Yahoo Tuning Groups Ultimate Backup tuning Re: TD 874 -- two different applications of Phi (for Jon Wild)

Hello, there, and thank you, Jon Wild, for an opportunity to clarify
my views on the distinction between the Phi-based Noble Mediant as a
scale generator applied to logarithmic fractions of an octave, and as
a "Noble Mediant of complexity" applied to two simple interval ratios.

With Dave Keenan and Paul Erlich, I agree that it is very important to
keep these distinctions clear -- and very easy for the distinction to
get blurred, which is a good reason to emphasize it early and often.

Your helpful questions may illustrate a kind of corollary to
Murphy's Law: if a text leaves any room for ambiguity on this kind of
distinction, astute and interested readers are likely to pick up on
this and get the impression that the two applications may be
synonymous.

As you pointed out, there are passages in what Dave and I have written
by no means excluding such a reading, and I could cite some remarks in
one of my recent papers where I might wisely have added some specific
disclaimers. Dave commented on this, and your inquiry shows the wisdom
of his advice.

Specifically, the use of Phi with logarithmic fractions of the octave,
as in both Thorwald Kornerup's Golden Meantone and Keenan Pepper's
neo-Gothic counterpart, may often result in strikingly "simple" and
"concordant" interval ratios.

As Paul observed, Kornerup's Golden Meantone is very "concordant" in
an historical 5-limit sense, being a kind of compromise between
1/4-comma meantone (pure major thirds) and Zarlino's 2/7-comma (major
and minor thirds equally tempered by 1/7-comma).

At the most obvious level, Keenan Pepper's neo-Gothic Phi-based tuning
produces very "concordant" fifths and fourths only about 2.14 cents
from pure. As in Pythagorean tuning, these stable concords provide
resolutions for all the unstable intervals. Like Kornerup's Golden
Meantone in a 5-limit setting, Pepper's tuning in a 3-limit setting
provides near-ideal stable concords.

Further, as I've suggested in some recent posts, the _differences_
between the ratios of certain unstable intervals in Keenan Pepper's
tuning and Dave Keenan's "Noble Mediants of complexity" for these same
intervals can be significant.

For example, the regular major third in this tuning is around 416
cents, a bit smaller than 14:11, and about 6 or 7 cents smaller than
the "Noble Mediant of maximum complexity" we get from 5:4 and 9:7
(~422.487 cents). This means that the interval is somewhat _milder_
than in 17-tET, for example.

Both Dave and I have agreed that if one wishes to maximize the
complexity -- and here we might also both say "dissonance" or
"tension" -- of major thirds, or the complexity of thirds and sixths
generally, then 17-tET would be an excellent choice. In contrast,
something like Keenan Pepper's tuning (or the region in the vicinity
of around 29-tET to 46-tET in general) represents a shading somewhere
between Pythagorean and 17-tET.

Of course, Paul, your example of Golden Meantone really makes the
distinction stand out. The same variety of octave-logarithmic
Phi-based approach can produce either a Renaissance-style meantone
(with the ratio of diatonic to chromatic semitones equal to Phi) or a
neo-Gothic style tuning (with the ratio of chromatic to diatonic
semitones equal to Phi).

Anyway, as Dave has commented, it's very important to make this
distinction between Phi in relation to octave-fractions, and Phi in
relation to simple interval ratios as a "Noble Mediant of complexity."

Sometimes there can be fortuitous overlaps between the two approaches,
as when Keenan Pepper's tuning (Phi-based logarithmic division of
octave) happens to include a close approximation of the interval ratio
Phi (~833.1 cents, maybe a flavor of "supraminor sixth").

However, there are lots of other tunings which may approximate
Phi-based mediants of "maximum complexity" between two simpler
interval ratios much more closely than tunings built from Phi-based
divisions of the octave.

For example, with our 5:4-9:7 region, the mediant of complexity is
around 422.49 cents. Kornerup's Golden Meantone gives us a diminished
fourth of around 430.28 cents, while Pepper's counterpart gives us a
regular major third at around 416.38 cents. However, a 17-tET major
third gets us much closer (~423.53 cents), and a diminished fourth in
2/9-comma meantone even closer (~422.59 cents).

Note that this leaves open the question of possible affinities between
these two different approaches; my point is to emphasize that they are
conceptually distinct.

In Kornerup's Phi-based tuning, we get near-pure thirds and slightly
impure fifths (as in any meantone tuning in this portion of the spectrum);
in Pepper's Phi-based tuning, we get near-pure fifths and relatively (but
not maximally) complex thirds, as with other neo-Gothic tunings in this
portion of the spectrum. Incidentally, the latter tuning also gives a
minor sixth very close to the somewhat-simple ratio of 11:7, with either
Pythagorean tuning of 17-tET appearing somewhat more "complex."

Most respectfully,

Margo Schulter
mschulter@value.net

Re: TD 874 -- two different applications of Phi (for Jon Wild)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Joseph Pehrson <pehrson@pubmedia.com>