Re: At the archicembalo

[This article was originally posted yesterday on rec.music.theory, where
it was addressed especially to Dave Webber, a contributor there. I hope
that it may be of interest here also.]

An open letter is a curious format for Usenet, but somehow a "harmonic
letter" (was that Banchieri's term?) seems a more natural form of
expression than some pretence of a treatise with arcane terminology
announcing a Grand Unified Theory of music. Maybe the restraining
factor is that at least I know that my theoretical perspective is
peculiar enough that it's very unlikely to be confused with a globally
valid, let alone imperative, theory of any kind <grin>.

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1. Reflections on a virtual archicembalo
----------------------------------------

Anyway, this is written about 24 hours after my first session at a
24-note archicembalo in more or less the style of Nicola Vicentino --
actually two keyboards in 1/4-comma meantone with those pure and
delicious thirds, both with the usual Eb-G# arrangement -- but the
upper one tuned a fifth of a tone higher! It's all the beauties of
what I'd call "conventional modern music" -- meaning the 16th century
-- plus fifthtones, minimal thirds and "proximate minor" thirds (which
Don Nicola describes), chromatic and "enharmonic" (microtonal)
progressions, and so on. By the way, a fair caution before I end this
paragraph: it's a digital setup, with a synthesizer and two keyboards,
rather than some reconstruction of a 16th-century design.

It's so exciting in part, maybe, because conventional 16th-century
music is like a "natural language" for me, the "modern" side of what I
usually play or improvise. It's really awesome to sit at a keyboard
with Vicentino's fifthtones (24 out of a potential 31 not quite
equally tempered steps), and with a sense of being "at home" with
Vicentino's everyday theory and musical approach -- making the journey
beyond 12 or even 19 notes per octave and inside the semitone even
more adventurous.

To me, anything outside Eb-G# is a "special accidental," and maybe
this makes Vicentino's new world of music all the more breathtaking.
To describe how I conceive of modes, consonances, cadences, and
patterns is a difficult task, one sometimes attempted here with a
realization of the imperfect nature of words to describe music -- but I
suspect that it is rather different than the way many people orient
themselves to keys, chords, modulations, and "harmonic structure." Is
it possible that Vicentino might actually be a greater adventure for
someone unacquainted with 16th-century style, when the everyday
assumptions of the music as well as the "avant garde" part might be a
new revelation?

Anyway, while we both seem to find some humor in the way that physics
analogies get (mis)applied to just about any discipline or argument, I
sense a real kinship between the sense of awe one can get from both
physics and music. The feeling of sitting at an archicembalo realizing
an open universe of 16th-century microtonalism -- however virtual the
instrument, however reasonable the facsimile -- reminds me of how I
felt reading a 17th-century text (translated from the French by Aphra
Behn, famed as possibly the first English woman to be recognized as a
professional writer) telling how inhabitants of another solar system
would see our Sun as a star in some constellation like the Bear in our
own evening sky. There are also passages in Nicolas of Cusa which have
this effect on me -- a 15th-century Bishop and theologian who
discussed an unbound universe, heliocentrism, and extraterrestrial
inhabitants.

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2. Enharmonic connotations: fifthtones, or musical homonyms?
------------------------------------------------------------

Anyway, getting back for a moment to the geometric/musical dialogue
that has been going on, I thought I might add a few odd comments.

First, to me, the term "enharmonic" has a rather specific meaning --
or two not unconnected meanings. In an ancient (i.e. Greek/Roman)
context, it means a genus which divides the tetrachord (four steps and
three intervals making up a fourth) into two dieses of about half a
semitone each, plus a major third or ditone. In a "modern"
(i.e. Renaissance) context, it means a style of music using intervals
of a rather similar size, typically the diesis of around a fifth of a
tone. This is equal to the difference in tertian just intonation or
meantone between G# and Ab, for example -- in 1/4-comma meantone, as
in JI, 128:125, or about 41 cents.

Thus to me, as to Vicentino, "enharmonic" means more or less
"involving intervals or motions of around a fifth of a tone" just as
"chromatic" means "involving (in tertian JI or meantone) the chromatic
or minor semitone" (e.g. G-G#).

The enharmonic diesis (e.g. G#-Ab, with the flat _higher_ in meantone)
is itself measured in effect by an interesting parameter -- the
craving for pure or near-pure major thirds in the 16th century. If a
pure 5:4 major third is made the measure of the system, with all
regular fifths the same, then G# and Ab will be about 41.06 cents
apart, or more exactly 128:125, the difference between three pure
major thirds and a pure 2:1 octave.

(To describe two different note spellings for the same tone, I might
use the term "homonym" -- thus "G#/Ab are homonyms in 12-tet" (12-tone
equal temperament) or "Fbb and D## are homonyms in 31-tet, and
_almost_ so in pure 1/4-comma meantone." It's curious how the same
term, "enharmonic," can carry the association either that two notes
such as G#-Ab differ very tangibly by a diesis or fifthtone, or that
they are realized by a single tone -- either connotation being
empirically verifiable if one selects the right keyboard tuning.)

What Vicentino does is to take this simple fact and make it a virtue,
indeed a radically new form of musical expression. His keyboard scheme
features ranks tuned a diesis apart -- or a "fifthtone," as I call it
in a friendly manner in English. Using an asterisk (*) to show a note
raised by such a fifthtone -- he uses a dot above the note, an ideal
convention but more difficult in ASCII -- I would diagram the basic
five ranks of his instrument as follows, with a dotted line showing
the division between his first and second manuals (with split keys,
and an extra sixth rank for some pure fifths to supplement the 31
notes shown here):

Db* Eb* Gb* Ab* Bb* 5
C* D* E* F* G* A* B* 4
...................................................................
Db D# E# Gb Ab A# B# 3
C# Eb F# G# Bb 2
C D E F G A B C 1

My 24-note subset uses two regular keyboards in 1/4-comma meantone
like this, with the second simply tuned a diesis or fifthtone higher:

Db Eb* Gb Ab Bb*
C* D* E* F* G* A* B* C*
...................................................................
C# Eb F# G# Bb
C D E F G A B C

The white keys of my two 12-note keyboards are identical to
Vicentino's first and fourth ranks, and the black of the first manual
to his second rank, with the black keys of the second manual including
the extra flats of his third rank (Db, Gb, Ab) plus two enharmonic
accidentals of his fifth rank (Eb*, Bb*).

Happily, these 24 notes are sufficient to play the two complete
enharmonic or fifthtone pieces I know from the 16th century -- with
one modification for Vicentino's _Musica prisca caput_, changing the
Eb* to a rather less exotic but still somewhat unusual D#. Anthoine de
Bertrand's _Ie suis tellement amoureux_ (1578) uses Eb*, but not D#.

------------------------------
3. Not-quite-equal temperament
------------------------------

While Vicentino's full set of an octave divided into 31 notes a
fifthtone apart is often equated with 31-tet -- I've done it myself --
I'm tempted to put my proverbial Foote down (are you reading this,
Ed?) and point out that 1/4-comma meantone (pure major thirds) would
seem easier to tune using 16th-century methods than a precise division
of the octave into 31 equal steps.

In fact, Vicentino himself tells us that one should start out by
tempering the first two ranks in the usual manner "as the good tuners
do it, with the fifths somewhat blunted," and 1/4-comma meantone would
produce an _almost_ equal 31-note division.

This is basically how I went about it digitally, starting with a
default 1/4-comma meantone (Eb-G#), and then editing the synthesizer's
tuning table for the second manual by raising each of the 12 notes by
the diesis or fifthtone of 128:125 or around 41 cents (expressed
almost exactly in digital terms as 35 tuning units of 1/1024 octave).

Either with Vicentino's original two-manual scheme, or with this
24-note variant, maybe the most striking feature is that one can move
from a note on the first manual to its counterpart on the second --
thus making a direct jump of a fifthtone, e.g. C-C*. This can be done
not only with individual notes, but with full four-voice sonorities,
as Vicentino does (I use MIDI-style notation, with C4 as middle C and
higher numbers showing higher octaves):

E4 E*4
G3 C*4
C4 G*3
C3 C*3

On a keyboard, the voice-crossing is a feature that gets "neutralized"
(to borrow a linguistic term) -- one simply leaps from C3-G3-C4-E4 on
the lower manual (Vicentino's first rank) to C*3-G*3-C*4-E*4, the same
notes on the upper manual (Vicentino's fourth rank).

Might one call this a "quantum leap"? Completing a familiar
16th-century progression of consonances or cadence -- but with the
shift of a diesis -- is another technique that both Vicentino and
Bertrand use (here the latter):

*
C5 D5
E4 F4*
G3 Bb*3
C3 Bb*2

Here my * above the highest voice suggests an editorial accidental --
although leaving the D unaltered would produce Vicentino's "proximate
minor third" (intermediate between minor and major) which he notes has
a ratio of around 11:9, and finds to be rather concordant. As always
with Renaissance accidentals, decisions, decisions. Maybe such
progressions are like meeting an old and welcome friend -- in some
kind of intriguingly warped interstellar space.

* * *

This leads me to a few observations on the art of metaphor in music,
especially physics and geometry metaphors and the like.

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4. Geomorphic tuning?: Time zones and paired antipodes
------------------------------------------------------

One metaphor that occurred to me for my 24-note version of Vicentino's
tuning was a "geomorphic" music, the idea being that just as the Earth
is not a perfect sphere, but slightly oblate or pear-shaped, so
1/4-comma meantone in 31 notes is _not quite_ circular.

We actually get two slightly different flavors of fifthtones, with [a
whole-tone consisting of] three usual dieses of 128:125 or 41.06 cents
plus two smaller fifthtones of around 34.99 cents, the difference between
a regular diesis and a minor semitone in 1/4-comma meantone of about 76.04
cents. I like this -- maybe it's a bit like flavors of quarks or
something, and more interesting than just five identical steps. I'm
tempted to call them major and minor fifthtones.

While Vicentino's full 31-note system is _almost_ circular, a 24-note
subset isn't, of course. Then, again, the idea of circumnavigating the
whole system is something I hear about rather than do, being
accustomed to open-ended modal systems and tunings based on
Pythagorean or meantone where the usual range of accidentals is Eb-G#;
actual 16th-century enharmonic (fifthtone) pieces that I'm aware of
stay within a 24-note range rather than attempting such closed
circularity in practice.

As a metaphor, I'm tempted maybe to compare a "24 out of 31-quasi-tet"
instrument to a planet where you can only see one side at a time --
considerably more than half (with the Moon, I seem to recall that
about 58% is visible from Earth).

Then the planetary metaphor suggested to me 24 "time zones" --
although this suggests a closed tuning, not even approximated in 24
notes. (For the curious, the ends of the chain at Fbb or Eb* and G#
form what I might call in Vicentino's terms the "proximate major
third," equal to a pure major third plus a fifthtone, here the smaller
one of 35 cents, or about 421 cents in all.)

A striking feature is that making an enharmonic shift of a fifthtone
like C-C* involves leaping 12 of our 24 time zones, if we arrange them
in a tuning chain of fifths. (We could also write C* as Dbb -- part of
a discussion on the "spatial metric" of 1/4-comma meantone which I'll
save for another post. Note that moving 12 fifths in the flat
direction of the tuning chain takes us _up_ a diesis.)

Is this like a sudden shift of "night and day," or like antipodes --
an attractive metaphor, given the way in which these keys are paired
with their counterparts in Vicentino's first and fourth ranks, for
example, or more generally throughout my lower and upper manuals (if
we write Gb as F#*, for example, this generalized feature of my
24-note arrangement may become clearer).

Taking a medieval or Renaissance perspective, incidentally, I consider
circles, chains, spheres, or other metaphors for fifths to be of
interest especially for tuning, but hardly as the main model for
vertical progressions or overall organization. True, Renaissance
cadences often involve the motion of the bass by a fifth or fourth,
etc. -- but this is just one of many ways to connect sonorities or to
relate a pattern of centers or progressions in the course of a
piece. Bass motion by a second or third seems to me equally
significant, and beautifully fluid. Often I tend to look to
progressions of consonances by stepwise contrary motion in describing
a vertical event or gesture, and the kinship of sonorities connected
by the bass motion of a third is a common feature of Renaissance style
exploited by various composers, both in its gentle shadings and its
ear-catching chromatic versions.

(My friend Peter van Marissing, an esteemed music teacher, organist,
and scholar of historical tunings, suggests that the pure or near-pure
thirds of meantone may foster such thirdwise kinship in Renaissance
music, a topic for some interesting discussions.)

Anyway, the various dialogues in progress here may suggest that just
as music can have many styles and tunings, so it can have many
"roadmaps" and metaphors.

Most appreciatively,

Margo Schulter
mschulter@value.net