Re: Response to Dave Hill on JI and European composition

Hello, there, and it's a great pleasure to respond to a question from
our noted pianist Dave Hill on the role of just intonation (JI) in
historical European compositional practice and theory.

Here there are certainly some polarized views: while a theorist such
as Zarlino and some 20th-century counterparts would answer that JI is
"certainly" the best paradigm at least for vocal intonation, in theory
and practice, others including Paul Erlich would answer "Absolutely
not" if the use of integer ratios for all melodic as well as vertical
intervals is meant.

My own conclusion is a bit less categorical: it depends in part on
taste, and in part on the music. In the 16th century, as now, one
theorist could assert that classic JI was the rule for voices, another
would offer a demonstration of its impossibility, and a third would
advocate the kind of scheme which Erlich has termed "adaptive JI."

First, a semantic but for me nontrivial matter of definition. As a
medievalist, I regard Pythagorean intonation as one form of JI, a form
providing a framework for Continental European polyphony during the
varied era of about 850-1450 -- or 850-1400, if we treat the early
15th century as a kind of special transitional period leading to
meantone for keyboards and 5-limit JI (at least in the ideal) for
voices.

If JI means a system where all intervals are defined as integer
ratios, and all stable concords (in styles like those of medieval
part-music where "stable/unstable" is a useful contrast) are realized
as "low-integer" ratios, then Pythagorean tuning could be a paradigm
case of a robust JI system.

Thus I would say that the first 550 or 600 years of composed European
polyphony were based on 3-limit JI -- not a bad track record.

The early 15th century (maybe starting in areas such as Florence a bit
earlier, 1370 or 1380) represents, at least on keyboards, a special
kind of JI approach exploiting the contrast in Pythagorean tuning
between regular thirds and sixths (e.g. M3 at 81:64, m3 at 32:27) and
what are now often termed "schisma" intervals a Pythagorean comma
narrower or wider (e.g. d4 at 8192:6561, ~5:4; A2 at 19683:16384,
~6:5). Since the tuning is still derived from integer ratios, with all
regular octaves, fifths, and fourths pure, it is still 3-limit JI.

From an aesthetic viewpoint, I would prefer to describe this as
modified 3-limit rather than 5-limit, because regular Pythagorean
thirds and sixths remain the rule for intervals not involving written
sharps (e.g. A3-C#4, realized as A3-Db4) -- I use C4 for middle C, and
higher numbers to show higher octaves. Mark Lindley speaks of "modal
color" contrasts analogous to the "key color" contrasts of unequal
well-temperaments of the 18th-19th centuries, and I have proposed for
the 17-note Pythagorean tunings (Gb-A#) proposed in the early 15th
century by Prosdocimus of Beldemandis and Ugolino of Orvieto the term
"well-untemperament." This may be a technically inexact usage, since
the chain of fifths remains open, but may suggest an ample gamut with
alluring contrasts of color.

Incidentally, to distinguish this type of early 15th-century tuning
from certain schemes (e.g. Helmholtz/Ellis) which seek _consistently_
near-pure thirds and sixths, I would propose the medieval Latin
spelling "schismatic" for the first type of tuning, and the
Helmholtz/Ellis spelling "skhismatic" for the second.

By the later 15th century on the Continent -- and possibly also
through much of the medieval era in at least some parts of England --
we appear to enter a period when tertian JI (pure thirds as well as
fifths) does define the ideal for voices and other non-fixed-pitch
instruments. The "simplified" monochord of Ramos (1482), with its
5-limit tuning, is one possible landmark.

In my view the novelty here is not JI -- deriving intervals from
integer ratios, and seeking the purest tunings for stable concords --
but the application of these ideals to a new musical style (especially
on the Continent) where thirds and sixths are among the full or stable
concords.

While Ramos set a possible precedent with his monochord of 1482,
16th-century theorists such as Fogliano and Zarlino championed the
syntonic diatonic of Ptolemy with its unequal 9:8 and 10:9 whole-tones
as the natural intonation for unaccompanied voices.

Granted that a 5-limit system based exclusively on such integer ratios
runs into such complications as impure fifths and fourths (the
syntonic comma problem) and comma shifts or drifting pitches, this
does not mean that for all musics it is as impossible a task as
squaring the circle.

Using a 15-note tuning based on a keyboard instrument described by
Zarlino (with two versions a syntonic comma apart of D, Bb, and F#), I
have performed some straightforward 16th-century pieces with pleasure,
and without any obviously insuperable difficulties. This included
pieces requiring me to move between two manuals, one with a just fifth
G-D and the other with a just D-A; while I was quite engaged in
placing my hands on the right manuals at the right time, any comma
shifts did not distress me.

As Zarlino and other theorists favoring JI for voices and other
non-fixed-pitch instruments recognize, JI for keyboards is an
experimental technique, and meantone temperaments are generally much
less complicated for the player to manage. However, it's a kind of
experimentation that I'd highly recommend, not as a substitute for
meantone but as a fascinating and moving experience in its own right.

In a recent article on "Pythagoras at the forge: tuning in early
music," in _Companion to Medieval and Renaissance Music_, ed. Tess
Knighton and David Fallows (New York, Schirmer Books, 1992),
pp. 316-326, Rogers Covey-Crump suggests that many vocal consorts
"tend towards Mean Tone tuning, but the best performers, when
thoroughly rehearsed, are capable of Just or Pythagorean tuning." He
recommends Pythagorean for most medieval styles through Machaut and
his late 14th-century successors, and Just (i.e. 5-limit) for
Renaissance music. For some English or Anglo-Norman medieval styles of
the 13th and 14th centuries, he notes the possibility of "alternative
or mixed tunings of thirds" which might lean toward a kind of 5-limit
JI (as suggested around 1300 by Walter Odington).

At the same time, he observes (like Paul Erlich here) that "tuning is
mainly about vertical chording and not about solo melodic lines," and
that "consistency of interval size ... is a feature of the best
western European singers."

This suggests that either Renaissance or 20th-century singers
approaching the 16th-century repertory may in fact be doing at least
some adaptive JI with tempering of melodic intervals.

One 16th-century theorist, Giovanni Battista Benedetti (c. 1563),
argued that comma drift made 5-limit JI impossible in practice for
voices, and proposed a system of temperament which Claude Palisca has
interpreted as 12-tone equal temperament (12-tet).

Interestingly, Vincenzo Galilei concluded in the 1580's that
unaccompanied singers favored neither 3-limit nor 5-limit JI, but
rather a form of temperament with unequal semitones (unlike 12-tet on
lutes) but equal whole-tones, rather resembling that of keyboards, for
which he recommended a 2/7-comma tuning like that favored by Zarlino.

However, from an adaptive JI viewpoint, Nicola Vicentino's alternative
tuning for his 36-note archicembalo (1555) presents a kind of
synthesis of the opposed viewpoints later expressed by Zarlino and
Benedetti or Galilei.

Describing the common practice as "tempered and mixed music," and
basing the tuning of the first 19 notes of his instrument on usual
meantone (likely 1/4-comma with pure major thirds), he directed that
the 17 notes of his second manual should be tuned in perfect fifths
with those of the first manual. The result, assuming a basic 1/4-comma
temperament, would be sonorities with pure fifths and minor thirds, as
well as major thirds, if the performer can reach the appropriate keys
on both manuals simultaneously.

While Vicentino's advice to singers to rely on his instrument for
finding their intervals would most obviously apply to his first tuning
with 31 notes arranged to divide the octave into more or less equal
dieses or fifthtones (the remaining 5 notes used for adaptive JI for a
few frequent sonorities), it could also apply to this second tuning.
In other words, singers might have emulated this tuning to realize a
kind of JI based on melodic intervals at or very close to those of
1/4-comma meantone, the general variety of tuning which Galilei later
suggested that unaccompanied singers tend to practice.

(From a melodic standpoint, one might argue that 1/4-comma would be a
somewhat preferable model to 2/7-comma, since it permits wider
whole-tones and narrower diatonic semitones.)

From the viewpoint of practical composition, the issue of classic
vs. adaptive JI for voices (or either model vs. keyboard meantone)
might not have much observable impact. Zarlino does say that certain
parallel progressions between imperfect consonances (thirds and
sixths) may have some redeeming variety because of the different sizes
of whole-steps, but generally such distinctions might be considered
details of performance rather than composition.

However close the "just" vocal intonation of the 16th century may have
been to the classic model of Zarlino (based on Ptolemy's syntonic
diatonic) or the adaptive tuning of Vicentino, an interesting
discussion showing both the continued recognition of Zarlino's model
in the 18th century and the need for some flexibility in its
application appears in Johann Philipp Kirnberger's treatise of 1771,
_The Art of Strict Musical Composition_, tr. David Beach and Jurgen
Thym (New Haven: Yale University Press, 1982).

In a discussion of "Scales and Temperament," pp. 20-21, in which he
rejects 12-tet for keyboards as harming "the diversity of keys,"
Kirnberger also notes that in vocal intonation, "entirely pure melodic
progressions in two parts produce various tempered or not quite pure
harmonic thirds."

Maintaining that melodic fourths and fifths "cannot be sung other than
pure," he gives an example, which I notate as if in 2/2 although it
might be read in 4/4 also:

1 & 2 & | 1 ||
C5 G4 D5
C5 F4 Bb4

Here each note is sustained until the next note in that part or the
end of the example, a convention that avoids the complications of ties
in ASCII.

The two parts begin at a unison, and melodic motions in the upper part
of a pure fourth down and pure fifth up necessarily place it at a 9:8
major second above the lower part. From this point, Kirnberger shows
that the melodic motions of a pure fifth down and a pure fourth up in
the lower voice must result in a concluding vertical major third
Bb4-D5 of 81:64 rather than 5:4. Although this third is a syntonic
comma larger than pure, "it is not to be discarded for that reason."
In his view, the just tuning of melodic fourths and fifths must take
precedence, especially in vocal music.

While concluding that "progressions by pure intervals result in thirds
that are sometimes larger and sometimes smaller," ibid. at p. 21,
Kirnberger nevertheless states in his discussion of "The Scale, and
the Keys and Modes Derived from It" that "[t]he basis of the tone system
currently in use throughout Europe" is "the basic diatonic scale
... said not to have come into use until Zarlino's time." (p. 315).

This is indeed Zarlino's syntonic diatonic, and Kirnberger remarks
that "before him the steps had these older Greek proportions," and
gives the ratios for a standard Pythagorean tuning.

If we define "JI" to include Pythagorean in a medieval context as well
as 5-limit in a Renaissance-Classic context, then both of Kirnberger's
basic diatonic scales are instances of just tuning.

To conclude, I would agree with Dave Hill that both 3-limit and
5-limit JI have been immensely influential in the European tradition
of composition, while at the same time recognizing (along with Nicola
Vicentino and Paul Erlich) the appeal of adaptive 5-limit JI.

On a personal note, I might add that while playing a keyboard in
classic 5-limit JI is an exacting experience, it is also a unique one,
and that I do not find unequal 9:8 and 10:9 whole-tones unpleasant. At
least for some music some of the time, this experience is not only
possible but most rewarding.

For me, "playing in JI" means especially 3-limit or Pythagorean, a
tuning system with five or six centuries of repertory including the
awesome polyphony of composers such as Leonin, Perotin and Machaut.
If we follow the definition of JI advocated here, then this is the one
JI system which has won acceptance as a standard keyboard tuning.

Finally, I might suggest a pragmatic study of intonation as practiced,
for example, by vocal groups famed for their "just" tuning of
16th-century music. Would we find some approximation of Zarlino's
syntonic diatonic, or of a Vicentino/Erlich adaptive tuning, or
something else?

Most respectfully,

Margo Schulter
mschulter@value.net