Re: tritone (was: Digest Number 1881)

Hello, there, Gerry and Monz and everyone.

On the tritone question, I would emphasize that the tritone or
diminished fifth occurs in medieval music as well as in the
Renaissance era and, of course, the major/minor tonality of the era
from about Corelli to Wagner (say 1680-1900).

As early as 1357, Johannes Boen gives a rule specifically approving the
use of the tritone as a _consonantia per accidens_ or "situational
consonance" above a minor third, while around 1325, Jacobus of Liege has
noted that such intervals occur, although rarely, in the ecclesiastical
chant.

Jacobus distinguishes between the _tritonus_ proper at 729:512 (about
611.73 cents) and the _semitritonus_ at 1024:729 (~588.27 cents),
finding the latter somewhat less dissonant, and noting therefore that
there are actually 14 diatonic intervals within the inclusive range of
unison to octave. Typical 13th-century discussions of polyphony list a
set of 13, counting the _tritonus_ but not the distinct _semitritonus_
of Jacobus.

This confirms a point you make, Monz: the term "tritone" is routine in
a medieval European setting where Pythagorean intonation is the norm,
with this interval equal to three 9:8 whole-tones.

In the Renaissance, by the time of Vicentino (1555) and Zarlino
(1558), the use of the diminished fifth in progressions by stepwise
contrary motion to a major third is standard in theory as well as
practice, and Zarlino notes that the older composers as well as the
"moderns" do this -- possibly referring, for example, to the
generation of Josquin des Prez around 1500. Such resolutions are
important in 16th-century style, and become a feature of late
17th-century tonality also.

For more on this, please see

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

Here I might add that 13th-14th century cadences seem to me just as
compelling as 18th-century cadences: each language has its own
beauty. Paul Erlich's theory of "strong" or "tonal" progressions, by
the way, nicely covers both types of progressions, and seems correctly
to predict the strongest 14th-century progressions.

Gerry, while noting that the tritone gets used in medieval music, I'd
much agree that it takes on a more central role in the Renaissance,
and yet more so in 18th-19th century tonality. The explanation I might
offer is simply that the most compelling progressions in composed
European music often seem to involve resolutions from an unstable to a
stable interval by stepwise contrary motion.

In a Gothic or neo-Gothic setting, this means such expanding
resolutions as major third to fifth, major sixth to octave, and major
second to fourth, for example; or contracting ones such as minor third
to unison and minor seventh to fifth.

Curiously, my rationale for often leaning toward ratios of 7 for these
intervals (9:7, 12:7, 8:7; 7:6, 7:4) is that cadences become more
efficient, with less motion needed to reach the stable goal. This is
hardly original: a medieval theorist such as Marchettus of Padua
(1318) refers to it as "closest approach."

On one level, this means that thirds expanding to fifths or sixths to
octaves should be major -- with accidental inflections, if needed. To
Marchettus, it additionally means that these cadential major thirds
and sixths should be extra-wide, so as more closely to "approach" the
interval of resolution, the fifth or octave.

Thus in various neo-medieval tuning systems, I might use a regular
major third around the Pythagorean 81:64 (~407.82 cents) or 14:11
(~417.51 cents), for example, but a larger major third around 9:7
(435.08 cents) for such cadential progressions to a fifth.

What happens around the late 15th century is that major and minor
thirds become increasingly stable intervals -- a change in musical
style accompanied by an intonational shift toward meantone temperament
on keyboards, and likely by vocal ensembles toward some kind of
meantone-like or adaptive 5-limit JI tuning.

In such a style, the diminished fifth can contract to a major third,
or an augmented fourth or tritone expand to a minor sixth, very
efficiently -- and reach a stable goal.

The open question, however, is how important "closest approach" at the
level of fine tuning would be to 16th-century ensembles. Curiously,
Vicentino (1555) has some "enharmonic" cadences in his 31-note
meantone system using the step of a chromatic semitone (2/5-tone in
this system) rather than the usual diatonic semitone (3/5-tone). This
seems more a special effect than an emulation of vocal intonation,
although we can hardly exclude the possibility that something similar
sometimes happened in performance. Vicentino's cadence calls for a
shift by a diesis (about 128:125 or ~41.06 cents), which is a main
point of the progression as one facet of his effort to adapt "ancient
theory" to "modern practice" -- the polyphonic use of the Greek
chromatic and enharmonic genera.

Zarlino (1558) tells us that vocal ensembles strive toward pure
5-based ratios (those included in the _senario_ of natural numbers
from 1 to 6, plus the 8:5 minor sixth), and can make adjustments to
avoid complications of the syntonic diatonic such as 40:27 fifths.
However, to my best knowledge, he does not address the use of smaller
cadential semitones than the 16:15 (~111.73 cents) of Ptolemy's
syntonic diatonic.

A final, personal, note on the tritone: I "discovered" it in 1966, on
the piano in the classroom where my Music Appreciation course was
taught, when I was delightedly playing parallel fourths or fifths and
suddenly ran into an "out of tune" interval. I suggested to someone
that the instrument must be out of tune, and was helpfully informed
that I had hit a tritone, B-F, quite distinct from my desired perfect
fifth.

In this kind of musical context, a tritone can be a very obtrusive
"Wolf," which could explain the eager desires -- not only mine, but
those of some medieval clerics -- to avoid its unintended appearance.

By around 1200, however, Perotin uses the tritone as well as the minor
second or major seventh -- all "perfect" or acute discords in this era
-- to beautiful effect. In a style with a continual mix of stable and
unstable sonorities, it helps diversify the continuum.

In this kind of style, the tritone or diminished fifth often behaves
much like a usual fourth or fifth, resolving by parallel or similar
motion to a stable fourth or fifth. I like this effect very much
either in the historically accurate Pythagorean tuning, or in a range
of systems such as 29-EDO, 46-EDO, and indeed 20-EDO (where the
"diminished fifth" in my quasi-diatonic modes is 540 cents).

Anyway, while I can say that the Renaissance use of the tritone as an
interval resolving to a stable third by stepwise contrary motion
developed in a meantone type of keyboard orientation, just how singers
or other flexible-pitch performers negotiated these progressions
remains an open question.

Most appreciatively,

Margo Schulter
mschulter@value.net

On 2/13/02 1:54 PM, "tuning@yahoogroups.com" <tuning@yahoogroups.com> wrote:

>
> Message: 7
> Date: Wed, 13 Feb 2002 09:22:09 -0800 (PST)
> From: "M. Schulter" <MSCHULTER@VALUE.NET>
> Subject: Re: tritone (was: Digest Number 1881)
>
> Hello, there, Gerry and Monz and everyone.
>
I'm delighted to find your very informative (as usual) response to my
off-hand reference to the "tritone." I learn so much from every post you
contribute here. I feel I'm growing like a teenager when I read your
eloquent discourse. In fact, the prose is often so beautiful in itself that
I'm temporarily distracted from your meaning and have to read it again. :-)
>
> A final, personal, note on the tritone: I "discovered" it in 1966, on
> the piano in the classroom where my Music Appreciation course was
> taught, when I was delightedly playing parallel fourths or fifths and
> suddenly ran into an "out of tune" interval. I suggested to someone
> that the instrument must be out of tune, and was helpfully informed
> that I had hit a tritone, B-F, quite distinct from my desired perfect
> fifth.

How interesting! That is precisely how I demonstrate the uniqueness (and
importance) of the tritone to my students, as well as any readers of Music
101 text "The Sounds of Music: Perception and Notation." I'm experiencing
goosebumps to see that this exercise was important in your formative past.
>
> In this kind of musical context, a tritone can be a very obtrusive
> "Wolf," which could explain the eager desires -- not only mine, but
> those of some medieval clerics -- to avoid its unintended appearance.

And this amplifies my point--that 17th century composers, having discovered
the strength of the tritone, used it *intentionally* to effect their
kaleidoscopic modulations.
>
> By around 1200, however, Perotin uses the tritone as well as the minor
> second or major seventh -- all "perfect" or acute discords in this era
> -- to beautiful effect. In a style with a continual mix of stable and
> unstable sonorities, it helps diversify the continuum.

Wouldn't it be fun to jump into a time machine and go back to Notre Dame and
listen to what was happening then. All we have now is the report of what
musicians *thought* they were doing. As we know from Paul's responses to my
own posts of what I *think* I'm doing, we have to be careful that we don't
take the word of the medieval "Gerald Eskelins" as to what they were
actually hearing. :-)
>
> Anyway, while I can say that the Renaissance use of the tritone as an
> interval resolving to a stable third by stepwise contrary motion
> developed in a meantone type of keyboard orientation, just how singers
> or other flexible-pitch performers negotiated these progressions
> remains an open question.

Again, wouldn't it have been great to have been there.

But in any case, isn't it reasonable to suppose that the physical functions
of medieval ears were pretty much the same as our own? I have to believe
that cultural influences don't affect the perception of "pure" acoustic
tonal relations differently today than in any previous time. On the other
hand, when we consider stylistic (or artistic) deviation for expressive
effect--the sky's the limit!

> Most appreciatively,

The appreciation is all mine!
>
> Margo Schulter

Gerald Eskelin
> mschulter@value.net

--- In tuning@y..., Gerald Eskelin <stg3music@e...> wrote:

> And this amplifies my point--that 17th century composers, having discovered
> the strength of the tritone, used it *intentionally* to effect their
> kaleidoscopic modulations.

i think you, margo, and i all share an unusual appreciation for the importance of this point. it's not emphasized enough.