Yahoo Tuning Groups Ultimate Backup tuning Re: Happy Holidays Margo! -- harmonic entropy in action

Hello, there, everyone, and please let me reply to some remarks about
intervals such as 6:7, 9:11, and 7:9 by Paul and Gene among others,
specifically responding to a question about Nicola Vicentino's
appraisal of these intervals which may suggest how tastes can vary.

While addressing this question, I would like to suggest a line of
concrete experimentation that might at once promote an interest in
"harmonic entropy" concepts and more intimately tie the theory to a
burgeoning and engaged process of making microtonal music. While my
idea may hardly be new, the wider application of it could generate
lots of excitement in our community.

First, to make my own leanings clear, I should affirm that I tend to
consider 6:7 as an _optimized_ minor third, naming one of my favorite
just tunings in honor of this ratio. Describing the same interval as a
"bruised" minor third is a rather different viewpoint, one which both
Paul and I seem both to find curious -- but which could be quite
descriptive in a musical context where some other size is expected.

Specifically, musicians and listeners accustomed to styles where minor
thirds are tuned at or near 5:6 might consider 6:7 "bruised" -- and
one place to look for such a viewpoint is in 16th-century Europe, the
milieu of the great enharmonicist, or as we might now say
"microtonalist," Vicentino (1511-1576).

As you noted, Paul, Vicentino indeed finds the "proximate minor third"
of about 9/31 octave (approximately in a 31-note circulating system of
1/4-comma meantone, precisely in 31-tET) more "consonant" than the
"proximate major third" at about 11/31 octave. He identifies these
ratios as approximately "5-1/2:4-1/2" or 11:9, and "4-1/2:3-1/2" or
9:7, and we can confirm these approximate sizes by looking at the
intervals of either 1/4-comma or 31-tET, both fitting his description
of a division of the whole-tone into five more-or-less equal dieses.

He remarks that while the near-11:9 is "rather consonant," tending
toward the consonance of the usual major third, the near-9:7 tends
more toward the fourth, an interval of somewhat equivocal status in
16th-century practice, and thus leaning more toward dissonance. He
urges more caution in treating the latter interval, although he finds
it acceptable in passing. In contrast, some of his four-voice
enharmonic (i.e. fifthtone) cadences feature an ornament in which one
of the voices moves in fifthtones from a minor third to a "proximate
minor" one to a concluding major third -- about 8/31, 9/31, and 10/31
octave respectively, or approximately 6:5, 11:9, 5:4.

From Vicentino's perspective, interestingly, the "minimal third" a
diesis smaller than the regular minor interval, or about 7/31 octave,
is indeed too narrow to serve as a persuasive consonance. He remarks
that the usual minor third is such that to tune it smaller is to make
it approach the "dissonance" of "a second" -- that is, the major
second. While he does not, as far as I am aware, give an approximate
integer ratio for this "minimal third," at 7/5-tone it would be very
close to 7:6 in either 1/4-comma or 31-tET.

Indeed "bruised minor third" might be a rather descriptive way of
expressing this kind of viewpoint. Whatever musical factors may have
contributed to his opinion, we know that he was not shy about
championing other "unconventional" intervals such as the 11:9 third.
Further, of course, he was famed for championing the melodic use of
the enharmonic or fifthtone step, which other musicians of the era
such as Gioseffo Zarlino and Vincenzo Galilei asserted was
ill-proportioned either to the ear (Galilei) or to modern part-music
(Zarlino, who accepted the enharmonic genus for the kind of monophonic
singing he attributed to the ancient Greeks).

However, even in settings where 6:5 minor thirds are taken as the
norm, some musicians are ready to accept a 6:7 or near-6:7 third as an
acceptable "equivalent." Thus Owen Jorgensen takes the view that a
narrow minor third in a meantone tuning does not become a "Wolf" --
unlike a wide major third, for example -- but rather approaches the
ratio of 6:7, which he considers concordant. This means, for example,
that Bb3-C#4-F4 (near-6:7 augmented second below) might serve in such
as setting as an acceptable "triad with narrow minor third" -- a
"subminor triad" as xenharmonicists often describe it -- where
F#4-Bb4-C#4 would be considered "unplayable" as an equivalent for a
regular major triad.

As someone involved with Gothic and neo-Gothic music where narrow
minor thirds ranging from around 7:6 to the Pythagorean 32:27 (~294.13
cents) are the rule, my leaning toward these thirds might not be
surprising. Here I can share an early impression in a 24-note version
of 1/4-comma meantone which could be considered a subset of
Vicentino's 31-note cycle.

Using a "buzzy" harmonic timbre somewhat like a crumhorn or regal
organ (both Renaissance double reed instruments), I found that the
near-11:9 seemed rather complex, and the near-7:6 rather smooth -- for
me, a reasonable equivalent for the usual near-6:5 minor third.

In this kind of musical setting, I found the near-9:7 (precisely 32:25
in 1/4-comma, ~427.37 cents) quite different from a usual pure 5:4:
very pleasing either in some idiomatic Renaissance treatments, or as a
"neo-Gothic" major third expanding to a fifth, but nonequivalent to a
regular major third in a 16th-century texture.

My interest in the near-7:6 was in part motivated by exigency. I have
taken a liking to a kind of fifthtone progression recommended by
Vicentino and mentioned above where a voice moves by these enharmonic
steps from minor through "proximate minor" or neutral third to major
third, here shown with an asterisk (*) indicating a note raised by a
diesis:

G3
Eb3 Eb*3 E3
C3

In 1618, Fabio Colonna carried this kind of progression further,
showing a "sliding of the voice" through fifthtones like the
following (not his precise four-voice example but a three-voice
progression along similar lines):

Bb3 Bb*3 B3 B*3 C4
F3 E3
D3 C3

With a 31-note cycle, as used by both Vicentino and Colonna on their
"superkeyboards," these progressions could be transposed to any step.
However, with my 24-note tuning set, I found that from some locations
my closest equivalent was something like:

F4
C#4 C#*4/Db4 D4
Bb3

Here the neutral third Bb3-Db*4 is not available, so instead I came up
with a variant of Vicentino's progression: start with a minimal or
subminor third, then move to a regular minor third, and from there to
a concluding major third. This is approximately 7:6, 6:5, 5:4, with
the last interval just in 1/4-comma tuning.

I liked it very much, although it is melodically as well as
harmonically different from Vicentino's progression: his features two
equal fifthtone steps (Eb3-Eb*3-E3), while mine has a fifthtone
followed by a chromatic semitone (C#4-C#*4/Db4-D4).

This kind of thing also led me to a variation of Colonna's longer
"slide" in fifthtones:

D5 E5
A4 B4
F#*4 F#4 F*4 F4 E4

Here the thirds between the lower voices are ~7:6, ~6:5, ~11:9, 5:4,
leading to a usual cadence with descending semitone and ascending
whole-tones in the other voices.

Thus I found pragmatically that in a texture based on more or less
"usual" 16th-century-style counterpoint -- doubtless with my own
idiosyncrasies and liberties -- 7:6 and 6:5 are readily interchanged.
In contrast, 5:4 and 32:25 (Vicentino's near-9:7) seem radically
divergent, the latter inviting either specific "diminished fourth"
treatments of the period, or a neo-Gothic context where it makes a
fine cadential major third.

One way to put this is that 7:6 has a "cool" and "sweet" effect, while
a large major third such as 9:7 belongs to a realm of "brilliance" and
energy quite apart from that of 5:4 in a harmonic timbre. That was my
impression as it might apply to a discussion like this.

In a neo-Gothic setting, those large major thirds seem to me "bright,"
"sunny," and "beautiful": it is a different world or quantum level.
However, a friend who heard me play a regular tuning with pure 11:14
major thirds found the thirds quite different from what she would
expected for this category, describing them as "dark," suggesting some
kind of deep forest.

This happened to be in my "Puff Pipes" timbre -- or actually Yamaha's
preset voice on the TX-802 -- which I consider quite mild or
"pastelized," the kind of thing that makes 17-tET very relaxed and
agreeable for 14th-century music with major thirds near the proposed
point of maximum entropy for a harmonic timbre.

For me, those thirds were routine and delightful; for her, they were
notably "different" and "dark." Here the instrument and timbre were
the same, showing that opinions can vary.

Maybe we should distinguish between three different problems, however
related:

(1) Who "likes" a given interval size, and why;

(2) Whether such a size is contextually "congruous"
or "incongruous" to a given listener in a given
style, and why -- with mileage apparently
varying on the 7:6 in a Renaissance meantone
kind of setting, for example; and

(3) How such opinions and conventions might tie in
with more general models of "harmonic entropy"
or the like.

To make this more concrete, and also especially exciting from a
viewpoint of practical musicmaking, one approach might be to document
how some of us combine tunings, timbres, and styles -- e.g. "I really
like a regular 11:14 tuning with PuffPipes," or "With this timbre I
can happily use a 9:7 in a Renaissance-stye texture as if it were a
5:4."

Then we might study the spectrum of the timbre -- itself a fluid
reality, as Jacky Ligon has observed -- and see how this might affect
the "minima" and "maxima" of an entropy model, or the "steepness" of
the curve.

One might ask, for example, whether an agreeably "pastelized" texture
shifts the points on the curve, or evens out the peaks and valleys, or
some combination of these effects.

We could also do comparisons. If I report: "Yes, I also love
sonorities with 11:14 in this harmonic timbre," then it might be a
question of a taste for relatively complex thirds, stylistically quite
congruous in a neo-Gothic setting, rather than simply of pastelized
textures.

Similarly, if other people share the reaction of my friend that thirds
of this size in a "pastelized" texture sound "unusual" or "dark" while
I find them "brilliant" and "accustomed," then something other than
timbre alone may be involved -- a matter of taste, either stylistic or
more idiosyncratic.

Concretizing harmonic entropy in this way, and getting people more
involved in comparing notes in actual tunings and timbres which might
be replicated, could be a way not only of developing the theory but of
making it of interest to more people -- and of fostering the
microtonal music which the theory addresses at the same time.

Most appreciatively, with happy holidays to all,

Margo Schulter
mschulter@value.net

Re: Happy Holidays Margo! -- harmonic entropy in action

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗genewardsmith <genewardsmith@juno.com>