bridges to 19/16 (was: Higher primes, not Greek)

> From: <BobWendell@technet-inc.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, August 09, 2001 9:29 AM
> Subject: [tuning] Re: Higher primes, not Greek
>
>
> The point is that this 19:16 minor 3rd NATURALLY came up with the
> advent of virtually universal acceptance of 12t-ET. It is already the
> 19th harmonic to such a close approximation that the difference is
> totally negligible (1 or 2 cents? I forget and don't have the means
> on hand to calculate it). Historically, at the same time composers
> began to employ the minor mode MUCH more fequently than before. The
> acquaintance I mentioned believes this is no mere coincidence.

Hi Bob,

Allow me to respectfully state that I have a few reservations about
what you say here.

I just posted something a few days ago about Eratosthenes's descriptions
of tetrachord tunings using 19 as a factor in the ratios:

/tuning/topicId_26618.html#26618
Date: Thu Aug 2, 2001 6:31 pm
Subject: ratios in ancient Greek tetrachords

Eratosthenes made use of what I call the "3==19 bridge",
the ratio 1216/1215 = ~1.424297941 cents. In other words,
he calculated 19-limit ratios for his tetrachord pitches
which are audibly indistinguishable from 3-limit "Pythagorean"
ratios already in common use.

And centuries or possibly millennia before Eratosthenes,
a music-theorist analyzing 19-limit ratios would have found
that 19/16 is very nearly the same as the common Pythagorean
"minor 3rd" with the ratio 32/27. This is another 3==19 bridge
which is a little over twice the size of Eratosthenes's
(or about 3&1/3 cents).

I would say that these 3==19 bridges take historical precedence
over the 2^(3/12)==19 bridge you discuss here, by well over
1000 years.

In response to your statement that "the difference is totally
negligible (1 or 2 cents? I forget and don't have the means on
hand to calculate it)": here's a table describing the deviation
of 19/16 from both the 12-EDO and the Pythagorean "minor 3rds":

ratio ~cents delta cents
of 19/16 from this

2^(3/12) 1.189207115 300 -2.486983868
19/16 1.1875 297.5130161
32/27 1.185185185 294.1349974 3.378018728

In *both* cases, the difference would be almost always be inaudible
(except for the rare musical contexts in which it could be heard clearly).

So what I'm saying here is that anyone who used a "minor triad"
with the ratios 1/1 : 32/27 : 3/2 would be presenting a sound
that's virtually indistinguishable from 1/1 : 19/16 : 3/2.

This could have happened as long ago as the Sumerian period
(c. 3000 BC), if my speculations about Sumerian tuning are correct.
http://www.ixpres.com/interval/monzo/sumerian/sumeriantuning.htm

(Of course, it could have happened even much earlier, but the
Sumerian and Babylonian tablets are the earliest written record
we've found so far, and will probably remain so.)

Also, while 12-EDO began to be advocated around 1600, primarily
for its facility in fretting the lute (which was by far the most
commonly used instrument at that time), the "virtually universal
acceptance of 12t-ET" didn't happen until only about 100 years ago.
Many composers and performers held on to meantones and
well-temperaments well into the 19th century. So can you say
more about the "time composers began to employ the minor mode
MUCH more frequently than before"?

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com