Yahoo Tuning Groups Ultimate Backup mills-tuning-list Consonance: self-correction and further discussion

Hello, there, and this is a post first to ask pardon for my public
display of innumeracy in Tuning List Digest #1569 in an article on the
"Spiral of Fifths/Fourths," and secondly to respond to some very
interesting remarks on consonance concepts, JI, and 12-tet.

First, my blooper duly confessed and corrected:

> Yet a doubly diminished fourth (fifth #16, e.g. d#-gb) of about 271
> cents is only about 3.8 cents from a pure 7:6, which I suspect many
> people on this list would regard as more concordant than a comma
> (fifth #12) or even a major seventh (fifth #5)...

In fact, the first quoted line should have said "fourth #15," which
does yield an interval very close to 7:6. My twofold error was think
in terms of 16 _notes_ (actually 15 tuning intervals), and to write
"fifth" when "fourth" is correct (i.e. 15 fourths up or fifths down).

Now we move to a much more delicate question of definition and taste
raised in a post on "Mathematical explanation of consonance" from
monz@juno.com:

> How does one explain the fact that so many musicians who are less
> "well tuned" than us will find 12-equal "3rd"s to be consonant, for
> instance? Most of us on this List wouldn't say that, but lots of
> other listeners would.

Personally, I might rephrase this a bit: "How does one explain the
fact that so many musicians ... apparently assume that 12-tet 3rds are
the 'ideal' and 'natural' versions of these intervals to use for all
periods and styles of music, as if no other tunings were known?"

This whole question could lead into lots of areas, and here I'll just
try to address a few.

First of all, as someone working mainly with just (3-based or
Pythagorean) and "semi-just" (1/4-comma meantone) tunings, and also
with 17-tet, I'd say that 12-tet 3rds can and do serve as full
consonances in triadic or post-triadic music, and have done so at
least since the middle of the 16th century, when 12-tet was already
being described as a standard tuning for the lute.

This doesn't mean that 16th-century musicians, who were likely _not_
generally accustomed to 12-tet but rather to 5-based just intonation
for voices and meantone approximations of it on keyboard, found a
12-tet third _ideally_ "concordant." Vincenzo Galilei, in his
_Fronimo_, speaks as both an accomplished lutenist and a tuning
theorist when he recognizes that major thirds with the standard 12-tet
fretting _are_ larger than the Renaissance ideal of 5:4, but
nevertheless finds them acceptably concordant.

Interestingly, he explains this acceptability by pointing to the
quality of the lute's strings, which tend to "soften" the sharpness of
these thirds. I don't know if this means a new trend with people
wearing t-shirts saying: _Vincenzo Galilei was a proto-Setharian_, but
he did distinguish between the timbre of a lute and a harpsichord
(customarily tuned in meantone), the latter being more intonationally
sensitive.

At the same time, he pokes a bit of fun at people who add _tastini_ or
"little frets" to the lute in order to get more just thirds. His
argument is basically: (1) These people often in practice wind up
playing intolerably out-of-tune fifths when the system gets too
complicated; and (2) A true virtuoso (like Galilei himself) doesn't
need such extra gadgets to impress an audience.

What we're dealing with, then, is what has been called "categorical
perception": in music, as in natural language, a vowel, consonant, or
harmonic consonance for that matter may vary somewhat in its
pronunciation or size and still be recognizable to the listener.

Differences between dialects of various languages might be analogous
to variations in tuning a "major 3rd," for example: it might be 386
cents in 5-based JI or 1/4-comma meantone, 379 cents in 1/3-comma
meantone or 19-tet (proposed in the 16th-century), 400 cents in
12-tet, or shift depending on the key between values ranging anywhere
from 386 cents to around 408 cents in the various well-temperaments
standard for harpsichords and soon also pianos during the era of about
1680-1880.

For a major third that the listener is expected to recognize as fully
concordant or stable in a Renaissance-Romantic setting, there do seem
to be outer tolerances. One line of demarcation is that an interval
much larger than the regular Pythagorean M3 (81:64 or 408 cents) may
be heard as a quite "dissonant" or "Wolvish" interval obviously "out
of tune." Another line, proposed by Easley Blackwood, is around 406
cents, the point where he feels that major thirds are just subdued
enough to form stable triads. By his standard, 12-tet is clearly
within this range.

However, "acceptable" and "ideal" are two different questions. For
16th-century music, I'd agree with Mark Lindley and _lots_ of attuned
listeners that 12-tet on an organ-like or harpsichord-like keyboard
would be a very serious compromise; which doesn't mean that one
couldn't enjoy the music, only that it would be an arrangement of
necessity rather than ideal beauty. In fact, I'd tend to stick pretty
much with 1/4-comma meantone (or 1/3-comma/19-tet for at least one
chanson by Costeley where it is explicitly indicated, and maybe
Zarlino's 2/7-comma for certain kinds of pieces), going to 13 notes
per octave or 15 or even 17 or 19 where required. Such are the
advantages of microtunable synthesizers.

Also, as has been pointed out by others in this thread, consonance is
a complex matter of stylistic expectations and "acclimatization" as
well as tuning ratios. In 13th-14th century European music, major
thirds are stylistically treated as "partial concords," and
Pythagorean tuning does a nice job of putting a bit of an accent on
the _partial_ in that statement. However, even if I played Perotin in
1/4-comma meantone with pure 5:4 major thirds, these intervals would
sound "_relatively_ concordant but unstable" to me, because this is
the musical definition of the language around 1200.

While I'd agree with 16th-century theorists that 5:4 and 6:5 are ideal
ratios for thirds in a style based on concordant triads, maybe a view
fitting my preference for Renaissance and Manneristic music as my
favorite triadic styles, lots of other approaches are possible. For
example, as Ed Foote has discussed, well-temperaments with their "key
color" exploit a subtle scale of microtonal consonance and dissonance,
and one I might add which seems based both on categorical recognition
(even a triad in one of the most remote keys is still recognizable as
a triad) and on tangible distinctions.

Having become acquainted with the experiments of Gary Morrison with
88-cet, for example, where even his "supermajor" thirds of 440 cents
(roughly 9:7) may successfully be treated as "consonant," I'd say that
there is indeed a xenharmonic continuum, and that 400 cents is just
one point on that continuum, neither especially bad nor necessarily
the best.

There are circumstances where 400 cents might be an ideal choice,
specifically late Romantic pieces which seem to assume a close
equivalence of all transpositions on the circle of fifths, and also
12-tone serialist works (yes, 11-tet serialism is a viable
alternative, too). Maybe this just goes to prove what I'd call
Darreg's Law: _all_ n-tet's offer a basis for beautiful music.

Anyway, I'd say that musical grammar, categorical perception, and
custom can explain a lot about why perceptions of consonance are fluid
and flexible, as well as often a matter of some divergence in views,
among n-limit JI and n-tet and "non-just, non-equal" enthusiasts on
this list as well as among other people.

Most respectfully,

Margo Schulter
mschulter@value.net

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End of TUNING Digest 1570
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Consonance: self-correction and further discussion

🔗"M. Schulter" <mschulter@...>

🔗"Paul H. Erlich" <PErlich@...>