Re: Gregorian chant intonation study: first reactions

Hello, there, and these are just some first reactions to a very
interesting spectrographic study of a Gregorian chant recording
presented here. This study suggests that at least some vocal ensembles
tend to tune major and minor thirds at around 5:4 and 6:5 when singing
such monophonic chants.

This empirical approach merits much praise, and the finding seems
unexpected not only from the viewpoint of a 3-limit medievalist
approach, but from the viewpoint of much modern tuning theory which
proposes that musicians tend to prefer 3-limit (Pythagorean) just
intonation (JI) for unaccompanied melody where harmonic considerations
do not apply.

There are a number of hypotheses which might be or indeed have been
proposed here to explain the finding of this study:

(1) One might suggest that in fact 5:4 and 6:5 thirds are "natural
attractants" to singers in "pure melody" as well as in a wide range of
styles of polyphony.

(2) One might suggest more generally that singers tend to seek
small-integer ratios. This raises a question as to whether the 7:4
minor seventh, for example, might in practice occur when singers are
performing repertories usually considered "3-limit" or "5-limit."

(3) One might argue, as people already have, that certain kinds of
acoustical spaces in effect transform monophonic chant into a kind of
polyphony by prolonging the reverberation of the notes and their
partials. It then becomes of interest to compare vocal intonations,
possibly by the same ensemble, in different environments.

(4) One might raise the question as to whether intonational style in
Gregorian chant or other monophonic singing may considerably vary
between ensembles, possibly influenced by such factors as whether a
group also performs medieval and/or later polyphony, and so on. Of
course, in any case, the fact that an ensemble performs monophonic
chant with a "5-limit" tuning is itself very interesting.

(5) One might join Bill Alves in seeking more specifics about ranges
of intonational variation, etc. It's good in any case to be reminded
early and often that the voice need not, and indeed _cannot_, be a
fixed-pitch instrument. The variation on the order of magnitude of 15
cents which he suggests is roughly 2/3 of a syntonic comma (81:80,
~21.51 cents).

A final, more personal note. Early this evening, while reflecting on
this very engaging empirical approach, I was improvising in
more-or-less late 14th-century style on a Pythagorean keyboard, one
manual like a usual "harpsichord" sound and the other maybe a bit like
a harpsichord "harp stop" (or lute stop). It was wonderful, and I
could move around the JI system without any of the complications often
discussed here for 5-limit.

What struck me is that whether or not medieval singers actually came
very close to 3-limit in their plainsong and polyphony, the theorists
have described a _beautiful_ just intonation system for keyboards. It
was almost like going back and sitting at what might have been one of
the first "harpsichord-like" instruments around the end of the 14th
century, not that those instruments offered anything as luxurious as
two four-octave manuals.

If I had wanted to use ab-eb -- it didn't occur to me to get this
adventurous -- I would have split the manuals between the usual g# and
an ab, but for my usual improvisation, the issue didn't even occur to
me. Would people here describe a 3-limit system as a plane or a
cylinder? -- anyway, it was a delight.

At one point, for fun, I tried c#-f (8192:6561 in 3-limit, only 1.95
cents from 5:4), and found that it sounded a bit "strange." This was a
bit like being on the other side of some "lattice barrier" or the
like, although I'm still not quite sure just what that tagline means.

Anyway, with all the JI systems which are reportedly much more
difficult to implement on a fixed-pitch keyboard than to approximate
with voices, it would be curious if classic 3-limit JI turns out to be
easier for keyboards than for singers, maybe even medieval singers.

Possibly it was even symbolic in a way that I chose to improvise in a
style inspired by recordings of late 14th-century music, an epoch
often termed the _Ars subtilior_ or Manneristic epoch (of the Gothic
era, that is, there also being the Mannerism of the 16th and 17th
centuries).

Mannerism has sometimes been defined as the technique of artfully
distorting rather than simply imitating nature. Could 3-limit tuning
be in this category: a deviation from the likeliest vocal intonation
which has a special beauty both mathematical and musical for one of
the most exquisite polyphonic repertories of all time?

Most respectfully,

Margo Schulter
mschulter@value.net

Margo Schulter wrote,

>Would people here describe a 3-limit system as a plane or a
>cylinder?

It's just a line, of course (think about it!), unless you use the schisma
thirds for 5-limit, in which case it's a cylinder.