyone who received perplexing e-mails from me based on this error.

I won't go into my historical theories in detail today, but I am
interested in exactly where the major scale came from. The usual
explanation is that it originated in Europe, based on Ancient
Greek ideas. This evidence appears to totally contradict that.
It's absolute dynamite!


Paul Erlich wrote:

> Certainly the evidence suggests that India achieved the 5-limit
> before the West.

Does "the West" include Greece and Byzantium here? Xenakis
describes Byzantine tetrachords in terms of 7-limit frequency
ratios, but I don't know how reliable his sources are. Every now
and then, discussions of Ancient Greek music arise on the list,
but they are too detailed for me to draw general conclusions from
them. I expect coverage of this in the FAQ we've been promised.



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