JI, tritones, etc. -- at Kraig's request

Hello, there, everyone, and at Kraig's request, please let me cover a
few points of history related to the "JI" concept, and also gladly
agree with Dave Keenan that his perceptual definition of JI represents
a different direction than any of the five definitions I gave, which
all presuppose rational ratios. Thus I should have stated: "Here are
five possible definitions treating JI as either synonymous with RI, or
as a subset; however, Dave Keenan has often discussed a different type
of definition focusing on audible purity rather than mathematics."

Turning to the historical discussion, I might first comment that the
real horrors of burning at the stake for heresy in later medieval
Europe, and also for witchcraft on a widespread basis in early modern
or Renaissance Europe, need to be remembered early and often. The year
2008, for example, will mark the 800th anniversary of the so-called
"Albigensian crusade," actually an act of genocide (1208-1242), which
by around 1230 led to the launching of the Inquisition.

However, if discussing or advocating the use of the tritone had then
actually been considered a form of heresy -- as opposing war, capital
punishment, or more specifically the execution of those convicted of
heresy and so sentenced often was, by the way -- then a number of
theorists would have been in danger, ranging from Jacobus of Liege
(c. 1325) to the famous Zarlino himself (1558 and later).

For more detail on this, see my Tritone FAQ at Todd McComb's fine site
of the Medieval Music and Arts Foundation:

http://www.medieval.org/emfaq/harmony/tritone.html

To give a brief summary from an intonational perspective, I might say
here that Jacobus of Liege in the early 14th century generally
regards the tritone as a strong discord, but observes that this
interval does sometimes occur melodically in certain ecclesiastical
chants. He also distinguishes this tritone equal to three whole-tones
at 9:8, or 729:512 (~611.73 cents), from the _semitritonus_ or
diminished fifth at 1024:729 (~588.27 cents), finding the latter
somewhat less dissonant. He says that these intervals rarely occur
because they are dissonant and awkward to sing, but that their theory
is beautiful.

In 1357, Johannes Boen specifically approves of the tritone as a
"consonance by situation" (_consonantia per accidens_) if it is placed
in a three-voice sonority above a minor third (e.g. D3-F3-B3, with C4
showing middle C as in MIDI notation).

Both melodic and vertical tritones occur in 13th-14th century music,
and while some of these intervals may have been altered to perfect
fourths or fifths by performers using unwritten accidentals (e.g. the
usual or _musica recta_ step of Bb in place of B-natural, or
additional "invented" or _musical ficta_ accidentals such as F#), they
could not well all have been avoided. Scholars such as Hans Tischler
and Gordon Anderson have emphasized this point.

Both Jacobus and Boen are writing in a setting of Pythagorean or
3-limit JI. By the way, Jacobus discusses ratios of 5 and 7,
concluding that they might well be concordant to a considerable degree
on instruments built to use them, but do not fit a musical system
based on well-defined Pythagorean steps. This could be taken as a
statement not of perceived theological evil, but of intonational
inertia, much like: "Ratios of 7 are not necessarily displeasing, but
we don't use them because they don't fit well with our 12-tone scale."

In fact, Jacobus demonstrates such things as the 28:27 and 135:128
semitones, and the commas of 81:80 and 64:63. Part of his interest
might relate to Greek theory; in any event, he doesn't consider these
things unmentionable. They are interesting, and potentially
concordant, but inconsistent (begging pardon of Paul Erlich, Dave
Keenan, and others who have made this a term of art) with the
intonational structure he takes as given in practice.

Johannes Boen also briefly discusses such ratios as 5:1 and 7:1,
showing that they do not coincide with usual Pythagorean intervals.
His approach generally seems pragmatic, and he is ready to describe
the pleasing practical use of such intervals as the tritone and also
the diminished fourth.

It is interesting that he takes the latter, described as a comma
smaller than the usual major third, as an interval which can be a
"consonance by situation" specifically when it has a regular major
third placed below (e.g. E3-G#3-C4). Thus in the mid-14th century,
this interval at 8192:6561 (~384.36 cents) was evidently regarded as a
"special" interval, much like a diminished fourth in a Renaissance
setting, rather than an independently concordant "schisma third,"
which it became by the early 15th century.

In the early Renaissance, the increasing treatment of thirds and
sixths as near-stable or even stable concords means that the tritone
or diminished fifth can resolve to such a concord by stepwise contrary
motion, giving it a new significance. By 1558, Zarlino recommends
basic progressions such as B3-F4 to C3-E3, which are indeed
characteristic both of late modal styles (16th and early 17th
centuries) and of major/minor tonality (established by around 1680,
the era of Corelli and Werckmeister).

Now for the matter of adaptive 5-limit JI and meantone in the
Renaissance. In one case, we have singers splitting or finessing
commas while maintaining or closely approximating just intonation
(under Dave Keenan's definition of "audible purity," these two might
be synonymous). In the other, we have a fixed-pitch instrument tuned
to divide the commas -- and possibly to permit an adaptive JI
technique, as with Nicola Vicentino's two manuals likely tuned at a
distance of 1/4-comma apart (his second archicembalo tuning of 1555).

Reading around and between the lines, we find that two theorists of
the 16th century reached similar conclusions. Zarlino (1558) remarks
that singers seem to seek pure concords without running into the
commas that one encounters on a keyboard tuned to the syntonic
diatonic; and he later specifically notes that singers can avoid such
difficulties as the 40:27 fifth (around 680 cents, an 81:80 narrower
than 3:2) which would occur in a single 7-note scale of this type.

His erstwhile student Vincenzo Galilei, however, looks at similar
pragmatic observations of vocal style in another way, suggesting that
such intonation is closest to a meantone tuning, and somewhere between
the tuning with equal semitones (12-EDO) for a lute and the 2/7-comma
temperament he, like Zarlino, prefers for keyboards. While Zarlino
sees this style of intonation as an adaptation of the syntonic
diatonic, Galilei looks at it as more of a quasi-regular tuning.

Yet Galilei, also, says that while he regards vocal intonation as
different from the syntonic diatonic, it is very similar to this
system. Thus the two famous rivals, Zarlino and Galilei, seem largely
at agreement on this point, at least.

From this theory we go back to a topic you have raised, Monz, the
reported "rusticities" in Gregorian chant around 800, or in the 9th
century, the same era in which the _rustica lingua Romana_ or Romance
vernacular (e.g. Old French) was winning recognition as the
appropriate language for sermons to the common people, and getting
recorded in written prose and poetry.

Here there might be a very wide scope for guesses, with 5-limit
intonation only one possibility. How about something like 12:11:10:9,
for example, a kind of intonation associated with Byzantine chant and
also with Near Eastern traditions, and documented in the equable
diatonic of Ptolemy? How about enharmonic or other "microtonal"
ornaments which wouldn't fit a diatonic organ tuning?

Of course, I would agree that there is very appreciable evidence for a
5-limit or similar tradition among the Vikings, for example -- the
later Hymn to St. Magnus from the Norwegian-controlled Orkney Islands,
and reports by Giraldus Cambrensis around 1200 of a distinctive kind
of part-singing in the North and West of England, which he attributes
to the influence of Scandanavian occupiers.

Certainly the 9th century was an era of raids from the North, with the
peace settlement in France making the warlike Rollo the Duke of
Normandy coming in 911, as I recall. The acceptance by King Pepin in
757 of an organ as a gift from Constantinople might likewise symbolize
the Byzantine influence, which could have brought a range of
alternative tunings as well, ironically, as the instrument which could
have served to "standardize" a diatonic Pythagorean framework for the
chant.

What I do want to clarify is that persecution or burning at the stake
for _theological_ heresy should not be equated with the fame or
notoriety that one could earn for advocating unorthodox musical
doctrine. Thus Marchettus of Padua (1318), by introducing a fivefold
division of the tone which some interpreters took to be geometric, won
both the theoretical scorn of Prosdocimus about a century later, who
regarded him as a very bad mathematician although a good practical
musician; and the approval of some other 15th-century writers.
However, I am not aware of anyone making an issue of faith out of
this -- although the same era, sadly, witnesses such events as the
statutory ratification of the burning of heretics in England (1401)
and the burning of Jan Hus at the Council of Constance (1415).

Similarly, when Ramos published his famous treatise of 1482 poking fun
at the Guidonians, as well as introducing a 5-limit monochord for the
easier instruction of young students confused by intricate Pythagorean
ratios and divisions, he drew much criticism -- but not theological
censure, as far as I am aware.

Incidentally, Ramos is a defender of the tritone in due place, for
example in defining the medieval Lydian mode with a lower ascending
pentachord of F-G-A-B-C. He quotes his friend Tristan de Silva to say
that a tritone is not the worst musical sin -- a metaphor, but not
evidently one to be taken too literally, given not only the
theoretical recognition of this interval by writers in earlier
centuries, but its occurrence in practice.

In Zarlino's scheme, I suspect that stylistic tidiness rather than any
theological consideration excludes 7 from the set of "concordant"
numbers, just as Kepler early in the next century excludes 9, earlier
included by Jacobus of Liege. The stylistic predilection of this
Renaissance/Manneristic era for saturated 5-odd-limit sonorities is
reflected in theological images and allegories, which one can also
find a bit later in Werckmeister.

Rather than assume that Zarlino was somehow constrained by theology to
regard 4:5:6:7 or 12:14:18:21 as a dissonance, or Kepler so to regard
6:8:9, I would suggest that this was a matter of stylistic perception
or "good taste," which could, of course, be treated as a basis for
philosophical or theological explanation or allegory.

Likewise, in 1555, Vicentino, who has few inhibitions about advocating
new ratios such the "proximate minor third" which he describes as an
approximate 11:9 and finds finds rather concordant, considers narrow
or minimal thirds and sevenths which we should describe at near 7:6
and 7:4 as on the discordant side. Factors such as certain categorical
perceptions of basic consonance/dissonance categories, which Vicentino
and Zarlino may have shared despite dramatic differences on other
issues, might explain why neither includes a ratio such as 7:4 among
the concords, although Vicentino is ready so to regard 11:9 to at
least a degree.

It is significant to recognize that in the year 1600, Giordano Bruno
indeed _was_ burned for heresy; the fact that in that very same year,
Giovanni Maria Artusi could take for granted the acceptability of the
tritone or diminished fifth when treated in the proper fashion, as
opposed to the unconventional one of an unnamed composer easily
recognizable as Claudio Monteverdi, should not distract us from this
act of cruelty and brutality.

To sum up: the tritone, although considered a dissonance (at least in
itself, as a simple dyad), was recognized as an interval in use by
both medieval and Renaissance theorists. People who want to explain
the apparently rather frequent appearance of the tritone in music
around 1200 from a "low integer ratio" or "audible purity" hypothesis
might argue that the Pythagorean ratios of 729:512 and 1024:729 are
only about 5.76 cents (5120:5103, the 5-7 schisma) from pure ratios of
10:7 and 7:5. Such people might propose that singers would lean toward
these simple ratios.

Of course, such a hypothesis should take note that in a typical Gothic
style, these intervals resolve to stable fifths and fourths, rather
than to the fully concordant thirds and sixths of a Renaissance style
which set a pattern of tritone tension and resolution also obtaining
in later tonal schemes.

More generally, I would suggest that musical predilections could
influence intonational preferences in medieval Europe as well as
elsewhere. Thus in England, where some 13th-century pieces have an
audibly different quality than typical Continental pieces since they
treat thirds as fully concordant and indeed sometimes conclusive,
these thirds tend to be associated with ratios of 5:4 and 6:5
(Theinred of Dover, Walter Odington). Odington, in contrast, regards
4:6:9 or 6:8:9 as dissonant, although recognizing that other musicians
may disagree; in the same era Jacobus finds these sonorities
relatively concordant.

Yet earlier, around 1100, John of Afflighem (sometimes known as John
Cotton) remarks that different people treat organum or polyphony in
different ways, but that he will offer a straightforward approach
based on the principle of contrary motion between the two voices.

Going yet further back, we find that Guido (c. 1030) describes the
kind of polyphony that he prefers and finds most pleasing, recognizing
not only the basic stable concords or _symphoniae_ (for him especially
the unison, fourth, and in a three-voice texture also the octave, and
only secondarily the fifth), but also major seconds and major or minor
thirds.

Certainly there are assumptions here about what sounds pleasing or
otherwise, and indeed some passages showing that musicians not only
could differ, but were sometimes aware of these differences and
controversies.

It is a tragic reality of history that relatively free musical
interchange and dialogue can occur at the same time as mass killings
and even genocide -- consider also the assertive musical independence
of a composer such as William Billings in the 18th century on the
continent traditionally known to some of its indigenous nations as the
Great Turtle Island, and the genocide against many of those nations in
progress during that same era.

Some medieval music theory has a very refreshingly experimental
quality, reminding me a bit of some Tuning List dialogues, albeit at a
slower pace imposed by the technologies then available. Now, as then,
musical pluralism may or may not imply respect for the basic human
rights guidelines of Amnesty International -- a lesson of which we
might take note at the opening of a new century.

Most appreciatively,

Margo Schulter
mschulter@value.net