Re: Nature, nature, and style

------------------------------------
Nature, Nurture, and Style
------------------------------------

Hello, everyone.

Recently there has been a great deal of discussion on the Tuning List
about "following Mother Nature" in vocal intonation, and about the
possible influence of keyboard tunings on singers. This discussion may
raise some basic issues not only about the compromises of keyboard
tunings as they may influence singers and performers on other
non-fixed-pitch instruments, but about how ideals for intonation may
vary along with the stylistic context.

In writing on these subjects, I should make clear my own orientation
and biases: my main focus in on medieval, Renaissance, and Manneristic
music of Europe during the era of around 1200-1650, an era in which
3-limit and 5-limit just intonation (JI) systems define the vocal
ideal. For keyboards, 3-limit JI (Pythagorean) systems are the norm
during the medieval era, with meantone coming into vogue around 1450
and serving as one index of the transition from late medieval to early
Renaissance style often taken to begin around the era of the young
Dufay (c. 1420).

Here I might emphasize that my views on this music -- my "Top 40" as
it were -- derive not only from reading treatises, as important as
these are, but from many years of listening, playing, and
improvising. It also fair to point out that my intense interest in
intonational issues is quite recent, and that many people here have
far more experience in such issues than I can claim. My special
interest has been in the techniques of medieval and Renaissance
verticality: the ways in which three or more voices form vertical
combinations and progress from one combination to the next.

While the treatment of sonorities around 1600 featuring bold use of
the seventh is a topic for an article in itself, I would like here to
set a framework for such an article by considering some general
issues.

-----------------------------------------------------
1. Tonality and verticality in Western European music
-----------------------------------------------------

From my perspective, an important observation about "tonality" in
Western European music is that major-minor key tonality is only one of
several historical styles providing material for today's practices --
along with more recent developments beginning around the epoch of
Debussy (c. 1900).

If we survey the history of Western European composition for three or
more voices, starting around 1200 with the epoch of Perotin -- about
800 years of history -- we find that the era of major-minor key
tonality (often called the "common practice period") runs roughly from
Stradella or Corelli to Wagner, let us say 1670-1900. This era covers
something like 230 years out of our 800 years -- only about 25%-30% of
the total.

Of course, the term "tonality" can mean different things to different
people. While the idea of a vertical center, corresponding for example
to a modal final, can be just as relevant to Gothic as to Classic
music, the stable and unstable sonorities and progressions defining
this "centrality" can vary radically. Each system is just as "natural"
as another -- or, from another viewpoint, each is equally a product of
art and style.

-------------------------------------
2. Combinative vs. chordal approaches
-------------------------------------

One way of distinguishing medieval-Manneristic practice of the era
1200-1650 or so with the "tonal" practice of 1670-1900 might contrast
the "combinative" vs. "chordal" nature of these practices -- both
approaches playing a significant role in 20th-century practice and
theory.

In a combinative approach, the "elementary" units are two-voice
intervals and progressions, which are artfully combined or united to
form compelling multi-voice sonorities and cadences. This approach
applies roughly from Perotin through Monteverdi, and is reflected, for
example, by the remark of Agostino Agazzari in 1607 that to master the
new art of continuo playing, a performer must be familiar with the
rule of counterpoint calling for progression to "the nearest
consonance."

In such a combinative approach, not only the plurality of modes or
octave-species but the rich vocabulary of vertical progressions often
lead to a delightful variety and fluidity of motions -- in
16th-century styles, for example, to diverse progressions of the bass
by seconds, thirds, fourths, and fifths. As the Monteverdi brothers
point out in their manifesto of 1607, the same year as Agazzari's
treatise, a piece may move through a variety of modes.

By around 1670 or 1680, however, a new practice has emerged in which
the "elementary" units seem to be not intervals, but chords forming
keys, a paradigm codified by Rameau (1722) and his successors. This
paradigm remains the norm until the epoch of Debussy around 1900, when
"post-tonal" (including "neo-modal") approaches begin to set the
direction of much new music.

While the era of 1670-1900 is often called the "common practice
period," a better term might be "last common practice period," since
after around 1900, there seem to be a number of competing models: for
example the neo-modality of Debussy or Bartok or Vaughan Williams; the
12-tone pantonalism of Schoenberg; various approaches using a range of
equal temperaments (n-tet's) and extended JI systems, etc.

These newer styles may draw both on the medieval, Renaissance, and
Manneristic heritage of intervals, combinations, and modes; and on the
late Baroque, Classic, and Romantic heritage of chords and keys.

--------------------------------------------------
3. Concords, discords, and dual-purpose sonorities
--------------------------------------------------

In considering intonational ideas as well as the pragmatic compromises
which may be required for keyboard tunings and the like, we should
focus on the stylistic expectations which can color intonational
preferences.

For example, to a singer of 16th-century polyphony, a pure 5:4 major
third might serve as one definition of ideal "concord," and likewise a
7:4 minor seventh to a singer of barbershop harmony.

However, from a 13th-century Continental European perspective, a major
third of _any_ proportion is an inherently unstable interval, although
a _relatively_ blending one. Tuning this interval as a Pythagorean
ditone, equal to two 9:8 whole-tones or 81:64, fits both the melodic
regularity of the music and the "dual-purpose" nature of this vertical
interval, neither stable nor urgently discordant.

From a 16th century perspective, where a 5:4 major third is regarded
as a richly stable concord (like a 3:2 fifth in Gothic practice), the
81:64 tuning would be out of place, indeed "discordant" not only in an
acoustical sense with its prominent beating, but in a _stylistic_
sense with its beating in a place where smooth and restful blending is
desired.

From this same 16th-century perspective, however, a 7:4 minor seventh
may have an effect similar to that of a minor seventh at around 16:9
or 9:5, or at the intermediate tunings between these latter ratios
likely on a 12-tet lute or a meantone keyboard -- a clear discord,
serving as a moment of urgent tension rather than independent euphony
or "concord." To propose that either vocal intonation or keyboard
tunings be altered in order to maximize the acoustical consonance of
this interval may seem less than intuitively obvious from this
stylistic viewpoint.

For a performance of 20th-century music based on relatively concordant
or even stable sonorities involving minor sevenths, and arguably also
for some later 19th century pieces where pervasive seventh sonorities
may be felt to _approach_ stability, a 7:4 tuning becomes as "natural"
as the ideal tuning of a Gothic fifth at 3:2 or a Renaissance major
third at 5:4. Indeed, pianist Dave Hill has very aptly used the
augmented sixths available on an instrument tuned in a 12-note version
of 1/4-comma meantone to approximate the 7:4 minor seventh in a
rendition of the Blues.

--------------------------------------------------
4. Nature and style: trines, triads, tetrads, etc.
--------------------------------------------------

Mother Nature sets some parameters and constraints on musics and
tuning systems, but it is the musicians who determine style.

Thus Mother Nature has established a state of affairs where, in most
timbres at least, a 5:4 major third will be smoothest in theory and
practice, and an 81:64 major third much more "beatful," with other
alternatives such as 1/6-comma meantone or 12-tet falling somewhere
between in their "restful" or "active" effect. However, it is musical
style which may help to determine which effect may be more fitting.

A very crude possible outline of the standards of stability in
European and related compositional practices might be something like
the following, with examples of saturated sonorities indicated using a
"MIDI-like" system of notation where C4 is middle C, and higher
numbers show higher octaves.

-----------------------------------------------------------------------
Era Rough dates Sonority/ String/Frequency
Ideal tuning ratios
=======================================================================
Trinic 1200-1450 Trine (D3-A3-D4) 6:4:3/2:3:4
3-limit
-----------------------------------------------------------------------
(Transition, c. 1400-1450: modified Pythagorean tuning
with some schisma thirds and sixths near 5-based ratios)
-----------------------------------------------------------------------
Early 1450-1670 Triad (G3-B3-D4) 15:12:10/4:5:6
tertian 5-limit
-----------------------------------------------------------------------
(Transition, c. 1640-1670: late modal/combinative style
leads to chordal/key system as modified meantone leads to
well-temperaments)
-----------------------------------------------------------------------
Later 1670-1900 Triad (G3-B3-D4) 15:12:10/4:5:6
tertian 5-limit
-----------------------------------------------------------------------
(Transition, c. 1840-1900: late Romantic style uses
pervasive seventh chords as dual-purpose sonorities,
possibly inviting 7-limit interpretations)
-----------------------------------------------------------------------
"Modern" 1900-? Various choices:
e.g. tetrads
G3-B3-D4-F4 105:84:70:60/4:5:6:7
E3-G3-B3-D4 21:18:14:12/12:14:18:21
E3-A3-D4-G4 64:48:36:27/27:36:48:64
-----------------------------------------------------------------------

This table may reflect two artistic processes in various periods. The
first is a stylistic tendency to treat certain unstable intervals as
more and more independently euphonious until they eventually become
stable concords -- or, as in the 20th century, until the distinction
of stability/instability may itself yield place to other concepts.

The second process is a tendency for intervals attaining a status of
independent euphony or even stability to invite a more acoustically
blending intonation, in contrast to the more "active" or complex
ratios which may have previously prevailed.

These tendencies do not apply in a simple or linear fashion to all
eras and intervals, of course. Thus, although unstable, combinations
such as G3-D4-A4 (9:6:4) or G3-C4-F4 (16:12:9) are evidently often
deemed relatively concordant "dual-purpose" sonorities in the music of
Perotin around 1200, but treated as clear discords in Renaissance
style around 1500. As it happens, these sonorities receive an ideally
euphonious tuning in medieval Pythagorean intonation, in contrast with
the greater compromises of Renaissance meantone -- and the "near-pure"
intonation of 12-tet on 20th-century keyboards in an era where these
quintal/quartal sonorities are treated as "concordant" and even
stable.

------------------------------------------------
5. Intonational ideals and pragmatic compromises
------------------------------------------------

In discussing "Mother Nature" and style, we might distinguish between
two factors: (1) The musical and intonational ideals of a given period
or style; and (2) The specific compromises which may be required in
tuning fixed-pitch instruments, for example.

While the element of compromise involved in most fixed-pitch tunings
is well-recognized, the room for differences in intonational ideals
themselves may be an equally important factor influencing vocal
tunings as well as keyboard tunings. Let me consider a striking
example that seems to fit with my own experience.

In 1555, as part of his treatise on the adaptation of the Greek
chromatic and enharmonic genera to modern practice, Nicola Vicentino
surveys the intervals available on his archicembalo or "superharpsi-
chord" dividing each whole-tone into five parts.

Here we have an intrepid musical explorer by no means bashful in
reporting, for example, that he finds the "proximate minor third" a
diesis (about 1/5-tone) wider than the usual meantone minor third,
which he describes as approximating a ratio of "5-1/2:4-1/2" (11:9),
as rather concordant.

However, Vicentino finds what might be called the "minimal seventh" a
diesis narrower than the usual minor seventh (e.g. Eb-C# in a usual
12-note meantone tuning) -- very close, we might note, to 7:4 -- as
quite dissonant, more so than the "proximate major sixth" a diesis
larger than major (e.g. C#-Bb), which we might add is not far from
12:7.

Indeed in a typical 16th-century setting where sevenths are typically
treated quite cautiously, with the suspension the most prominent
conventional usage, a near-7:4 minor seventh may suggest another
flavor of dissonance rather than any obvious "concord." This statement
may also hold for the late combinative style of the early 17th
century, where bold seventh combinations are a kind of "special
effect"; and even for 18th-century style, where combinations of this
kind are set in deliberate contrast to restful concord.

Similarly, the late 16th-century theorist Vincenzo Galilei, himself a
radical theorist and noteworthy advocate of bold seventh combinations,
declares that just as nature has made intervals such as a 2:1 octave
consonant, so nature has made other intervals such as a 9:5 seventh
dissonant.

Thus to seek a 7:4 minor seventh in a Renaissance or Manneristic
setting seems to me a less than obvious ideal, unless one is
experimenting with creatively anachronistic elements.

In contrast, if we ask why all concords on a 16th-century meantone
keyboard are not pure -- fifths and minor thirds, for example, in a
1/4-comma tuning -- then the pragmatics of keyboard tuning with a
limited number of fixed pitches per octave may be of special
relevance. Theorists such as Zarlino recognize this distinction when
they write that voices lean toward the pure ratios of JI (in this era
5-limit JI), while keyboard instruments generally are tempered by
irrational ratios in deference to practical considerations.

Even in the case of voices or other flexible-pitch instruments,
intonational ideals may be in tension. For example, singers of the
late 15th century may well have gravitated at once to the newer ideal
of smooth vertical thirds and sixths, and to the older ideal of narrow
cadential semitones. This dilemma might occur in a passage such as the
following

G4 F#4 G4
D4 D4 D4
B4 A3 G3
... G3 D3 G2

Here we have a suspension in the highest voice of 7-6 in relation to
the tenor, leading to a cadential progression of M6-8 -- and also a
4-3 (or 11-10) suspension between this voice and the bass. From the
viewpoint of tertian euphony -- a concern reflected in this epoch by
the adoption of meantone for keyboards -- the suspended fourth should
resolve to a smooth major third with the bass (and major sixth with
the tenor) tuned as pure as possible.

Thus the G4-F#4 resolution in this voice might from this viewpoint
involve a semitone of 16:15, or ~112 cents, permitting a resolution
from a 4:3 fourth (or 8:3 eleventh) with the bass to a pure 5:4 major
third (or 5:2 major tenth) -- and likewise from a 16:9 minor seventh
to a pure 5:3 major sixth with the tenor.

However, with freely intoning voices as well as meantone keyboards,
such pure or near-pure thirds and sixths imply rather wide cadential
semitones. In medieval practice, where active and unstable thirds and
sixths are the norm, narrower diatonic semitones for voices and
keyboards alike are practical, at or near the Pythagorean 256:253 (~90
cents). In Renaissance practice, the shift to meantone on keyboards
suggests more generally an intonational ideal in which smoothly
blending thirds and sixths become a priority, with diatonic semitones
accordingly widened.

However, as Mark Lindley notes, a survey of theoretical writings
around 1500 on the topic of large and small semitones may suggest not
only a certain degree of confusion to be expected in such a
transitional era, but also an artistic ambivalence about changing
intonational ideals.

"Mother Nature" -- and the applicable mathematics -- remain the same;
styles and artistic choices change.

------------------------------------------------
6. Intonational pluralism in theory and practice
------------------------------------------------

Like the participants on this Tuning List, 16th-century theorists
often note the plurality of tunings and temperaments in actual use. A
typical categorization might go like this:

1. VOICES. Theorists such as Zarlino regard just
intonation as natural for voices, in contrast
to the temperaments customary for fixed-pitch
instruments. In such an ideal scheme, the
concordant intervals of Zarlino's _senario_
(with ratios defined by the "sonorous numbers"
or integers from 1 to 6, plus the minor sixth
at 8:5) tend always to occur in their pure
ratios; other intervals such as whole-tones
(9:8 or 10:9) or minor sevenths (9:5 or 16:9)
may vary in size.

2. LUTES. By around 1545, a 12-note tuning in
equal semitones is regarded as "standard"
for these fretted instruments, although
certain more complicated tunings might
sometimes be applied.

3. KEYBOARDS. Meantone tuning with narrowed
fifths is regarded as the norm, with
1/4-comma tuning (pure 5:4 major thirds)
evidently one common choice and Zarlino's
2/7-comma tuning (major and minor thirds
equally compromised) another.

In keeping with these standards, writers note, for example, the
intonational difficulties of managing an ensemble where lutes and
keyboards sound together.

Also, as we might expect, there are some contrasting views both about
what people _are_ singing and playing in practice, and about what they
_should_ be singing or playing.

Thus Vincenzo Galilei, while agreeing with his former teacher and
frequent adversary Zarlino that there are three categories of tuning,
argues that unaccompanied voices in fact tend toward a kind of
intonation with equal whole-tones ("interval consistency," as Paul
Erlich might say) somewhat closer to a keyboard than to a lute. In
other words, he proposes that voices lean to a kind of meantone
temperament not too far from 2/7-comma, which he agrees with Zarlino
in preferring for keyboards.

Vicentino suggests two possible paradigms, both realized in his
archicembalo tunings. For his "enharmonic" music using novel melodic
and sometimes also vertical intervals based on dieses or fifthtones,
he urges singers to take his instrument (in its 31-tone division of
the octave) as the norm. However, his alternative tuning would provide
a basis for what has here been termed "adaptive JI," with very slight
shifts from a usual meantone intonation in order to make concords pure
-- fifths and minor thirds as well as major thirds (already at 5:4 in
a 1/4-comma temperament).

One curious statement by Zarlino about the comma seems of special
interest here[1]:

"This interval would be very annoying to the listener,
more so because such an interval is not heard in
voices, which can tune intervals higher or lower
as desired and through this bring perfection to
any composition. This is not true with artificial
instruments, because art can never equal nature.
Voices do tend to approximate more closely those
instruments that are tuned to the Pythagorean
steps, in which these minute intervals are not
present, than those that are perfectly tuned in
accord with the perfection of the harmonic numbers.
Thus it might be said that temperament is more
useful to the musician than perfect tuning. For
scientific purposes the latter is more useful
because from it the true ratio of each interval
may be determined, and, most important, because
voices (as I have said elsewhere) seek the
perfection of intervals."

It might be argued, for example, that a system of "adaptive JI" like
that of Vicentino or Erlich combines some traits mentioned by Zarlino
and Galilei. Vertical intervals are pure, while melodic intervals are
_almost_ uniform, varying by only 1/4 syntonic comma or ~5.38 cents if
the tuning is based on 1/4-comma meantone, for example.

-------------
7. Conclusion
-------------

To say that "Mother Nature" dictates musical style or intonation would
be rather like saying that the laws of physics dictate architectural
style. Nature indeed in each case offers a range of structural and
aesthetic choices while imposing certain limitations, but the
architect or musician operating within a certain cultural background
and setting selects from among these choices.

One problem in generalizing about intonational ideals over an era such
as the Western European key tonality period of around 1670-1900 is
that such an era includes many epochs and stylistic currents. While
Vivaldi and Wagner are often seen as belonging to the same "common
practice," I would not necessarily assume that the same aesthetics of
tuning for minor sevenths would apply to both composers and epochs.

From another point of view, the practices of 1670-1900 represent only
what I call the "later tertian" era of composed European music; the
Gothic multi-voice polyphony of 1200-1420, for example, represents an
era of around equal duration.

One advantage of the Tuning List is that we can indeed compare
practices and intonations, thus getting a wider view of what nature
permits and artistic choice may favor.

----
Note
----

1.Gioseffo Zarlino, _The Art of Counterpoint: Part Three of Le
Istitutioni harmoniche, 1558_, trans. Guy A. Marco and Claude Palisca
(W. W. Norton, 1976), ISBN 0-393-00833-9, Chapter 17, at p. 35.

Most respectfully,

Margo Schulter
mschulter@value.net