Yahoo Tuning Groups Ultimate Backup tuning Re: Stylistic consonance and categorical entropy

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Stylistic consonance and categorical entropy:
An essay for Pierre Lamothe
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"Language is system."
-- Ferdinand de Saussure

While recent discussions on "harmonic entropy" and related subjects
have often discussed acoustical factors which may influence interval
perception, I would like here to focus on two concepts which may lend
an interesting perspective: stylistic consonance and categorical
entropy or ambiguity.

Emphasizing that I find the harmonic entropy inquiries of Paul Erlich
and others both fascinating and valuable, I would add that such
studies have a sometimes express but often implicit musical
qualification: "all things being equal both as to the timbres used,
and as to the stylistic expectations of the musical style itself."

Before delving more into these factors, I would like strongly to
emphasize that contributors such as Paul Erlich and David Keenan, to
mention only two, _have_ taken an interest in contextual and
historical factors which can shape the choice of intervals. My purpose
here is not to suggest that these factors have been ignored, but
possibly to highlight some of their aspects at more length.

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1. Stylistic dissonance and consonance
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A bit more than 30 years ago, I met three singers who very kindly
offered to perform a three-part conductus I had written in a more or
less conventional 13th-century style. Such an impromptu sight-reading
was hardly the easiest task, especially at an unexpected time and
place for this kind of musicmaking.

At the end of the first phrase, where I had written a typical cadence
something like this (here given in MIDI-style notation, with C4 as
middle C):

D5 E5
B4 A4
G4 A4

one of the singers happened to arrive at the note C#4 rather than A4
or E4 -- or, at any rate, at some third. At that moment, I vividly
learned the nature of "stylistic dissonance."

Whether that C#4 had been tuned as a 5:4, an 81:64, a 23:18, or a 9:7,
it would have simply been "out of place" in a sonority meant to be
stable -- in this style of course, not necessarily in the style of the
next piece you happen to be performing or listening to, or maybe the
new style you're now (re)inventing for the 21st century.

Looking back on this episode, in which the generosity of those three
singers is much more important than their fortuitous "slip" which
taught me an important musical lesson, itself a precious gift, causes
me to reflect on a curious question.

If someone requested that I conclude a piece in Gothic or neo-Gothic
style with some sonority more complex than a 2:3:4 trine (outer
octave, lower fifth, upper fourth, e.g. D3-A3-D4) which sonority would
I choose?

As first choice, and with some theoretical medieval sanction, I would
close with a 1:3:9 or 1:6:9 (e.g. D3-A4-E6 or D3-A5-E6), the latter
having the advantage of being playable in real time by one person on a
conventional keyboard instrument. To my ears, curiously, this sonority
seems almost as "concordant" as a usual 2:3:4 trine, although more
complex: somehow the notes occupy the same aural space without the
kind of tension which in a trinic context suggests "instability."

Since Jacobus of Liege (c. 1325) lists 9:1 as a "perfect concord"
along with more traditional stable intervals such as octaves, fifths,
and fourths, my choice would have some historical as well as aesthetic
basis.

If requested to remain within a typical medieval compass of around a
twelfth, then to a complete 2:3:4 trine (e.g. D3-A3-D4) I would add a
note forming a sonority maintaining a quintal/quartal color such as
6:8:9:12 (e.g. D3-G3-A3-D4), 8:9:12:16 (e.g. D3-E3-A3-D4), or 4:6:8:9
(e.g. D3-A3-D4-E4).

Again, although such a usage goes beyond known Gothic practice, there
are possible constructions of theory one might borrow to explain it:
for example, the famous four hammers or strings 6:8:9:12 of Pythagoras
described by various medieval authors. However, my main argument would
be that, if we are to go beyond the historical bounds of the stable
trine, sonorities of this kind seem the most "trinelike" or
"stylistically concordant."

Whether or not 9:8 or 9:4 is regarded as the most "sensorily
consonant" ratio beyond 2:3:4, it seems to me the most stylistically
fitting, or the least unfitting, interval to add to a final sonority
maintaining a "Gothiclike" -- or "neo-Gothic" -- flavor.

Other questions remain. For example, does the very "concordant" effect
for me of 1:3:9 or 1:6:9 relate in some way to the low products of the
three terms (1*3*9 = 27; 1*6*9 = 54)? Or is it in large part of matter
of octave spacing diluting the tension of the major 23rd by placing
any beating between partials outside the "critical band?"

Of course, with these extraordinary sonorities and with the more usual
6:8:9 or 8:9:12, etc., the predominantly "quintal/quartal" color seems
a main attraction.

This discussion brings me to a related but slightly different question
of "stylistic consonance," the question of the 16:9.

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2. The status of 16:9 -- placing "entropy" in musical context
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One feature of certain 13th-14th century as well as 20th-century
European and related styles is a penchant for freely treated minor
sevenths tuned at a just 16:9 (~996.09 cents), or at the modern
approximation of 1000 cents.

Thus one scholar of the style of Guillaume de Machaut (c. 1300-1377)
remarked that he seems at times to treat minor sevenths as "imperfect
consonances," and 20th-century practice and theory often shows a
similar preference.

Around 1200, this "partially concordant" quality is especially notable
in 9:12:16 with its two pure 4:3 fourths complementing the outer 16:9
(e.g. G3-C4-F4). In one late 14th-century piece I find that sonorities
combining a minor seventh, fifth, and minor third, e.g. D3-F3-C4 at
27:32:48, can also have a pleasingly "mild" quality, a "jazzlike"
or "floating" effect.

Interestingly, recent discussions on harmonic entropy have suggested
that a minor seventh at a pure 16:9 (two pure 4:3 fourths), or at the
1000 cents of 12-tET, is in a local region of maximum complexity or
"entropy," in comparison to the simpler regions around 7:4 or 9:5.

A region of maximal complexity, indeed, has been placed by Paul
Erlich's sophisticated computer routine at around 999 cents; the Noble
Mediant or "two Keenans function" published here by Keenan Pepper and
applied to harmonic entropy by Dave Keenan suggests a maximum at
around 1001 cents. By such measures, a pure Gothic 16:9 is quite
"dissonant," and the interval of 10/12 octave or 1000 cents so beloved
in many 20th-century styles is almost exactly at the point of "maximum
dissonance."

Further, it appears that 16:9 may be less clearly defined from an
acoustical point of view than its octave complement 9:8 -- an
asymmetry which may also apply to intervals such as the 5-limit major
third at 5:4 and considerably less "sensorily consonant" minor sixth
at 8:5.

Happily, however, whatever the acoustical complexity of a minor
seventh at 16:9 or 1000 cents, styles which treat this interval as to
some degree "compatible" or even "concordant" remain free to do so.

Through the years, I have regarded 16:9 as an ideal ratio for a minor
seventh because of the derivation from the two pure fourths, a
derivation perfectly realized in Pythagorean tuning and approximated
in temperaments such as 12-tET or the yet closer 29-tET.

The observation that these intervals are actually near a local maximum
of complexity is certainly a noteworthy one, in part because it
illustrates how musical styles have their own sense of system and
"concord" in which complexity or entropy from one point of view may
represent just proportion and harmoniousness from another point of
view.

To say that a minor seventh around 16:9 is near a point of "maximal
entropy" may be a bit like saying that Paris, in comparison to a place
high in the Alps, is at a point of "maximal gravity" because closer to
the Earth's center. The latter remark is a very accurate statement of
physics, yet should not be taken to imply that Parisians find it
difficult to walk or climb gentle hills because of the greater
acceleration of gravity.

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3. Entropy and categorical perception
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The phenomenon of "harmonic entropy" might usefully be compared with
another stylistically-related concept which I term, after Brian
McLaren for example, _categorical perception_ or "categorical
ambiguity." These two different kinds of concepts sometimes, but not
always, may be synonymous for a given region of the continuum of
intervals in a given style.

For example, while a 16:9 minor seventh or one at 1000 cents may be in
a region of "maximal entropy," it is, at least in many medieval and
20th-century Western European styles, near the very center of the
range of categorical recognition or "minor seventhness." However
acoustically simple or complex, it is an ideal minor seventh in these
stylistic settings.

In contrast, another "entropy maximum" is located in one of Erlich's
computer-generated mappings at around 457 cents, a region which is
also categorically ambiguous in many of these same styles as a kind of
transitional zone between a large major third and a narrow fourth.

Thus from a neo-Gothic perspective, I use a minor seventh at or close
to 16:9 as a "normal" interval, one realizing or approximating the
ideal of Pythagorean just intonation. In contrast, an interval of
around 450-455 cents (e.g. 11/29 octave or 9/24 octave) is a "special
effect," a deliberate blurring of familiar categories.

Please let me emphasize that to focus on the important stylistic
factor of categorical perception or ambiguity is not to argue that any
given style should be taken as a "natural" or universal standard of
musical perception or propriety. It is simply to assert that a given
style, like a given natural language, may have its own patterns of
musical "phonology" and "grammar."

The ancient Greek historian Herodotus, citing the poet Pindar, once
remarked that "_nomos_ is monarch of all," _nomos_ meaning custom or
convention. This observation can apply to musical styles also.

Far from being made irrelevant by such considerations, harmonic
entropy theory can serve to illuminate the element of stylistic
variety and choice, the many patterns of acoustical simplicity and
complexity which take shape in a given style or tuning.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Stylistic consonance and categorical entropy

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Monz <MONZ@JUNO.COM>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Pierre Lamothe <plamothe@aei.ca>

🔗Joseph Pehrson <pehrson@pubmedia.com>