Re: Just temperament -- another meaning (Paul, Gene)

Hello, there, everyone, and thanks to Paul and Gene for your responses
to my posts on "Just Temperament -- Another Meaning."

Paul, you've raised an important point, to which there might not be
any definitive solution: at what point do ratios become complex enough
that "rational intonation" (RI) would be more descriptive to many
people than "just intonation" (JI)?

Of course, this is relevant to the concept of "just temperament,"
which implies that certain intervals are being slightly "compromised"
while others remain "pure" -- using integer ratios only.

For example, we might have no problem agreeing that something like a
fifth at 182:121 (wide by 364:363) or 176:171 (wide by 352:351) is
rationally or virtually tempered by not quite five cents (actually
about 4.76 cents or 4.93 cents respectively).

With the kind of "just temperament" I describe, the "pure" intervals
sought are the stable 3:2 or 4:3, and the mildly unstable 14:11 and
13:11. These intervals all are within Partch's 17-odd-limit system,
and less complex than 17:13 (a*b=221) at one point proposed as an
example of a ratio tuneable by ear.

The complication is that with 14:11 and 13:11, the fifth's complements
have considerably more complex ratios of 33:28 and 33:26, raising the
limit from 11-odd or 13-odd to 33-odd for something like 22:28:33 or
22:26:33.

Since I routinely conceptualize about things like "a pure 33:42:56,"
my perspective on this type of "just temperament," as I call it, could
be quite different than that of someone not so oriented to these types
of ratios for regular thirds and sixths.

What such an otherwise inclined listener might notice, however, is the
way that some fifths and fourths are pure while others are not quite
five cents wide of pure.

Whether one takes this as an effort to get some "pure" ratios in
mildly unstable sonorities like 22:28:33 or 22:26:33, or simply to
have some pure 2:3:4 sonorities along with regular thirds somwhere
around 413-418 cents and 284-289 cents, it's the contrast in the just
or virtually tempered quality of the fifths and fourths that makes
this approach distinct from a regular temperament yielding similar sizes
of thirds.

Gene, I agree that "just temperament" can have different
interpretations, with the reading of a "virtually just" tuning (or
"perceptually just" one, to use Dave Keenan's explanation) as one
approach.

Sometimes the same problem might have solutions using "just
temperament" in more than interpretation.

For example, to get pure or "virtually pure" ratios of 3 and 7, we
might either use Graham's 1/14-schisma temperament (i.e. the 3-7
schisma of ~3.80 cents) realized almost precisely by 135-EDO; or place
two Pythagorean chains at the precise distance of 64:63, "virtually
tempering" one fifth by the full schisma, and giving it a size very
close to 10/17 octave.

Most appreciatively,

Margo Schulter
mschulter@value.net